Theory of Computing Report

Authors: Anurag Anshu, Shankar Balasubramanian, Jonas Haferkamp, Aram W. Harrow, Xinyu Tan

We construct a constant-degree and constant-gap quantum expander on $n$ qubits where each unitary can be implemented by a depth-$1$ and 1D circuit of Pauli or CNOT gates. We provide two applications of this expander. First, we use it to construct a family of frustration-free 1D Hamiltonians whose ground states obey the entanglement-gap relation $S = Θ(Δ^{-1/2})$; this is believed to be optimal, but achieving it had been open. Second, we use it to provide a streaming protocol that tests for closeness to a class of 1D volume-law entangled states. Moreover, we extend our quantum expander to a constant-degree and constant-gap expander on the unitary group where each unitary is a single $T$ gate, a single $T^{\dagger}$ gate, or a depth-$1$ Clifford circuit. This implies that a random sequence of unitaries from the expander yields a gapped walk on a dense subgroup of the unitary group. This improves upon previous work by Bourgain and Gamburd which did not control the dependence of the gap on the dimension.

Authors: Anurag Anshu, Shankar Balasubramanian, Jonas Haferkamp, Aram W. Harrow, Xinyu Tan

We construct a constant-degree and constant-gap quantum expander on $n$ qubits where each unitary can be implemented by a depth-$1$ and 1D circuit of Pauli or CNOT gates. We provide two applications of this expander. First, we use it to construct a family of frustration-free 1D Hamiltonians whose ground states obey the entanglement-gap relation $S = Θ(Δ^{-1/2})$; this is believed to be optimal, but achieving it had been open. Second, we use it to provide a streaming protocol that tests for closeness to a class of 1D volume-law entangled states. Moreover, we extend our quantum expander to a constant-degree and constant-gap expander on the unitary group where each unitary is a single $T$ gate, a single $T^{\dagger}$ gate, or a depth-$1$ Clifford circuit. This implies that a random sequence of unitaries from the expander yields a gapped walk on a dense subgroup of the unitary group. This improves upon previous work by Bourgain and Gamburd which did not control the dependence of the gap on the dimension.

Wednesday, September 02 2026, 00:00

SVP Is NP-Hard for Some Rank-2 Cyclotomic Modules

Authors: Jiaqi Liu, Yansong Feng, Yanbin Pan

Let $q$ range over primes congruent to $3$ modulo $4$. Let $ζ_q$ be a primitive $q$th root of unity, and put $K=\mathbb{Q}(ζ_q)$, with ring of integers $\mathcal{O}_K=\mathbb{Z}[ζ_q]$. We prove that the decision version of the Shortest Vector Problem ($\mathrm{SVP}$) in the $\ell_2$-norm is $\mathrm{NP}$-complete on full-rank free submodules of $\mathcal{O}_K^2$ by a deterministic polynomial-time many-one reduction from Exact Cover by 3-Sets (X3C). The module rank is fixed at two. As a $\mathbb{Z}$-lattice, the module has rank $2(q-1)$, which grows with $q$. The main obstacle is closure under the action of $\mathcal{O}_K$. A module containing a nonzero vector also contains every scalar multiple of that vector by a nonzero element of $\mathcal{O}_K$, and some of these multiples may be shorter. Three ideas overcome this obstacle. First, we map the Bennett--Peikert Reed--Solomon lattice to a principal cyclotomic ideal and use Wan's point-count estimates to prove that a coset of this ideal contains many binary coefficient representatives. Second, a checker based on a quadratic Gauss sum turns the X3C equations into a canonical squared norm. Third, the checker and a second module coordinate combine with a separation bound for ideal cosets to rule out every unintended vector created by the $\mathcal{O}_K$-action. Each constructed instance consists of a prime $q\equiv3\pmod4$, two integral generators whose $2\times2$ generator matrix has nonzero determinant, and an integer squared threshold. The construction also gives $\mathrm{NP}$-hardness of search-$\mathrm{SVP}$ under polynomial-time Turing reductions.

Authors: Jiaqi Liu, Yansong Feng, Yanbin Pan

Let $q$ range over primes congruent to $3$ modulo $4$. Let $ζ_q$ be a primitive $q$th root of unity, and put $K=\mathbb{Q}(ζ_q)$, with ring of integers $\mathcal{O}_K=\mathbb{Z}[ζ_q]$. We prove that the decision version of the Shortest Vector Problem ($\mathrm{SVP}$) in the $\ell_2$-norm is $\mathrm{NP}$-complete on full-rank free submodules of $\mathcal{O}_K^2$ by a deterministic polynomial-time many-one reduction from Exact Cover by 3-Sets (X3C). The module rank is fixed at two. As a $\mathbb{Z}$-lattice, the module has rank $2(q-1)$, which grows with $q$. The main obstacle is closure under the action of $\mathcal{O}_K$. A module containing a nonzero vector also contains every scalar multiple of that vector by a nonzero element of $\mathcal{O}_K$, and some of these multiples may be shorter. Three ideas overcome this obstacle. First, we map the Bennett--Peikert Reed--Solomon lattice to a principal cyclotomic ideal and use Wan's point-count estimates to prove that a coset of this ideal contains many binary coefficient representatives. Second, a checker based on a quadratic Gauss sum turns the X3C equations into a canonical squared norm. Third, the checker and a second module coordinate combine with a separation bound for ideal cosets to rule out every unintended vector created by the $\mathcal{O}_K$-action. Each constructed instance consists of a prime $q\equiv3\pmod4$, two integral generators whose $2\times2$ generator matrix has nonzero determinant, and an integer squared threshold. The construction also gives $\mathrm{NP}$-hardness of search-$\mathrm{SVP}$ under polynomial-time Turing reductions.

Wednesday, September 02 2026, 00:00

Subgroup Accessibility in Group Order Logic

Authors: Anatole Dahan

We investigate the expressive power of fixed-point logics (FP) and their extensions in defining generating sets for accessible subgroups of definable permutation groups. This operation, computable in polynomial time via the Schreier-Sims algorithm, plays a central role in the group-theoretic approach to Graph Isomorphism and Graph Canonisation. In particular, it underpins polynomial-time canonisation for bounded colour-class graphs--a class for which no natural logic capturing P is currently known. We first show that this operation cannot, in general, be expressed in any logic for P. This limitation arises from the fact that accessible subgroups need not admit symmetric generating sets of polynomial size. However, we prove that when the base group admits a definable ordered generating set, the accessible subgroup operation becomes definable in fixed-point logic with the group order operator (FP + ord). This is achieved by partially simulating the Schreier-Sims algorithm within FP + ord. As a corollary, we show that fixed-point logic with counting (FPC) can also define the operation when the base group is abelian. In particular, FPC can define the automorphism group of any graph with abelian colours--despite being unable to canonise such graphs.

Authors: Anatole Dahan

We investigate the expressive power of fixed-point logics (FP) and their extensions in defining generating sets for accessible subgroups of definable permutation groups. This operation, computable in polynomial time via the Schreier-Sims algorithm, plays a central role in the group-theoretic approach to Graph Isomorphism and Graph Canonisation. In particular, it underpins polynomial-time canonisation for bounded colour-class graphs--a class for which no natural logic capturing P is currently known. We first show that this operation cannot, in general, be expressed in any logic for P. This limitation arises from the fact that accessible subgroups need not admit symmetric generating sets of polynomial size. However, we prove that when the base group admits a definable ordered generating set, the accessible subgroup operation becomes definable in fixed-point logic with the group order operator (FP + ord). This is achieved by partially simulating the Schreier-Sims algorithm within FP + ord. As a corollary, we show that fixed-point logic with counting (FPC) can also define the operation when the base group is abelian. In particular, FPC can define the automorphism group of any graph with abelian colours--despite being unable to canonise such graphs.

Wednesday, September 02 2026, 00:00

Bounded Relative Boundary Implies Narrow DNF Approximation

Authors: Chenghua Liu, Boning Meng

Friedgut conjectured that an increasing family in the $p$-biased discrete cube with bounded relative boundary can be approximated arbitrarily well by one whose minimal elements have bounded size, with a bound independent of the dimension and the bias (J. Amer. Math. Soc. 12 (1999)). We prove this conjecture by showing that, for $0

Authors: Chenghua Liu, Boning Meng

Friedgut conjectured that an increasing family in the $p$-biased discrete cube with bounded relative boundary can be approximated arbitrarily well by one whose minimal elements have bounded size, with a bound independent of the dimension and the bias (J. Amer. Math. Soc. 12 (1999)). We prove this conjecture by showing that, for $0

Wednesday, September 02 2026, 00:00

A Dichotomy for Complex Boolean Holant with Binary Disequality

Authors: Chenghua Liu, Boning Meng

We prove a complexity dichotomy for Boolean Holant problems defined by arbitrary finite sets of algebraic complex-valued signatures when binary disequality is available. The tractable cases are characterized by an explicit, decidable criterion.

Authors: Chenghua Liu, Boning Meng

We prove a complexity dichotomy for Boolean Holant problems defined by arbitrary finite sets of algebraic complex-valued signatures when binary disequality is available. The tractable cases are characterized by an explicit, decidable criterion.

Wednesday, September 02 2026, 00:00

Sierpiński--Knopp Wasserstein Distance for Persistence Diagrams and Applications to 2-Wasserstein Approximation

Authors: Sebastien Tchitchek, Julien Tierny

This paper introduces the Sierpiński-Knopp (SK) Wasserstein distance, a fast metric between persistence diagrams. The SK-Wasserstein distance, denoted $d_{\mathrm{SK}}$, maps diagram points and their diagonal projections to the unit interval via the Sierpiński-Knopp space-filling curve on the upper diagonal triangle. The encoded point sets are then efficiently matched via one-dimensional optimal assignment, in $O(N\log N)$ steps, yielding an explicit diagonal-aware point assignment between the two input persistence diagrams. We show that the SK-Wasserstein distance controls the classical $2$-Wasserstein distance between diagrams, admits an explicit isometric embedding into a Hilbert space, and induces a positive-definite Gaussian kernel, making the resulting geometry directly compatible with Euclidean and kernel-based learning methods. A tighter surrogate dissimilarity, noted $W_Γ$, is also introduced based on the point assignments along the curve. Experiments on 12 scientific collections comprising 227 diagrams show median per-collection speedup of $d_{\mathrm{SK}}$ over state-of-the-art approximations of $W_2$ is $626\times$, while the aggregate speedup over the full benchmark is $2100\times$. Average-linkage partitions obtained from $d_{\mathrm{SK}}$ and $W_Γ$ each exactly match the corresponding $W_2$ partition on 8 of the 12 collections. Hilbert $k$-means and Gaussian spectral clustering, both based on $d_{\mathrm{SK}}$, achieve mean adjusted Rand indices (ARI) of $0.756$ and $0.800$, respectively, with respect to the benchmark reference partitions, compared to $0.750$ obtained by average linkage on $W_2$. The Gaussian $d_{\mathrm{SK}}$ kernel supports other kernel-based analysis tasks, as illustrated by its use for contiguous segmentation of ordered diagram collections in our experiments.

Authors: Sebastien Tchitchek, Julien Tierny

This paper introduces the Sierpiński-Knopp (SK) Wasserstein distance, a fast metric between persistence diagrams. The SK-Wasserstein distance, denoted $d_{\mathrm{SK}}$, maps diagram points and their diagonal projections to the unit interval via the Sierpiński-Knopp space-filling curve on the upper diagonal triangle. The encoded point sets are then efficiently matched via one-dimensional optimal assignment, in $O(N\log N)$ steps, yielding an explicit diagonal-aware point assignment between the two input persistence diagrams. We show that the SK-Wasserstein distance controls the classical $2$-Wasserstein distance between diagrams, admits an explicit isometric embedding into a Hilbert space, and induces a positive-definite Gaussian kernel, making the resulting geometry directly compatible with Euclidean and kernel-based learning methods. A tighter surrogate dissimilarity, noted $W_Γ$, is also introduced based on the point assignments along the curve. Experiments on 12 scientific collections comprising 227 diagrams show median per-collection speedup of $d_{\mathrm{SK}}$ over state-of-the-art approximations of $W_2$ is $626\times$, while the aggregate speedup over the full benchmark is $2100\times$. Average-linkage partitions obtained from $d_{\mathrm{SK}}$ and $W_Γ$ each exactly match the corresponding $W_2$ partition on 8 of the 12 collections. Hilbert $k$-means and Gaussian spectral clustering, both based on $d_{\mathrm{SK}}$, achieve mean adjusted Rand indices (ARI) of $0.756$ and $0.800$, respectively, with respect to the benchmark reference partitions, compared to $0.750$ obtained by average linkage on $W_2$. The Gaussian $d_{\mathrm{SK}}$ kernel supports other kernel-based analysis tasks, as illustrated by its use for contiguous segmentation of ordered diagram collections in our experiments.

Wednesday, September 02 2026, 00:00

Efficient K-Visibility Query in Polygons

Authors: Yeganeh Bahoo, Roni Sherman

This paper investigates $k$-visibility, where a line of sight can penetrate up to $k$ obstacles. While computing the $k$-visibility polygon from a single query point is well-studied, existing spatial preprocessing approaches rely on full $O(n^2)$ line arrangements through all vertex pairs without characterizing the minimal set of topological boundaries. We present a refined cell decomposition framework that isolates the exact geometric events governing $k$-visibility: primary vertex horizon lines and secondary mutually critical hinge lines. We prove that this minimal set of partition lines yields a spatial decomposition of $Θ(n^4)$ cells within which the combinatorial structure of the $k$-visibility polygon remains strictly invariant. By leveraging a combinatorial $δ$-compression scheme across cell boundaries, we achieve an overall storage complexity of $\mathcal{O}(n^4)$ while supporting optimal $\mathcal{O}(\log n + m)$ query time to reconstruct explicit $k$-visibility polygons of size $m$. Our framework naturally extends to polygons containing holes.

Authors: Yeganeh Bahoo, Roni Sherman

This paper investigates $k$-visibility, where a line of sight can penetrate up to $k$ obstacles. While computing the $k$-visibility polygon from a single query point is well-studied, existing spatial preprocessing approaches rely on full $O(n^2)$ line arrangements through all vertex pairs without characterizing the minimal set of topological boundaries. We present a refined cell decomposition framework that isolates the exact geometric events governing $k$-visibility: primary vertex horizon lines and secondary mutually critical hinge lines. We prove that this minimal set of partition lines yields a spatial decomposition of $Θ(n^4)$ cells within which the combinatorial structure of the $k$-visibility polygon remains strictly invariant. By leveraging a combinatorial $δ$-compression scheme across cell boundaries, we achieve an overall storage complexity of $\mathcal{O}(n^4)$ while supporting optimal $\mathcal{O}(\log n + m)$ query time to reconstruct explicit $k$-visibility polygons of size $m$. Our framework naturally extends to polygons containing holes.

Wednesday, September 02 2026, 00:00

Exact curve counting of given word length on the once-punctured torus

Authors: Filippo Baroni, David Fisac, Mingkun Liu

On the once-punctured torus, we give an exact formula for the number of curves in any given mapping class group orbit of given word length. This settles a conjecture of Chas in [Cha16].

Authors: Filippo Baroni, David Fisac, Mingkun Liu

On the once-punctured torus, we give an exact formula for the number of curves in any given mapping class group orbit of given word length. This settles a conjecture of Chas in [Cha16].

Wednesday, September 02 2026, 00:00

The Discrete Harmonic Center of a Quadrilateral

Authors: Marc Alexa

Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.

Authors: Marc Alexa

Triangulate a simple quadrilateral by connecting all vertices to an additional point. If the vertices carry values, the piecewise linear function can be assigned a Dirichlet energy. We show that the minimal Dirichlet energy as a function of the location of the inserted point is convex, and the location of the minimum is independent of the values at the corners - a quadrilateral has a discrete harmonic center, characterized by an equilibrium of currents across the inserted edges. It turns out that the fixed points of the Möbius involution swapping opposite corners of the quadrilateral are critical points of this energy, so the discrete harmonic center is Möbius-covariant. For tangential and cyclic quadrilaterals the center admits simple closed forms related to the circle centers. The center and its data-independence generalize to polytopes with d + 2 vertices in dimension d, but the conformal characterizations are special to four points in the plane.

Wednesday, September 02 2026, 00:00

A Dimension-Reducing Fréchet Simplification Oracle

Authors: Boris Aronov, Tsuri Farhana, Matthew J. Katz, Indu Ramesh

Let $P$ be a polygonal curve with $n$ vertices in the plane. We construct a data structure of size $O(n \log n)$ suited for simplification queries of the following kind. Given a query line $\ell$ and an integer $k\ge1$, find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to $P$, among all such curves. Using our data structure, a query can be handled in $O(k^2 \log^3 n + k\log^4 n)$ time. More generally, a geometric tree $T$ on $n$ vertices in the plane can be preprocessed into a near-linear-size structure so that, given a pair $u$, $v$ of its vertices, a line $\ell$, and an integer $k\ge1$, one can find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to the path from $u$ to $v$ in $T$, in time $O(k^2 \mathop{polylog} n)$. For the general dimension-reduction problem, where $P$ is a curve in $\mathbb{R}^d$ ($d \ge 3$), $0 < \varepsilon_0 < 1$ is a real parameter, and a query specifies a $g$-flat $h$ ($1 \le g \le d-1$) and an integer $k \ge 1$, we construct a data structure of size $O(n\log n + f(\varepsilon_0) n)$, where $f(\varepsilon_0)=(1+1/\varepsilon_0)^{(d-1)/2}$, that allows us to find a curve $Q$ on $h$ with at most $k$ vertices, whose discrete Fréchet distance to $P$ is at most $1+\varepsilon_0$ times the distance of $Q^*$ to $P$, where $Q^*$ is such a curve that minimizes the distance to $P$. The query handling time is $O(f(\varepsilon_0) k^2 \log^2 n)$.

Authors: Boris Aronov, Tsuri Farhana, Matthew J. Katz, Indu Ramesh

Let $P$ be a polygonal curve with $n$ vertices in the plane. We construct a data structure of size $O(n \log n)$ suited for simplification queries of the following kind. Given a query line $\ell$ and an integer $k\ge1$, find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to $P$, among all such curves. Using our data structure, a query can be handled in $O(k^2 \log^3 n + k\log^4 n)$ time. More generally, a geometric tree $T$ on $n$ vertices in the plane can be preprocessed into a near-linear-size structure so that, given a pair $u$, $v$ of its vertices, a line $\ell$, and an integer $k\ge1$, one can find a curve $Q$ on $\ell$ with at most $k$ vertices that minimizes the discrete Fréchet distance to the path from $u$ to $v$ in $T$, in time $O(k^2 \mathop{polylog} n)$. For the general dimension-reduction problem, where $P$ is a curve in $\mathbb{R}^d$ ($d \ge 3$), $0 < \varepsilon_0 < 1$ is a real parameter, and a query specifies a $g$-flat $h$ ($1 \le g \le d-1$) and an integer $k \ge 1$, we construct a data structure of size $O(n\log n + f(\varepsilon_0) n)$, where $f(\varepsilon_0)=(1+1/\varepsilon_0)^{(d-1)/2}$, that allows us to find a curve $Q$ on $h$ with at most $k$ vertices, whose discrete Fréchet distance to $P$ is at most $1+\varepsilon_0$ times the distance of $Q^*$ to $P$, where $Q^*$ is such a curve that minimizes the distance to $P$. The query handling time is $O(f(\varepsilon_0) k^2 \log^2 n)$.

Wednesday, September 02 2026, 00:00

Disproving the Greedy Superstring Conjecture

Authors: Hiroki Shibata

The shortest common superstring problem is to find the shortest string that contains every string in a given set as a substring. It is conjectured that the greedy algorithm that repeatedly selects a pair of strings with maximum overlap and merges them is a $2$-approximation algorithm, and this conjecture had remained open for nearly four decades. In this paper, we disprove this conjecture and show that the approximation ratio of this algorithm is at least $9/4$.

Authors: Hiroki Shibata

The shortest common superstring problem is to find the shortest string that contains every string in a given set as a substring. It is conjectured that the greedy algorithm that repeatedly selects a pair of strings with maximum overlap and merges them is a $2$-approximation algorithm, and this conjecture had remained open for nearly four decades. In this paper, we disprove this conjecture and show that the approximation ratio of this algorithm is at least $9/4$.

Wednesday, September 02 2026, 00:00

Sensitivity Oracles for Matroid Packing, Matroid Covering, and Matching Problems with Applications

Authors: Keerti Choudhary, Amit Kumar, Lakshay Saggi

Sensitivity oracles preprocess a graph so that queries can be answered after any $f$ edge insertions and deletions, without recomputing from scratch. For structural optimization problems the known landscape is limited: for flows and cuts, all known compact oracles handle only $f\le2$ failures; existing oracles for $s$- and global min-cut apply only to undirected graphs; and for matchings, arborescence and spanning-tree packings, and arboricity, no efficient oracle is known for $f>1$. We present a unified algebraic framework based on sensitivity oracles for matroid packing, covering, and parity of sparse linear matroids, yielding the first oracles supporting an arbitrary number $f$ of updates across all of these problems (all constructions randomized Monte-Carlo). Concretely, we obtain efficient oracles for exact $(s,t)$-max-flow/min-cut, resolving an open problem of Baswana, Bhanja, and Pandey (ICALP'22) with near-optimal space; for all-pairs $k$-bounded flow, generalizing the near-optimal reachability oracle of Brand and Saranurak (FOCS'19, the case $k=1$); the first oracles for any $f$ for directed $s$- and global min-cut; oracles for $k$-disjoint arborescences, $k$-disjoint spanning trees, colorful spanning trees, and arboricity; and oracles for the existence of an $α$-factor, with perfect matching as the case $α=1$. We further introduce the \emph{subset sensitivity model}, in which updates are confined to a susceptible edge set of size $σ$ fixed during preprocessing. Here we decouple updates from the matroid representation and eliminate the dependence on $k$ and the matroid density altogether: all of the above are supported with $\widetilde O(f^ω)$ query time and $O(fσ^2)$ space. We also prove a matching $Ω(\min\{σ^2,n^2\})$-bit lower bound when $f\ge2$, establishing optimality.

Authors: Keerti Choudhary, Amit Kumar, Lakshay Saggi

Sensitivity oracles preprocess a graph so that queries can be answered after any $f$ edge insertions and deletions, without recomputing from scratch. For structural optimization problems the known landscape is limited: for flows and cuts, all known compact oracles handle only $f\le2$ failures; existing oracles for $s$- and global min-cut apply only to undirected graphs; and for matchings, arborescence and spanning-tree packings, and arboricity, no efficient oracle is known for $f>1$. We present a unified algebraic framework based on sensitivity oracles for matroid packing, covering, and parity of sparse linear matroids, yielding the first oracles supporting an arbitrary number $f$ of updates across all of these problems (all constructions randomized Monte-Carlo). Concretely, we obtain efficient oracles for exact $(s,t)$-max-flow/min-cut, resolving an open problem of Baswana, Bhanja, and Pandey (ICALP'22) with near-optimal space; for all-pairs $k$-bounded flow, generalizing the near-optimal reachability oracle of Brand and Saranurak (FOCS'19, the case $k=1$); the first oracles for any $f$ for directed $s$- and global min-cut; oracles for $k$-disjoint arborescences, $k$-disjoint spanning trees, colorful spanning trees, and arboricity; and oracles for the existence of an $α$-factor, with perfect matching as the case $α=1$. We further introduce the \emph{subset sensitivity model}, in which updates are confined to a susceptible edge set of size $σ$ fixed during preprocessing. Here we decouple updates from the matroid representation and eliminate the dependence on $k$ and the matroid density altogether: all of the above are supported with $\widetilde O(f^ω)$ query time and $O(fσ^2)$ space. We also prove a matching $Ω(\min\{σ^2,n^2\})$-bit lower bound when $f\ge2$, establishing optimality.

Wednesday, September 02 2026, 00:00

Kernelization of 2-Club Cluster Edge Deletion on Interval Graphs

Authors: Ajinkya Gaikwad

The \emph{$s$-Club Cluster Edge Deletion} problem asks whether, given a graph $G$ and an integer $k$, one can delete at most $k$ edges so that every remaining connected component has diameter at most~$s$. This generalizes the classical \emph{Cluster Edge Deletion} problem by permitting components of bounded diameter instead of requiring cliques. On general graphs, $2$-Club Cluster Edge Deletion is known to be fixed-parameter tractable when parameterized by $k$, but it remains open whether it admits a polynomial kernel, as posed in~\cite{ABUKHZAM2023113864}. Motivated by this question, we study the problem on interval graphs and obtain a polynomial vertex kernel of size $\mathcal{O}(k^{5})$. As a complementary result, we also show that the \emph{$s$-Club Cluster Edge Deletion} problem is polynomial time solvable on unit interval graphs. We also show that $2$-Club Cluster Edge Deletion is NP-hard even on split graphs.

Authors: Ajinkya Gaikwad

The \emph{$s$-Club Cluster Edge Deletion} problem asks whether, given a graph $G$ and an integer $k$, one can delete at most $k$ edges so that every remaining connected component has diameter at most~$s$. This generalizes the classical \emph{Cluster Edge Deletion} problem by permitting components of bounded diameter instead of requiring cliques. On general graphs, $2$-Club Cluster Edge Deletion is known to be fixed-parameter tractable when parameterized by $k$, but it remains open whether it admits a polynomial kernel, as posed in~\cite{ABUKHZAM2023113864}. Motivated by this question, we study the problem on interval graphs and obtain a polynomial vertex kernel of size $\mathcal{O}(k^{5})$. As a complementary result, we also show that the \emph{$s$-Club Cluster Edge Deletion} problem is polynomial time solvable on unit interval graphs. We also show that $2$-Club Cluster Edge Deletion is NP-hard even on split graphs.

Wednesday, September 02 2026, 00:00

Random-Priority Frontier Routing: Tight $Θ(n^c)$ Bounds Against $c$-Node Cartels

Authors: Krišjānis Petručena

We study path diversification in trusted-node networks, where sensitive material is relayed through intermediate nodes, some of which may be compromised. Our randomized routing rule assigns each vertex an independent random priority and repeatedly expands the highest-priority vertex on the global frontier of the explored region. Let $G$ have $n$ vertices, let $s,t$ be honest endpoints, and let $C$ be a set of $c$ compromised intermediate vertices, called a cartel, whose deletion leaves $s$ and $t$ connected. For every fixed $c$ and every fixed target probability $q\in(0,1)$, we prove that $Θ(n^c)$ independent executions are sufficient in the worst case for some route to avoid $C$ with probability at least $q$.

Authors: Krišjānis Petručena

We study path diversification in trusted-node networks, where sensitive material is relayed through intermediate nodes, some of which may be compromised. Our randomized routing rule assigns each vertex an independent random priority and repeatedly expands the highest-priority vertex on the global frontier of the explored region. Let $G$ have $n$ vertices, let $s,t$ be honest endpoints, and let $C$ be a set of $c$ compromised intermediate vertices, called a cartel, whose deletion leaves $s$ and $t$ connected. For every fixed $c$ and every fixed target probability $q\in(0,1)$, we prove that $Θ(n^c)$ independent executions are sufficient in the worst case for some route to avoid $C$ with probability at least $q$.

Wednesday, September 02 2026, 00:00

Prediction-Assisted Pricing and Admission for LLM APIs with Stochastic Token Consumption

Authors: Patrick Wong

An LLM application often sells or internally allocates several service products: a small or premium model, a short or long token cap, and possibly multiple posted prices. The operational decision is not merely which model answers a prompt. A price changes purchase probability, a token cap changes both user value and the tail of resource consumption, and accepted requests compete for shared compute and premium-model capacity. Demand and output length are initially uncertain, while an offline model may provide useful but imperfect predictions. We formulate sequential pricing and admission with stochastic resource consumption. Each arriving request belongs to an observable segment. The platform chooses a product--price pair or makes no offer; purchase, revenue, and resource use are then random. An offline predictor supplies a uniform, validated error radius for every segment--product cell. We propose Prediction-Clipped UCB (PCUCB), which intersects the offline prediction interval with an online confidence interval, evaluates products using resource shadow prices, and reserves a sample-path envelope before commitment. The prior gives a fast start when accurate, while online learning protects the platform when predictions are coarse. The analysis is modular. On a simultaneous confidence event, regret against a buffered fluid benchmark is bounded by a pacing term plus the cumulative diameter of the intersected intervals. For $J$ segment-product cells and prediction radius $\varepsilon$, this yields \[ \widetilde O\left( \sqrt{T}+(1+\barΛ) \min\{T\varepsilon,\sqrt{JT}\} \right), \] where $\barΛ$ bounds operational shadow prices. Thus the algorithm smoothly interpolates between an almost full-information regime and learning from scratch. Hard feasibility holds on every sample path through reservation envelopes.

Authors: Patrick Wong

An LLM application often sells or internally allocates several service products: a small or premium model, a short or long token cap, and possibly multiple posted prices. The operational decision is not merely which model answers a prompt. A price changes purchase probability, a token cap changes both user value and the tail of resource consumption, and accepted requests compete for shared compute and premium-model capacity. Demand and output length are initially uncertain, while an offline model may provide useful but imperfect predictions. We formulate sequential pricing and admission with stochastic resource consumption. Each arriving request belongs to an observable segment. The platform chooses a product--price pair or makes no offer; purchase, revenue, and resource use are then random. An offline predictor supplies a uniform, validated error radius for every segment--product cell. We propose Prediction-Clipped UCB (PCUCB), which intersects the offline prediction interval with an online confidence interval, evaluates products using resource shadow prices, and reserves a sample-path envelope before commitment. The prior gives a fast start when accurate, while online learning protects the platform when predictions are coarse. The analysis is modular. On a simultaneous confidence event, regret against a buffered fluid benchmark is bounded by a pacing term plus the cumulative diameter of the intersected intervals. For $J$ segment-product cells and prediction radius $\varepsilon$, this yields \[ \widetilde O\left( \sqrt{T}+(1+\barΛ) \min\{T\varepsilon,\sqrt{JT}\} \right), \] where $\barΛ$ bounds operational shadow prices. Thus the algorithm smoothly interpolates between an almost full-information regime and learning from scratch. Hard feasibility holds on every sample path through reservation envelopes.

Wednesday, September 02 2026, 00:00

Two-State Max-Plus Comparison Is Decidable

Authors: Keigo Oka

Daviaud, Guillon, and Merlet proved that comparison of max-plus automata is undecidable under a fixed state bound of 553 and explicitly left the range from 2 to 552 states open. We resolve the two-state endpoint. More strongly, given an arbitrary finite max-plus automaton $A$ and a max-plus automaton $B$ with at most two states, it is decidable whether $[\![A]\!](w)\leq [\![B]\!](w)$ for every word $w$. The structural reason is a one-dimensional projective normal form for two-state dynamics. Outside an effective bounded region, a transition has one of three tail behaviors: it propagates the unbounded projective gap with gap-independent height increment, forgets the gap with gap-independent height increment, or reads the gap magnitude into the height increment and then forgets it. In particular, any transition whose output depends on the unbounded gap necessarily destroys that gap. This yields an exact one-counter realization of $B$. Effective semilinearity of context-free Parikh images then reduces comparison to Presburger arithmetic. As a consequence, two-state max-plus comparison, equivalence, and positivity are decidable.

Authors: Keigo Oka

Daviaud, Guillon, and Merlet proved that comparison of max-plus automata is undecidable under a fixed state bound of 553 and explicitly left the range from 2 to 552 states open. We resolve the two-state endpoint. More strongly, given an arbitrary finite max-plus automaton $A$ and a max-plus automaton $B$ with at most two states, it is decidable whether $[\![A]\!](w)\leq [\![B]\!](w)$ for every word $w$. The structural reason is a one-dimensional projective normal form for two-state dynamics. Outside an effective bounded region, a transition has one of three tail behaviors: it propagates the unbounded projective gap with gap-independent height increment, forgets the gap with gap-independent height increment, or reads the gap magnitude into the height increment and then forgets it. In particular, any transition whose output depends on the unbounded gap necessarily destroys that gap. This yields an exact one-counter realization of $B$. Effective semilinearity of context-free Parikh images then reduces comparison to Presburger arithmetic. As a consequence, two-state max-plus comparison, equivalence, and positivity are decidable.

Wednesday, September 02 2026, 00:00

TR26-163 | Quantitative Results on Super-Ramanujan Graphs | Gil Cohen, Gal Maor

from ECCC Papers

This work studies super-Ramanujan graphs: $d$-regular graphs whose spectral expansion lies strictly below the Ramanujan threshold $2\sqrt{d-1}$. Although this threshold is asymptotically optimal for fixed $d$ as $n$ tends to infinity, it leaves open the finer question of how small the spectral expansion can be as a joint function of $n$ and $d$. Beyond their fundamental mathematical interest and potential applications to pseudorandomness, super-Ramanujan graphs may also have practical significance, since asymptotic analyses can obscure important finite-size effects. Our first result shows that, for every $d\geq 3$ and every even $n$, there exists a $d$-regular graph $G$ on $n$ vertices satisfying \[ \lambda_2(G) \leq 2\sqrt{d-1} - \frac{\sqrt{d}}{4n^{2/3}}. \] For each fixed $d$, an asymptotic version of this bound also follows from the recent breakthrough work of Huang, McKenzie, and Yau. Related edge-universality results of Huang and Yau (The Annals of Probability, 2026) imply a bound of the same order in the regime $ n^\varepsilon\leq d\leq n^{1/3-\varepsilon}, $ for any fixed sufficiently small $\varepsilon>0$, while He (Commun. Math. Phys., 2024) obtains a substantially larger advantage in the denser regime $d\gg n^{2/3}$. These works provide a much richer probabilistic description of the spectral edge, whereas our proof is considerably simpler and gives a nonasymptotic bound that is uniform in both $n$ and $d$. Turning to explicit constructions, it is folklore that the Lubotzky-Phillips-Sarnak graphs are super-Ramanujan, but their guaranteed advantage over the Ramanujan threshold is only exponentially small in $n$. Our second result gives a polynomial-time construction of bipartite super-Ramanujan expanders with advantage at least $ \frac{\sqrt{d}}{2n} $ over the Ramanujan threshold.

Tuesday, September 01 2026, 18:56

LLMs and self-referentiality

from Scott Aaronson

I woke up yesterday with the following thoughts, which are probably either obvious or dumb. A central thesis that many readers, including me, took from Douglas Hofstadter’s Gödel Escher Bach when young was that the secret of intelligence (and therefore, of AI) was going to have a lot to do with self-referentiality and “strange loops.” […]

I woke up yesterday with the following thoughts, which are probably either obvious or dumb.

A central thesis that many readers, including me, took from Douglas Hofstadter’s Gödel Escher Bach when young was that the secret of intelligence (and therefore, of AI) was going to have a lot to do with self-referentiality and “strange loops.”

Even Roger Penrose’s The Emperor’s New Mind, which in some ways was the anti-GEB, ironically agreed with GEB about the fundamental importance of self-reference to the success or failure of the whole AI project. It claimed (incorrectly, in my view and in most experts’) that AI could never work because there was something about Gödel’s Theorem and self-reference that no computer program could ever capture, but that could be captured by exotic physics accessible to the human brain.

Now, in 2026, we’ve succeeded at building AIs that outperform most humans at most intellectual tasks that are well-defined enough to judge. And at no point in the tech stack of those AIs — neither in the transformer neural nets, nor in the GPU clusters they run on, nor in the training process, nor anywhere else — did anyone need to build in anything about self-reference. (Excepting, eg, the system instructions that tell the model about its role and identity, which aren’t needed for intelligent behavior. Also, I’m not going to count the autoregressive nature of LLMs as “self-referential”; that’s just dynamical feedback.)

Of course, GPT 5.6 Pro and Fable can talk about themselves, about Gödel’s Theorem, about self-reference, about what we’re talking about right now, all of it, better than most humans. But at no point did anyone need to build self-referential abilities in. They popped out as a byproduct of the same pretraining that let the models talk about Pokémon and long-chain polymers and cognitive behavioral therapy and plate tectonics and everything else.

No wonder Hofstadter says he’s been stunned by the success of LLMs, and has seemed depressed about current AI capabilities in essays like this one. He’s way too smart to deny what’s happened or invent reasons why it doesn’t really count (the approach many have taken). But he realizes that we now have true conversational intelligence from a path that the GEB worldview would’ve regarded as far too cheap and simple, and that certainly has no “strange loops” built in anywhere.

Of course, a Hofstadterian could argue that a strange loop emerges in LLMs — indeed, nothing in GEB ever said that strange loops would need to be explicitly engineered at the outset. But would anyone who hadn’t been brought up on GEB arrive at this as a useful way of thinking about LLMs?

What can we say about this with hindsight? While the ideas of diagonalization and self-reference of course played a central role in the birth of modern mathematical logic and computer science, the most famous uses were negative: there is not a bijectjon between the natural numbers and the reals. There is not a complete sound proof system for arithmetic. There is not an algorithm to solve the halting problem.

If your goal was only to build the axioms of ZFC and the rules of first-order inference, or build an electronic computer, you wouldn’t explicitly need self-reference for that. You would just … start building, taking care that your instruction set didn’t fall short of universality.

Yes, ZFC can formalize and prove theorems about itself. Yes, electronic computers can run programs that take their own code as input. But no one ever needed to build those abilities in, any more than self-reference needed to be built in to the alphabet or the rules of grammar. It popped out as a free byproduct of universality.

In the same way, LLMs’ ability to talk about themselves popped out as a byproduct of their ability to talk about anything in the discourse universe they were trained on. The big, old ideas about intelligence that ended up basically vindicated were the ideas about how intelligence is about prediction, and prediction is about compression, and compression is about finding better and better upper bounds on Kolmogorov complexity. Not the self-reference stuff. (Although, if you wanted to know why Kolmogorov complexity can’t be computed perfectly, that negative statement would again require a self-referential argument.)

What’s left? Consciousness and subjective experience of course remain extremely mysterious. For all we know, Hofstadter could be right that those have something to do with self-reference. (For all we know, even Penrose could be right that they have something to do with exotic physics accessible to biological brains but not digital computers!)

But the idea that you’d need explicit self-referentiality before you could get convincing and world-changing conversational intelligence? Let it be buried in a Westminster Abbey or Arlington National Cemetery for the most important wrong ideas in human history — geocentrism, Aristotle’s teleological physics, aether, phlogiston, Freud’s psychology, Marx’s prediction of a workers’ uprising followed by a classless utopia, etc. But buried it needs to be.

By Scott

Tuesday, September 01 2026, 17:17

Halley The Superforecaster

from Ben Recht

Edmond Halley and the multiple purposes of forecasting.

Hi there, argmin readers! As the fall semester picks up, posting volume will, too. So I’m going to commit to writing short descriptive headers to help you sort through the different threads. Today’s post is a live blog of Lecture 2 of my graduate seminar “Forecasting: A Critical Retrospective.” A table of contents is here.

I’m always looking for how we did science and engineering before the social structures and norms were built up to normalize practice. So for a class on forecasting, let me ask how experts made forecasts before they had proper scoring rules, Bayesian statistics, and Stata.

Though there are plenty of places to start the search, why don’t we today look to the dawn of rationalism in the Enlightenment? Patterns, Predictions, and Actions opens and closes with stories about Edmond Halley. Halley was a master of prediction. He had a keen sense of statistical approximation, knowing how to round and manipulate data to explain the past and predict the future. Halley’s publication record showed a man obsessed with a wide range of forecasting applications.

Halley is best known for his comet, a celestial body that returns to our skies once every 76 years. [footnote: It’s due to return in 2061, but many people’s AGI timelines suggest we won’t be around to see it.] Halley, a strong proponent of Isaac Newton who helped fund the publication of the Principia, wanted to find definitive evidence to prove Newton right.

His proof would come from observations of a comet he had charted in his backyard in 1682. Using Newton’s rules, Halley computed the orbital parameters of the body. He found these parameters neatly matched those of comets observed by Johannes Kepler in 1607. Moreover, they matched those of one seen by German astronomer Petrus Apianus in 1531. Since the gaps between these observations were around 76 years, Halley forecast another observation in 1758. He’d die before its return, but he was right.

This successful celestial prediction is heralded as a crowning achievement of Enlightenment Science. In 1850, Yale astronomer Denison Olmsted, who observed the comet’s return in 1835, wrote, “The Return of Halley’s Comet in exact conformity with the predictions of astronomers established the truth of all those principles by which those predictions were made.” Explaining data we’ve already seen is fine, but there is nothing more convincing to scientists than when theory predicts the future.

The funny thing about this, and a theme we’ll frequently return to this semester, is that this conclusion is completely illogical. It’s a lovely example of affirming the consequent, a fallacy you’ll learn in an introductory logic course.

P implies Q
Q is true
Therefore, P is true.

This syllogism is clearly invalid. (“All the Rationalists live in Berkeley. Ben lives in Berkeley. Therefore, Ben is a Rationalist.” How dare you!) But this is how a lot of science works! Theory predicts a particular outcome. That outcome is observed. This makes scientists feel more convinced their theory is right.

We could go into a long rigmarole about logical positivism at this point, but I don’t want to argue with Bayesian epistemologists today. I just want to point out that even at the inception of the Enlightenment, science was based on the illogic of accurately divining the future. This is one of the things we love about forecasts. When we accurately predict the future, we feel like our internal narrative is true.

Halley’s intuition about forecasting extended far beyond the heavens. He was also a key contributor to modern demography and actuarial science. Protestant pastor Caspar Neumann had collected records of lives, births, and deaths in his hometown of Wroclaw in Poland. Neumann was apparently interested in using this data to disprove the existence of climacterics, where deaths were associated with specific ages like 63. Halley, who came across this data after Leibniz presented it to the Royal Society, had other predictive interests. He churned through Neumann’s data and produced his foundational actuarial life table.

The numbers here represent the counts of people of any given age at a particular snapshot. There were approximately 1000 infants between 0 and 1 (a number, perhaps a bit too convenient), and a total of approximately 34000 individuals in Wroclaw.

In presenting his table to the Royal Society, Halley saw numerous uses for it. He first explained how his table could be used to calculate the number of men draftable into the army. He computed this by counting the number of people aged 18 to 56 and dividing by two.

More relevant to our class, he also pioneered probabilistic forecasting. His second claimed use of the table was calculating the odds that someone might die in a particular time interval. To do this, he counted frequencies and assumed rates of the past were indicative of chance in the future. There were 567 people aged 25, and 560 aged 26. Therefore, the odds a 25-year-old lives to see 26 were 560 to 7 or, simplifying fractions, 80 to 1. Similarly, if you wanted to know the odds that person might live ten years, Halley advocated taking the number alive by age 35, 490, and computing odds: 490 to 77, or approximately 6 to 1.

Halley also used his table to estimate how long people would live by finding the point in a table at which the odds of living dropped below 2 to 1. He called this “The age to which it is an even wager.” Even back in the day before probability, people equated forecasts with fair betting odds.

Not surprisingly, if you could equate mortality forecasts with betting, you could price insurance. This was Halley’s fourth proposed use of his table. Similarly, for a fifth application, Halley worked out more sophisticated calculations and determined a clever scheme to value annuities, a popular means for the crown to raise money. Halley’s calculations set different prices for different ages, based on bets on how long an annuitant might live.

The life table is only one example of Halley’s keen sense that you could make forecasts without physics. Indeed, he seemed to appreciate that the key to forecasting was simply linking past observations with future extrapolations. These extrapolations could be used to confirm physics and create a shared model of reality. They could also guide the pricing of financial instruments wagering on matters of life and death. For Halley, as for us in this class, predicting the future served multiple purposes.

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By Ben Recht

Tuesday, September 01 2026, 14:31

On the Complexity of the Compatibility Problem for Succinctly Encoded Conditional Distributions

Authors: Guy Emerson

The motivation for this paper is the investigation of the trade-offs implicit in probabilistic models used in machine learning. Models are often used to make predictions in the form of conditional probabilities. However, a pair of conditional distributions p(x|y) and p(y|x) may not be compatible with any joint distribution p(x,y). Given two such conditionals, determining if there exists a compatible joint is known as the compatibility problem. For discrete random variables, when the conditionals are encoded as probability tables, the compatibility problem has a known solution, which is computationally tractable. In this paper, we formalise and study a succinct version of the problem, encoding conditional distributions as arithmetic circuits. This is applicable to practical applications of probabilistic modelling in high-dimensional settings, including neural network models. We show that, for succinct circuit representations of conditionals, the compatibility problem is intractable. In the case that all probabilities are non-zero, the problem is co-NP-complete. In the case that probabilities can be zero, we give examples to demonstrate that several notions of compatibility can be distinguished, and we prove that multiple versions of the problem are PSPACE-complete. Furthermore, we show that, assuming the polynomial hierarchy does not collapse, there exist compatible succinct conditionals whose joint cannot be expressed succinctly. Implications of these results for probabilistic modelling and machine learning are discussed.

Authors: Guy Emerson

The motivation for this paper is the investigation of the trade-offs implicit in probabilistic models used in machine learning. Models are often used to make predictions in the form of conditional probabilities. However, a pair of conditional distributions p(x|y) and p(y|x) may not be compatible with any joint distribution p(x,y). Given two such conditionals, determining if there exists a compatible joint is known as the compatibility problem. For discrete random variables, when the conditionals are encoded as probability tables, the compatibility problem has a known solution, which is computationally tractable. In this paper, we formalise and study a succinct version of the problem, encoding conditional distributions as arithmetic circuits. This is applicable to practical applications of probabilistic modelling in high-dimensional settings, including neural network models. We show that, for succinct circuit representations of conditionals, the compatibility problem is intractable. In the case that all probabilities are non-zero, the problem is co-NP-complete. In the case that probabilities can be zero, we give examples to demonstrate that several notions of compatibility can be distinguished, and we prove that multiple versions of the problem are PSPACE-complete. Furthermore, we show that, assuming the polynomial hierarchy does not collapse, there exist compatible succinct conditionals whose joint cannot be expressed succinctly. Implications of these results for probabilistic modelling and machine learning are discussed.

Tuesday, September 01 2026, 00:00

An Optimal Separation Between Certificate Complexity and Approximate Degree

Authors: Kaspars Balodis

We prove that certificate complexity can be quartically larger than approximate degree. More precisely, we construct a family of total Boolean functions $G$ with $$ C(G) = \tildeΩ(\tilde{deg}(G)^4), $$ where $C$ denotes certificate complexity and $\tilde{deg}$ denotes $1/3$-approximate degree. This is optimal up to polylogarithmic factors, since every total Boolean function $f$ satisfies $C(f)\le O(\tilde{deg}(f)^4)$ by the classical block-sensitivity bounds of Nisan and Nisan--Szegedy. Thus the result closes the gap between these two measures and improves the previously best known separation $C(f)=\tildeΩ(\tilde{deg}(f)^3)$ by Balodis, Ben-David, Göös, Jain, and Kothari. The construction starts from the partial function they used to quadratically separate $0$-certificate complexity from unambiguous $1$-certificate complexity. It already has the required certificate hardness, but its $0$-certificates are unstructured, which blocks the derivation of a low-degree verifier. We keep its $1$-condition and restrict the $0$-inputs to those certified by a structured family whose validity admits a low-degree approximant, while preserving the quadratic hardness. The partial function with its low-degree verifier is then fed through the cheat-sheet framework to yield the total function $G$ with the claimed separation. The main technical ingredient is an approximate polynomial that verifies the certificate in degree $\tilde{O}(\sqrt n)$. The verifier forms a low-degree count $W$ of the candidate $1$-certificates that remain compatible with the asserted $0$-certificate, and tests whether this count is zero. Crucially, the construction ensures that $W$ never exceeds $\tilde{O}(n)$, instead of the $Θ(n^2)$ candidate pairs it counts bringing the verification down to degree $\tilde{O}(\sqrt n)$.

Authors: Kaspars Balodis

We prove that certificate complexity can be quartically larger than approximate degree. More precisely, we construct a family of total Boolean functions $G$ with $$ C(G) = \tildeΩ(\tilde{deg}(G)^4), $$ where $C$ denotes certificate complexity and $\tilde{deg}$ denotes $1/3$-approximate degree. This is optimal up to polylogarithmic factors, since every total Boolean function $f$ satisfies $C(f)\le O(\tilde{deg}(f)^4)$ by the classical block-sensitivity bounds of Nisan and Nisan--Szegedy. Thus the result closes the gap between these two measures and improves the previously best known separation $C(f)=\tildeΩ(\tilde{deg}(f)^3)$ by Balodis, Ben-David, Göös, Jain, and Kothari. The construction starts from the partial function they used to quadratically separate $0$-certificate complexity from unambiguous $1$-certificate complexity. It already has the required certificate hardness, but its $0$-certificates are unstructured, which blocks the derivation of a low-degree verifier. We keep its $1$-condition and restrict the $0$-inputs to those certified by a structured family whose validity admits a low-degree approximant, while preserving the quadratic hardness. The partial function with its low-degree verifier is then fed through the cheat-sheet framework to yield the total function $G$ with the claimed separation. The main technical ingredient is an approximate polynomial that verifies the certificate in degree $\tilde{O}(\sqrt n)$. The verifier forms a low-degree count $W$ of the candidate $1$-certificates that remain compatible with the asserted $0$-certificate, and tests whether this count is zero. Crucially, the construction ensures that $W$ never exceeds $\tilde{O}(n)$, instead of the $Θ(n^2)$ candidate pairs it counts bringing the verification down to degree $\tilde{O}(\sqrt n)$.

Tuesday, September 01 2026, 00:00

Exact quantum splitting and the structure of finite algebras

Authors: Muhammad Imran

Berlekamp's algorithm factors a squarefree polynomial $f\in\mathbb{F}_q[x]$ by deterministic linear algebra, reducing the problem to splitting an explicit commutative algebra $B\cong\mathbb{F}_q^r$ into its $r$ simple factors. For large odd $q$, the standard efficient splitting step is randomized, while known derandomizations are conditional on the Extended Riemann Hypothesis. We give an unconditional exact quantum implementation in a circuit model permitting single-qubit rotations through efficiently computable angles. The construction uses an unconditional counting argument. For a block containing $s\ge2$ irreducible factors, a quadratic-character test in odd characteristic and an absolute-trace test in characteristic $2$ yield a nonconstant test element with probability $p_{q,s}\ge\tfrac12$, known exactly in advance and depending only on $q$ and $s$, not on the unknown factorization. Exact amplitude amplification therefore converts each randomized test into a procedure succeeding with certainty after one amplification iteration. The resulting algorithm uses exactly $r-1$ quantum splitting rounds and $O(n^3\log q)$ quantum $\mathbb{F}_q$-operations and $O(n^3)$ classical operations, requiring no primitive root, quadratic non-residue, or distinct-degree preprocessing. The method also splits arbitrary finite-dimensional separable commutative $\\mathbb{F}_q$-algebras given by structure constants. Combined with R'onyai's classical structure theory, which computes the radical deterministically and reduces the remaining tasks deterministically to polynomial factorization, it yields the radical and the Wedderburn decomposition of $A/\mathrm{Rad}(A)$ into minimal two-sided ideals, with certainty, for any $n$-dimensional associative $\mathbb{F}_q$-algebra given by structure constants, using $O(n^4\log q)$ quantum $\mathbb{F}_q$-operations.

Authors: Muhammad Imran

Berlekamp's algorithm factors a squarefree polynomial $f\in\mathbb{F}_q[x]$ by deterministic linear algebra, reducing the problem to splitting an explicit commutative algebra $B\cong\mathbb{F}_q^r$ into its $r$ simple factors. For large odd $q$, the standard efficient splitting step is randomized, while known derandomizations are conditional on the Extended Riemann Hypothesis. We give an unconditional exact quantum implementation in a circuit model permitting single-qubit rotations through efficiently computable angles. The construction uses an unconditional counting argument. For a block containing $s\ge2$ irreducible factors, a quadratic-character test in odd characteristic and an absolute-trace test in characteristic $2$ yield a nonconstant test element with probability $p_{q,s}\ge\tfrac12$, known exactly in advance and depending only on $q$ and $s$, not on the unknown factorization. Exact amplitude amplification therefore converts each randomized test into a procedure succeeding with certainty after one amplification iteration. The resulting algorithm uses exactly $r-1$ quantum splitting rounds and $O(n^3\log q)$ quantum $\mathbb{F}_q$-operations and $O(n^3)$ classical operations, requiring no primitive root, quadratic non-residue, or distinct-degree preprocessing. The method also splits arbitrary finite-dimensional separable commutative $\\mathbb{F}_q$-algebras given by structure constants. Combined with R'onyai's classical structure theory, which computes the radical deterministically and reduces the remaining tasks deterministically to polynomial factorization, it yields the radical and the Wedderburn decomposition of $A/\mathrm{Rad}(A)$ into minimal two-sided ideals, with certainty, for any $n$-dimensional associative $\mathbb{F}_q$-algebra given by structure constants, using $O(n^4\log q)$ quantum $\mathbb{F}_q$-operations.

Tuesday, September 01 2026, 00:00

A note on the $Σ_2^P$-completeness of the Frobenius number

Authors: Thomas Rothvoss

Given a finite set $A$ of natural numbers whose greatest common divisor is one, the Frobenius number $g(A)$ is the largest integer that is not a non-negative integer combination of the numbers in $A$. In a 2016 preprint, Matsubara states that given $A$ and $k$, deciding if $g(A) \geq k$ is $Σ_2^P$-complete. A decade has passed since without peer-reviewed publication of this result. At the same time, the community has found it difficult to verify this result. In this note, we give a write-up of the completeness proof based on Matsubara (2016).

Authors: Thomas Rothvoss

Given a finite set $A$ of natural numbers whose greatest common divisor is one, the Frobenius number $g(A)$ is the largest integer that is not a non-negative integer combination of the numbers in $A$. In a 2016 preprint, Matsubara states that given $A$ and $k$, deciding if $g(A) \geq k$ is $Σ_2^P$-complete. A decade has passed since without peer-reviewed publication of this result. At the same time, the community has found it difficult to verify this result. In this note, we give a write-up of the completeness proof based on Matsubara (2016).

Tuesday, September 01 2026, 00:00

On the Complexity of Bayesian Signal Processing

Authors: Yi Liu

We develop a computational framework for Bayesian decision-making. We show that as long as no action is optimal in every state, Bayes-optimal choice is intractable. This hardness need not arise from large action, state, or signal spaces, nor from a complicated represented utility function: extracting enough information from a hard-to-interpret signal to act optimally can itself be computationally hard. We also characterize tractability across approximation notions and identify their sources of difficulty. Under the probably approximately correct criterion, sample-based Bayesian learning is tractable if and only if the signal support is bounded. Our results provide justifications for bounded rationality, costly Bayesian inference, and sample-based Bayesian learning.

Authors: Yi Liu

We develop a computational framework for Bayesian decision-making. We show that as long as no action is optimal in every state, Bayes-optimal choice is intractable. This hardness need not arise from large action, state, or signal spaces, nor from a complicated represented utility function: extracting enough information from a hard-to-interpret signal to act optimally can itself be computationally hard. We also characterize tractability across approximation notions and identify their sources of difficulty. Under the probably approximately correct criterion, sample-based Bayesian learning is tractable if and only if the signal support is bounded. Our results provide justifications for bounded rationality, costly Bayesian inference, and sample-based Bayesian learning.

Tuesday, September 01 2026, 00:00

The Complexity of Coverability-Like Problems in Elementary Object Systems: Data-Nets to the Rescue

Authors: Francesco Di Cosmo, Soumodev Mal, Tephilla Prince

Elementary Object Systems (EOSs) are a model in the nets-within-nets (NWNs) paradigm, where tokens in turn can host standard Petri nets. We study the complexity of coverability-like problems, including termination and boundedness, over EOSs. Since coverability and boundedness are undecidable in general on EOSs, we focus on the relevant fragment of conservative EOSs (cEOSs). Our technique interprets cEOSs into the framework of data nets, whose tokens carry data from an infinite domain, thus bridging the nesting and the data-aware paradigms. Specifically, we show that cEOS coverability-like problems are equivalent to the coverability-like problems over an interesting fragment, called channel-$ν$PNs (c-$ν$PNs), of data nets that extends $ν$PN (featuring globally fresh name creation) with restricted forms of transfers with renaming. c-$ν$PNs remain less expressive than Unordered Data Nets, which feature lossy name creation as well as powerful forms of whole-place operations and broadcasts. These reductions allow us to analyze cEOS coverability taking advantage of known results on data nets. We conclude that the complexity of cEOS coverability is double-Ackermanian, $\mathcal{F}_{ω2}$-complete, while termination and boundedness are non-primitive recursive.

Authors: Francesco Di Cosmo, Soumodev Mal, Tephilla Prince

Elementary Object Systems (EOSs) are a model in the nets-within-nets (NWNs) paradigm, where tokens in turn can host standard Petri nets. We study the complexity of coverability-like problems, including termination and boundedness, over EOSs. Since coverability and boundedness are undecidable in general on EOSs, we focus on the relevant fragment of conservative EOSs (cEOSs). Our technique interprets cEOSs into the framework of data nets, whose tokens carry data from an infinite domain, thus bridging the nesting and the data-aware paradigms. Specifically, we show that cEOS coverability-like problems are equivalent to the coverability-like problems over an interesting fragment, called channel-$ν$PNs (c-$ν$PNs), of data nets that extends $ν$PN (featuring globally fresh name creation) with restricted forms of transfers with renaming. c-$ν$PNs remain less expressive than Unordered Data Nets, which feature lossy name creation as well as powerful forms of whole-place operations and broadcasts. These reductions allow us to analyze cEOS coverability taking advantage of known results on data nets. We conclude that the complexity of cEOS coverability is double-Ackermanian, $\mathcal{F}_{ω2}$-complete, while termination and boundedness are non-primitive recursive.

Tuesday, September 01 2026, 00:00

Separating Parsing Expression Grammars using Cell-Probe Lower Bounds

Authors: Jungyeom Kim, Jihyeok Park

We resolve three open problems concerning parsing expression grammars (PEGs). We construct a single language $C$ satisfying $C\in\mathsf{LIN}\cap\mathsf{PEG}$ and $C^R\in\mathsf{LIN}\setminus\mathsf{PEG}$. This proves that some linear context-free language is not a PEG language and that PEG languages are not closed under reversal, confirming a conjecture of Loff, Moreira, and Reis. Factoring the same witness resolves the concatenation-closure problem of Rubtsov and Chudinov negatively, in the strong form $\mathsf{PEG}\cdot\mathsf{REG}\not\subseteq\mathsf{PEG}$ despite $\mathsf{REG}\cdot\mathsf{PEG}\subseteq\mathsf{PEG}$. It also refutes closure under Kleene star, homomorphisms, and substitutions. Our main technique converts scaffolding automata (SCAs), which characterize reversals of PEG languages, into dynamic data structures in the cell-probe model. For any suitably local serialization of a problem with preprocessing, updates, and a final Boolean query, an SCA recognizer yields an exact deterministic cell-probe data structure whose operation costs are proportional to the corresponding encoding lengths. Cell-probe lower bounds can therefore prove SCA non-membership and, by reversal, PEG non-membership. We apply this transfer to Multiphase Inner Product using one-symbol update blocks and a query suffix of length $O(\log n)$, while keeping both the language and its reversal linear context-free. Ko's cell-probe lower bound then yields the witness above. The arguments are additionally formalized in Lean 4.

Authors: Jungyeom Kim, Jihyeok Park

We resolve three open problems concerning parsing expression grammars (PEGs). We construct a single language $C$ satisfying $C\in\mathsf{LIN}\cap\mathsf{PEG}$ and $C^R\in\mathsf{LIN}\setminus\mathsf{PEG}$. This proves that some linear context-free language is not a PEG language and that PEG languages are not closed under reversal, confirming a conjecture of Loff, Moreira, and Reis. Factoring the same witness resolves the concatenation-closure problem of Rubtsov and Chudinov negatively, in the strong form $\mathsf{PEG}\cdot\mathsf{REG}\not\subseteq\mathsf{PEG}$ despite $\mathsf{REG}\cdot\mathsf{PEG}\subseteq\mathsf{PEG}$. It also refutes closure under Kleene star, homomorphisms, and substitutions. Our main technique converts scaffolding automata (SCAs), which characterize reversals of PEG languages, into dynamic data structures in the cell-probe model. For any suitably local serialization of a problem with preprocessing, updates, and a final Boolean query, an SCA recognizer yields an exact deterministic cell-probe data structure whose operation costs are proportional to the corresponding encoding lengths. Cell-probe lower bounds can therefore prove SCA non-membership and, by reversal, PEG non-membership. We apply this transfer to Multiphase Inner Product using one-symbol update blocks and a query suffix of length $O(\log n)$, while keeping both the language and its reversal linear context-free. Ko's cell-probe lower bound then yields the witness above. The arguments are additionally formalized in Lean 4.

Tuesday, September 01 2026, 00:00

It's Hard to PArcK

Authors: Kyle Burke, Jeffrey Leman, Craig Tennenhouse

We show that Partizan Arc Kayles (PArcK), a generalization of Domineering to graphs, is PSPACE-complete via a reduction from Positive CNF and with recently-discovered techniques for creating PArcK positions with high temperature. The reduction uses only red and blue edges.

Authors: Kyle Burke, Jeffrey Leman, Craig Tennenhouse

We show that Partizan Arc Kayles (PArcK), a generalization of Domineering to graphs, is PSPACE-complete via a reduction from Positive CNF and with recently-discovered techniques for creating PArcK positions with high temperature. The reduction uses only red and blue edges.

Tuesday, September 01 2026, 00:00

Algorithmic threshold for high-dimensional projection pursuit I: general theory

Authors: Brice Huang, Mark Sellke, Nike Sun

We study a null model of high-dimensional projection pursuit: we are given $M$ points sampled i.i.d. from a standard gaussian in $N$ dimensions, where $M,N\to\infty$ with $M/N\toα\in(0,\infty)$. Our goal is to characterize the possible empirical distributions of these points' projections along a data-dependent direction $x$, which ranges over either the sphere $S_N=\sqrt{N}\mathbb{S}^{N-1}$ or cube $Σ_N=\{-1,+1\}^N$. We consider this problem in an algorithmic setting, where $x$ must be the output of an algorithm with dimension-free Lipschitz dependence on the input; this class of algorithms includes general gradient-based methods such as Langevin dynamics and approximate message passing (AMP). Our main result exactly characterizes the set of empirical distributions attainable by this class in terms of a one-dimensional stochastic control problem. As a consequence of our main result, we obtain exact algorithmic thresholds for optimizing the Hamiltonian of a spherical or Ising perceptron model with general bounded continuous activation. For the spherical problem, independent work of Montanari and Zhou (2024) characterized the empirical distributions attainable by a related two-stage AMP algorithm, also in terms of stochastic control. Our proof of hardness builds on the branching overlap gap property introduced in earlier work by the first two authors. Our main innovation is to develop stochastic control theory within the branching OGP framework, significantly expanding the settings in which it locates an exact algorithmic threshold. Notably, our methods apply even though the non-algorithmic problem of characterizing all feasible projections remains a major outstanding challenge. For the matching algorithmic result, we construct a new incremental AMP algorithm that acts on a Brownian-bridge revelation of the gaussian disorder and simulates the same family of controlled SDEs.

Authors: Brice Huang, Mark Sellke, Nike Sun

We study a null model of high-dimensional projection pursuit: we are given $M$ points sampled i.i.d. from a standard gaussian in $N$ dimensions, where $M,N\to\infty$ with $M/N\toα\in(0,\infty)$. Our goal is to characterize the possible empirical distributions of these points' projections along a data-dependent direction $x$, which ranges over either the sphere $S_N=\sqrt{N}\mathbb{S}^{N-1}$ or cube $Σ_N=\{-1,+1\}^N$. We consider this problem in an algorithmic setting, where $x$ must be the output of an algorithm with dimension-free Lipschitz dependence on the input; this class of algorithms includes general gradient-based methods such as Langevin dynamics and approximate message passing (AMP). Our main result exactly characterizes the set of empirical distributions attainable by this class in terms of a one-dimensional stochastic control problem. As a consequence of our main result, we obtain exact algorithmic thresholds for optimizing the Hamiltonian of a spherical or Ising perceptron model with general bounded continuous activation. For the spherical problem, independent work of Montanari and Zhou (2024) characterized the empirical distributions attainable by a related two-stage AMP algorithm, also in terms of stochastic control. Our proof of hardness builds on the branching overlap gap property introduced in earlier work by the first two authors. Our main innovation is to develop stochastic control theory within the branching OGP framework, significantly expanding the settings in which it locates an exact algorithmic threshold. Notably, our methods apply even though the non-algorithmic problem of characterizing all feasible projections remains a major outstanding challenge. For the matching algorithmic result, we construct a new incremental AMP algorithm that acts on a Brownian-bridge revelation of the gaussian disorder and simulates the same family of controlled SDEs.

Tuesday, September 01 2026, 00:00

Unconditional $V^0_1$-independence of a certified hitting-set principle

Authors: Martin Kolář

We show that a certified formalization of the hitting-set-existence axiom of Atserias and Tzameret, instantiated on the parity-based Nisan-Wigderson compression class of Khaniki, is independent of the two-sorted theory $V^0_1$ of $\mathrm{AC}^0$-reasoning, unconditionally: $V^0_1$ proves neither it nor its negation. The same holds for the corresponding certified dual weak pigeonhole principle, whose refutation is witnessed by a single seed that certified-computes every string of the model simultaneously. The mechanism is a bounded-arithmetic transfer of Atserias-Tzameret's reduction from hitting sets to the dual weak pigeonhole principle: the amplification half of that reduction, the sole source of its NP-oracle, is unnecessary at the native stretch of the Nisan-Wigderson map, and the compression half becomes a $V^0_1$-provable implication once circuit evaluation is replaced by its certified $Σ^B_0$ unfolding. This is, to our knowledge, the first independence result for a derandomization-flavoured existence principle at the $\mathrm{AC}^0$-reasoning level, and it makes explicit the bridge between the Khaniki Nisan-Wigderson line and the Atserias-Tzameret reverse mathematics of hitting sets.

Authors: Martin Kolář

We show that a certified formalization of the hitting-set-existence axiom of Atserias and Tzameret, instantiated on the parity-based Nisan-Wigderson compression class of Khaniki, is independent of the two-sorted theory $V^0_1$ of $\mathrm{AC}^0$-reasoning, unconditionally: $V^0_1$ proves neither it nor its negation. The same holds for the corresponding certified dual weak pigeonhole principle, whose refutation is witnessed by a single seed that certified-computes every string of the model simultaneously. The mechanism is a bounded-arithmetic transfer of Atserias-Tzameret's reduction from hitting sets to the dual weak pigeonhole principle: the amplification half of that reduction, the sole source of its NP-oracle, is unnecessary at the native stretch of the Nisan-Wigderson map, and the compression half becomes a $V^0_1$-provable implication once circuit evaluation is replaced by its certified $Σ^B_0$ unfolding. This is, to our knowledge, the first independence result for a derandomization-flavoured existence principle at the $\mathrm{AC}^0$-reasoning level, and it makes explicit the bridge between the Khaniki Nisan-Wigderson line and the Atserias-Tzameret reverse mathematics of hitting sets.

Tuesday, September 01 2026, 00:00

Unfolding Overlaps of the Exceptional Regular Polytopes

Authors: Satyan L. Devadoss, Matthew Harvey, David Richter

We find explicit ridge unfoldings of the three exceptional 4D polytopes (24-cell, 120-cell, 600-cell) that result in overlaps of their facets. These failures bring an end to the full classification of regular polytopes with the all-net property.

Authors: Satyan L. Devadoss, Matthew Harvey, David Richter

We find explicit ridge unfoldings of the three exceptional 4D polytopes (24-cell, 120-cell, 600-cell) that result in overlaps of their facets. These failures bring an end to the full classification of regular polytopes with the all-net property.

Tuesday, September 01 2026, 00:00

A unified geometric design framework for kirigami structures

Authors: Qinghai Jiang, Gary P. T. Choi

In recent years, kirigami metamaterials have been widely studied and applied in science and engineering. While various two- and three-dimensional kirigami design methods have been developed, most of them are only applicable to a limited class of kirigami structures. In this work, we develop a unified framework for kirigami design that encompasses a wide range of 2D-to-2D, 2D-to-3D, and 3D-to-3D shape-morphing effects, as well as additional geometric and physical properties such as compact reconfigurability and rigid deployability. In particular, by reformulating the design task as a length-based constrained optimization problem and solving it simultaneously for multiple target states of the kirigami structure, our unified design framework enables greater design flexibility and stronger theoretical support. Experimental results with a wide range of shape-morphing effects are presented to demonstrate the effectiveness of our framework. We further present a rigorous theoretical analysis of several key aspects of kirigami design, covering inertia transposition, aspect-ratio law, and angle defects, thereby elucidating important design rules and limitations. Altogether, our work paves a new way for the design of shape-morphing mechanical metamaterials.

Authors: Qinghai Jiang, Gary P. T. Choi

In recent years, kirigami metamaterials have been widely studied and applied in science and engineering. While various two- and three-dimensional kirigami design methods have been developed, most of them are only applicable to a limited class of kirigami structures. In this work, we develop a unified framework for kirigami design that encompasses a wide range of 2D-to-2D, 2D-to-3D, and 3D-to-3D shape-morphing effects, as well as additional geometric and physical properties such as compact reconfigurability and rigid deployability. In particular, by reformulating the design task as a length-based constrained optimization problem and solving it simultaneously for multiple target states of the kirigami structure, our unified design framework enables greater design flexibility and stronger theoretical support. Experimental results with a wide range of shape-morphing effects are presented to demonstrate the effectiveness of our framework. We further present a rigorous theoretical analysis of several key aspects of kirigami design, covering inertia transposition, aspect-ratio law, and angle defects, thereby elucidating important design rules and limitations. Altogether, our work paves a new way for the design of shape-morphing mechanical metamaterials.

Tuesday, September 01 2026, 00:00

Combinatorial maps for hierarchical splines

Authors: Caleb B. Goates, Kendrick M. Shepherd, Derek C. Thomas

Hierarchical splines are an important part of multiscale and adaptive isogeometric analysis formulations. The Bézier meshes of these splines are an essential part of their definition and of several important hierarchical spline algorithms, such as adaptive refinement and Bézier extraction. Topological data associated with the Bézier mesh-such as adjacency information-can be used to improve the performance of many of these algorithms as well as downstream applications of the splines, but typical hierarchical spline formulations do not compute the topological data, storing instead just a list of elements. In this work we present algorithms to build a performant topological data structure, namely the combinatorial map, to represent Bézier meshes of hierarchical splines over cubical cell complexes where the refinement levels have conforming Bézier meshes. This includes hierarchical and truncated hierarchical B-splines, as well as subsets of other hierarchical spline formulations. We show the performance characteristics of the construction algorithms of these hierarchical combinatorial maps, as well as an example use case, showing that the topological information can provide up to an order of magnitude reduction in computation time in downstream applications of the splines.

Authors: Caleb B. Goates, Kendrick M. Shepherd, Derek C. Thomas

Hierarchical splines are an important part of multiscale and adaptive isogeometric analysis formulations. The Bézier meshes of these splines are an essential part of their definition and of several important hierarchical spline algorithms, such as adaptive refinement and Bézier extraction. Topological data associated with the Bézier mesh-such as adjacency information-can be used to improve the performance of many of these algorithms as well as downstream applications of the splines, but typical hierarchical spline formulations do not compute the topological data, storing instead just a list of elements. In this work we present algorithms to build a performant topological data structure, namely the combinatorial map, to represent Bézier meshes of hierarchical splines over cubical cell complexes where the refinement levels have conforming Bézier meshes. This includes hierarchical and truncated hierarchical B-splines, as well as subsets of other hierarchical spline formulations. We show the performance characteristics of the construction algorithms of these hierarchical combinatorial maps, as well as an example use case, showing that the topological information can provide up to an order of magnitude reduction in computation time in downstream applications of the splines.

Tuesday, September 01 2026, 00:00

Elastic Triangle Splatting

Authors: Tian Shi, Shenhan Qian, Daniel Cremers

While neural rendering methods such as 3D Gaussian Splatting achieve remarkable visual fidelity, traditional polygonal meshes remain the backbone of established graphics pipelines. Triangle splatting bridges this gap by optimizing triangle primitives as differentiable splats, producing representations that are closer to mesh-based workflows. Central to these methods is the kernel function that softens triangle boundaries to propagate gradients to vertex positions. Existing triangle splatting methods make inconsistent choices of kernel functions, and analysis of these kernels' optimization behavior has been limited to unstructured triangle soups for novel-view synthesis. In this work, we consider triangle splatting as a generic tool for photometric optimization, comparing kernel properties through two complementary tasks: mesh optimization for shape reconstruction and triangle soup optimization for novel-view synthesis. Along with the analysis, we introduce an elastic kernel function that features bilateral gradient support across the boundary and an adaptive boundary value, which are shown to be essential for robust optimization. Under isolated comparison, our elastic kernel outperforms existing kernels on shape reconstruction and in the majority of novel-view synthesis benchmarks, demonstrating the importance of kernel design in the effectiveness and versatility of triangle splatting.

Authors: Tian Shi, Shenhan Qian, Daniel Cremers

While neural rendering methods such as 3D Gaussian Splatting achieve remarkable visual fidelity, traditional polygonal meshes remain the backbone of established graphics pipelines. Triangle splatting bridges this gap by optimizing triangle primitives as differentiable splats, producing representations that are closer to mesh-based workflows. Central to these methods is the kernel function that softens triangle boundaries to propagate gradients to vertex positions. Existing triangle splatting methods make inconsistent choices of kernel functions, and analysis of these kernels' optimization behavior has been limited to unstructured triangle soups for novel-view synthesis. In this work, we consider triangle splatting as a generic tool for photometric optimization, comparing kernel properties through two complementary tasks: mesh optimization for shape reconstruction and triangle soup optimization for novel-view synthesis. Along with the analysis, we introduce an elastic kernel function that features bilateral gradient support across the boundary and an adaptive boundary value, which are shown to be essential for robust optimization. Under isolated comparison, our elastic kernel outperforms existing kernels on shape reconstruction and in the majority of novel-view synthesis benchmarks, demonstrating the importance of kernel design in the effectiveness and versatility of triangle splatting.

Tuesday, September 01 2026, 00:00

Bellman--Shoreline Search in Arbitrary Dimension: Exponential Vector Oscillators, Active Memory, Precession, and Effective Computability

Authors: Florentin Koch

We study online search for an unknown affine hyperplane in $\mathbb{R}^D$, for arbitrary fixed finite dimension. Building on a companion self-similar cell reduction and support-function formulation, we ask how the mechanism changes as the normal space grows from $\mathbb{S}^0$ to $\mathbb{S}^{D-1}$. In $D=1$, alternation and productivity yield an equal-ripple principle and the exact stationary constant $9$. In $D=2$, the analogous relative equilibrium is a logarithmic spiral whose bottleneck chord imposes tangency and selects the pitch. For exponential orbits $Γ(σ)=e^{κσ}ω(σ)$, we develop log-directional geometry, exponentially discounted memory, gauges, and recursive hyperspherical parametrizations. Without a shape ansatz, the bottleneck admits a certificate supported by at most $D$ historical suppliers, and at globally worst phases the current point lies on the active face. Within regular chambers we derive exact variation, tangency, pitch, age, and, in $D=3$, delay-system identities. Odd-dimensional obstructions, antipodal subclasses, and harmonic towers provide constraints and explicit candidate families but are not claimed globally optimal. Finally, the N-COMP theorem shows that $C_D^*$ is a computable real for every fixed finite $D$ and that algebraic polygonal $\varepsilon$-optimal cells can in principle be synthesized. Numerical screening through $D=10$ is kept separate from the proved results.

Authors: Florentin Koch

We study online search for an unknown affine hyperplane in $\mathbb{R}^D$, for arbitrary fixed finite dimension. Building on a companion self-similar cell reduction and support-function formulation, we ask how the mechanism changes as the normal space grows from $\mathbb{S}^0$ to $\mathbb{S}^{D-1}$. In $D=1$, alternation and productivity yield an equal-ripple principle and the exact stationary constant $9$. In $D=2$, the analogous relative equilibrium is a logarithmic spiral whose bottleneck chord imposes tangency and selects the pitch. For exponential orbits $Γ(σ)=e^{κσ}ω(σ)$, we develop log-directional geometry, exponentially discounted memory, gauges, and recursive hyperspherical parametrizations. Without a shape ansatz, the bottleneck admits a certificate supported by at most $D$ historical suppliers, and at globally worst phases the current point lies on the active face. Within regular chambers we derive exact variation, tangency, pitch, age, and, in $D=3$, delay-system identities. Odd-dimensional obstructions, antipodal subclasses, and harmonic towers provide constraints and explicit candidate families but are not claimed globally optimal. Finally, the N-COMP theorem shows that $C_D^*$ is a computable real for every fixed finite $D$ and that algebraic polygonal $\varepsilon$-optimal cells can in principle be synthesized. Numerical screening through $D=10$ is kept separate from the proved results.

Tuesday, September 01 2026, 00:00

Bellman Search in Arbitrary Finite Dimension: A Self-Similar Cell Theorem and Effective Computability of Planar Shoreline Search

Authors: Florentin Koch

A shoreline-search path starts at the origin and must meet an unknown affine line, without knowing either its normal or its distance. We first establish a self-similar reduction theorem for homogeneous search problems whose historical information is a record profile updated by pointwise maximum. Two quasi-returns of the normalized state delimit a block that renews the required profile by itself; a short connector closes this block into a cell. Every finite-ratio path can therefore be approximated, with arbitrarily small loss, by repetitions of a single cell at all scales. The main chain is then made effective. A finite coding of the state space computably bounds the scale factor and normalized length of a nearly optimal cell. For planar Shoreline search, the support function of the convex hull gives an exact cell functional. A one-sided polygonalization then reduces the problem to a computable number of vertices, after which quantifier elimination decides whether a polygonal cell exists below a rational threshold. It follows that the optimal deterministic planar Shoreline value $C_2^*$ is a computable real: for every rational $ε>0$, an algorithm terminates with a rational interval of width at most $ε$ containing $C_2^*$. Additional results---sliding memory, Bellman transitions, deadlines, geometric filters, and relative equilibria---are presented separately as a toolbox for certified computation and for the study of spiral rigidity; they are not used in the computability proof.

Authors: Florentin Koch

A shoreline-search path starts at the origin and must meet an unknown affine line, without knowing either its normal or its distance. We first establish a self-similar reduction theorem for homogeneous search problems whose historical information is a record profile updated by pointwise maximum. Two quasi-returns of the normalized state delimit a block that renews the required profile by itself; a short connector closes this block into a cell. Every finite-ratio path can therefore be approximated, with arbitrarily small loss, by repetitions of a single cell at all scales. The main chain is then made effective. A finite coding of the state space computably bounds the scale factor and normalized length of a nearly optimal cell. For planar Shoreline search, the support function of the convex hull gives an exact cell functional. A one-sided polygonalization then reduces the problem to a computable number of vertices, after which quantifier elimination decides whether a polygonal cell exists below a rational threshold. It follows that the optimal deterministic planar Shoreline value $C_2^*$ is a computable real: for every rational $ε>0$, an algorithm terminates with a rational interval of width at most $ε$ containing $C_2^*$. Additional results---sliding memory, Bellman transitions, deadlines, geometric filters, and relative equilibria---are presented separately as a toolbox for certified computation and for the study of spiral rigidity; they are not used in the computability proof.

Tuesday, September 01 2026, 00:00

The (Parameterized) Complexity of Ordering a Graph While Avoiding a Forbidden Pattern

Authors: Thomas Depian, Simon D. Fink, Alexander Firbas, Robert Ganian, Martin Nöllenburg, Marie Diana Sieper

In this paper, we study the Pattern Avoidance problem of determining whether a given graph $G$ admits a linear vertex order which avoids a given pattern $P$, i.e., a vertex sequence with some forced and forbidden edges, on every suborder. Such patterns form a natural ordered counterpart to induced subgraphs in the order-invariant setting, and it is known that Pattern Avoidance captures a broad variety of graph problems including Bandwidth, Vertex Coloring, Queue Number, and extends to vertex-deletion problems such as Odd Cycle Transversal. We show that Pattern Avoidance is $Σ_2^{\textsf{P}}$-complete and furthermore remains intractable (in both the classical and parameterized sense) even under a variety of severe restrictions to both the pattern $P$ and the graph $G$. As our main contributions, we complement these lower bounds with the following tractability results, which provide a unifying framework for recognizing pattern-definable graph classes: - a fixed-parameter algorithm w.r.t. the vertex integrity of $G$ plus $|V(P)|$, - a fixed-parameter algorithm w.r.t. the neighborhood diversity of $G$ plus $|E(P)|$, and - a polynomial algorithm for Pattern Avoidance on forests for almost all constant-sized patterns.

Authors: Thomas Depian, Simon D. Fink, Alexander Firbas, Robert Ganian, Martin Nöllenburg, Marie Diana Sieper

In this paper, we study the Pattern Avoidance problem of determining whether a given graph $G$ admits a linear vertex order which avoids a given pattern $P$, i.e., a vertex sequence with some forced and forbidden edges, on every suborder. Such patterns form a natural ordered counterpart to induced subgraphs in the order-invariant setting, and it is known that Pattern Avoidance captures a broad variety of graph problems including Bandwidth, Vertex Coloring, Queue Number, and extends to vertex-deletion problems such as Odd Cycle Transversal. We show that Pattern Avoidance is $Σ_2^{\textsf{P}}$-complete and furthermore remains intractable (in both the classical and parameterized sense) even under a variety of severe restrictions to both the pattern $P$ and the graph $G$. As our main contributions, we complement these lower bounds with the following tractability results, which provide a unifying framework for recognizing pattern-definable graph classes: - a fixed-parameter algorithm w.r.t. the vertex integrity of $G$ plus $|V(P)|$, - a fixed-parameter algorithm w.r.t. the neighborhood diversity of $G$ plus $|E(P)|$, and - a polynomial algorithm for Pattern Avoidance on forests for almost all constant-sized patterns.

Tuesday, September 01 2026, 00:00

Upper and lower bounds on the OBDD-width of a special integer multiplication

Authors: Tong Qin

We consider the Boolean function ${\rm SMul}_{n-1}^n(\boldsymbol{x},\boldsymbol{y})$, which computes the middle bit of the multiplication of two natural numbers represented as $n$-bit binary strings $\boldsymbol{x}$ and $\boldsymbol{y}$, drawn from a restricted domain. We investigate the width of OBDDs computing ${\rm SMul}_{n-1}^n$. We introduce a combinatorially defined function $s_*(n)$ and show that the width of such OBDDs is $Θ(2^{s_*(n)})$.

Authors: Tong Qin

We consider the Boolean function ${\rm SMul}_{n-1}^n(\boldsymbol{x},\boldsymbol{y})$, which computes the middle bit of the multiplication of two natural numbers represented as $n$-bit binary strings $\boldsymbol{x}$ and $\boldsymbol{y}$, drawn from a restricted domain. We investigate the width of OBDDs computing ${\rm SMul}_{n-1}^n$. We introduce a combinatorially defined function $s_*(n)$ and show that the width of such OBDDs is $Θ(2^{s_*(n)})$.

Tuesday, September 01 2026, 00:00

Unrestricted Boolean Multiplicative Complexity of Four-Term Binary Polynomial Multiplication: Rational Places, Hasse Jets, and the Failure of Nonlinear Feedback

Authors: Gregory Morse

Classical lower bounds show that multiplying two degree-three polynomials over $\mathbb F_2$ requires nine scalar products in bilinear or quadratic models. They do not settle unrestricted Boolean multiplicative complexity: an XOR--AND circuit may reuse nonlinear intermediate wires, and Boolean equality is taken modulo $x_i^2=x_i$, so a multiplication can lower algebraic degree. Let $\operatorname{Mul}_4:\mathbb F_2^8\to\mathbb F_2^7$ output the seven coefficients of the product of two four-term binary polynomials. We prove that its unrestricted XOR--AND multiplicative complexity is exactly nine. This resolves, for a natural vector-valued quadratic function, the Boyar--Find question of whether a quadratic-circuit lower bound can persist against unrestricted nonlinear reuse. The proof is structural rather than exhaustive. A useful purely quadratic prefix is forced onto the three rational places of $\mathbb P^1(\mathbb F_2)$. In a hypothetical eight-AND circuit, the unique non-useful gate must carry a cubic high part. Any useful continuation then forces a rational tangent and exposes a first Hasse jet, while exterior jet separation together with Boolean idempotence prevents the same defect from exposing the second Hasse jet. The required useful suffix therefore cannot exist. A complete Lean 4 formalization verifies the Boolean-ANF semantics, the unrestricted circuit model, and the exact theorem; it uses no project-specific axiom or native decision procedure. The same zero-defect flag argument gives multiplicative complexity six for three-term multiplication, and the method isolates the multi-defect obstruction for five terms.

Authors: Gregory Morse

Classical lower bounds show that multiplying two degree-three polynomials over $\mathbb F_2$ requires nine scalar products in bilinear or quadratic models. They do not settle unrestricted Boolean multiplicative complexity: an XOR--AND circuit may reuse nonlinear intermediate wires, and Boolean equality is taken modulo $x_i^2=x_i$, so a multiplication can lower algebraic degree. Let $\operatorname{Mul}_4:\mathbb F_2^8\to\mathbb F_2^7$ output the seven coefficients of the product of two four-term binary polynomials. We prove that its unrestricted XOR--AND multiplicative complexity is exactly nine. This resolves, for a natural vector-valued quadratic function, the Boyar--Find question of whether a quadratic-circuit lower bound can persist against unrestricted nonlinear reuse. The proof is structural rather than exhaustive. A useful purely quadratic prefix is forced onto the three rational places of $\mathbb P^1(\mathbb F_2)$. In a hypothetical eight-AND circuit, the unique non-useful gate must carry a cubic high part. Any useful continuation then forces a rational tangent and exposes a first Hasse jet, while exterior jet separation together with Boolean idempotence prevents the same defect from exposing the second Hasse jet. The required useful suffix therefore cannot exist. A complete Lean 4 formalization verifies the Boolean-ANF semantics, the unrestricted circuit model, and the exact theorem; it uses no project-specific axiom or native decision procedure. The same zero-defect flag argument gives multiplicative complexity six for three-term multiplication, and the method isolates the multi-defect obstruction for five terms.

Tuesday, September 01 2026, 00:00

Hardness of Approximation of Rank Aggregation on Ulam Metric

Authors: Sk Ruhul Azgor, Diptarka Chakraborty, Le Van Cuong, Debarati Das, Tien Long Nguyen

We study the approximability of rank aggregation under the Ulam metric. In the \emph{Ulam median} problem, the goal is to find a permutation minimizing the sum of its Ulam distances to the input permutations, while in the \emph{Ulam center} problem the objective is to minimize the maximum such distance. Both problems are known to be NP-hard, but no explicit approximation hardness was previously known. We prove that, for every $\varepsilon>0$, it is NP-hard to approximate either Ulam median or Ulam center within a factor of $51/50-\varepsilon$, even when the input consists of only four permutations. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. The hardness result for Ulam median is established via a reduction from MAX-E3-LIN-2. The corresponding hardness for Ulam center is then obtained through a reduction from Ulam median.

Authors: Sk Ruhul Azgor, Diptarka Chakraborty, Le Van Cuong, Debarati Das, Tien Long Nguyen

We study the approximability of rank aggregation under the Ulam metric. In the \emph{Ulam median} problem, the goal is to find a permutation minimizing the sum of its Ulam distances to the input permutations, while in the \emph{Ulam center} problem the objective is to minimize the maximum such distance. Both problems are known to be NP-hard, but no explicit approximation hardness was previously known. We prove that, for every $\varepsilon>0$, it is NP-hard to approximate either Ulam median or Ulam center within a factor of $51/50-\varepsilon$, even when the input consists of only four permutations. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. The hardness result for Ulam median is established via a reduction from MAX-E3-LIN-2. The corresponding hardness for Ulam center is then obtained through a reduction from Ulam median.

Tuesday, September 01 2026, 00:00

Parameterized Complexity of Edge-Constrained Graph Partitioning

Authors: Ajinkya Gaikwad, Jan Pokorný, Tomáš Valla

We study the Edge-Constrained Graph Partitioning Problem (ECGP), which asks whether the vertices of a graph can be partitioned into r parts, each inducing at least gamma edges. We also consider a balanced variant (BECGP), requiring equal-sized parts, and signed variants, where the utility of a part is the difference between its numbers of positive and negative edges. We show that ECGP and BECGP remain NP-hard for fixed gamma, while BECGP is also NP-hard for fixed r. For the natural parameterization r+gamma, both problems admit polynomial kernels. We obtain FPT algorithms for ECGP and BECGP parameterized by maximum leaf number, vertex deletion distance to a clique, cluster vertex deletion number plus gamma, and vertex integrity. Furthermore, ECGP is FPT parameterized by vertex deletion distance to stars plus gamma and vertex deletion distance to paths plus gamma. On the negative side, ECGP and BECGP are W[1]-hard when parameterized by r together with several structural parameters. In particular, hardness holds for feedback edge set, vertex deletion distance to stars or paths, and modular width even when the corresponding parameter is zero. The problems are also W[1]-hard parameterized by cluster vertex deletion number plus r, and by clique-width even when gamma=3. For signed graphs, both variants are NP-hard even when r+gamma=3 and the input is a disjoint union of two cliques. Finally, the balanced signed variant is W[1]-hard parameterized by treedepth plus r, even when gamma=0.

Authors: Ajinkya Gaikwad, Jan Pokorný, Tomáš Valla

We study the Edge-Constrained Graph Partitioning Problem (ECGP), which asks whether the vertices of a graph can be partitioned into r parts, each inducing at least gamma edges. We also consider a balanced variant (BECGP), requiring equal-sized parts, and signed variants, where the utility of a part is the difference between its numbers of positive and negative edges. We show that ECGP and BECGP remain NP-hard for fixed gamma, while BECGP is also NP-hard for fixed r. For the natural parameterization r+gamma, both problems admit polynomial kernels. We obtain FPT algorithms for ECGP and BECGP parameterized by maximum leaf number, vertex deletion distance to a clique, cluster vertex deletion number plus gamma, and vertex integrity. Furthermore, ECGP is FPT parameterized by vertex deletion distance to stars plus gamma and vertex deletion distance to paths plus gamma. On the negative side, ECGP and BECGP are W[1]-hard when parameterized by r together with several structural parameters. In particular, hardness holds for feedback edge set, vertex deletion distance to stars or paths, and modular width even when the corresponding parameter is zero. The problems are also W[1]-hard parameterized by cluster vertex deletion number plus r, and by clique-width even when gamma=3. For signed graphs, both variants are NP-hard even when r+gamma=3 and the input is a disjoint union of two cliques. Finally, the balanced signed variant is W[1]-hard parameterized by treedepth plus r, even when gamma=0.

Tuesday, September 01 2026, 00:00

Structural Corrections to the Bethe Approximation of the Permanent

Authors: Ijay Narang, Will Perkins

We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.

Authors: Ijay Narang, Will Perkins

We study deterministic approximation algorithms for the permanent of a nonnegative matrix through the Bethe permanent, an approximation computable in polynomial time. The tight analysis of Anari and Rezaei gives a universal comparison between the permanent and the Bethe permanent within a factor $(\sqrt 2)^n$. The simple example of the unweighted $4$-cycle $C_4$ (or a union of disjoint $C_4$'s) shows that this bound is tight. We show that such $4$-cycle obstructions can be identified and exploited algorithmically. Given a Bethe optimizer, our algorithm identifies nearly isolated weighted $2\times2$ blocks and peels off a vertex-disjoint family of them. If the total weighted correction is large, we can improve the Bethe approximation; if it is small, we show that the Bethe permanent is within a factor of $(\sqrt2 - \varepsilon)^n$ of the truth. Combining these facts, we obtain a deterministic polynomial time $(\sqrt2-\varepsilon)^n$-approximation algorithm for the permanent of an arbitrary nonnegative $n\times n$ matrix, where $\varepsilon>0$ is some absolute constant.

Tuesday, September 01 2026, 00:00

Breaking the Exponential Barrier: The First Polynomial-Time Algorithm for the Győri-Lovász Theorem

Authors: Mohammad T. Hajiaghayi, Mahdi JafariRaviz, Alireza Kaviani, Soheil Mohammadkhani

We give the first polynomial-time algorithm, after half a century, for the celebrated Győri-Lovász theorem, which resolved a conjecture of Frank (1975). The theorem, one of the simplest existential theorems to explain, states that every $k$-connected graph can be partitioned into $k$ disjoint connected subgraphs of arbitrary prescribed positive sizes. This is a fundamental structural result with broad applications, such as flexible allocation of connected subnetworks of prescribed sizes in sufficiently connected cloud infrastructures. While Lovász (1977) gave a highly non-constructive proof for a stronger directed version using algebraic topology, Győri's original constructive proof (1976) requires exponential time. Despite more than 50 years of effort, no polynomial-time algorithm was known even for $k>4$. Determining the computational complexity of the Győri-Lovász theorem---whether it admits even a sub-exponential-time algorithm or is computationally hard (in particular, PLS-complete or PPAD)---has remained one of the central open problems in algorithmic graph theory. In this paper, we finally resolve this long-standing problem by a fundamentally new proof of the existential theorem via introducing the novel concept of \emph{flow-essential assignment}, which genuinely marries matching and cut structures and yields the first polynomial-time constructive algorithm for the Győri-Lovász theorem. In fact, we obtain a polynomial-time algorithm for Lovász's stronger directed version, whose proof was non-constructive even for DAGs; for DAGs, we further obtain a near-linear-time algorithm. We also develop polynomial-time algorithms for weighted generalizations where the seminal work of Chen, Kleinberg, Lovász, Rajaraman, Sundaram, and Vetta (JACM'07) on confluent flows established only existential non-constructive results.

Authors: Mohammad T. Hajiaghayi, Mahdi JafariRaviz, Alireza Kaviani, Soheil Mohammadkhani

We give the first polynomial-time algorithm, after half a century, for the celebrated Győri-Lovász theorem, which resolved a conjecture of Frank (1975). The theorem, one of the simplest existential theorems to explain, states that every $k$-connected graph can be partitioned into $k$ disjoint connected subgraphs of arbitrary prescribed positive sizes. This is a fundamental structural result with broad applications, such as flexible allocation of connected subnetworks of prescribed sizes in sufficiently connected cloud infrastructures. While Lovász (1977) gave a highly non-constructive proof for a stronger directed version using algebraic topology, Győri's original constructive proof (1976) requires exponential time. Despite more than 50 years of effort, no polynomial-time algorithm was known even for $k>4$. Determining the computational complexity of the Győri-Lovász theorem---whether it admits even a sub-exponential-time algorithm or is computationally hard (in particular, PLS-complete or PPAD)---has remained one of the central open problems in algorithmic graph theory. In this paper, we finally resolve this long-standing problem by a fundamentally new proof of the existential theorem via introducing the novel concept of \emph{flow-essential assignment}, which genuinely marries matching and cut structures and yields the first polynomial-time constructive algorithm for the Győri-Lovász theorem. In fact, we obtain a polynomial-time algorithm for Lovász's stronger directed version, whose proof was non-constructive even for DAGs; for DAGs, we further obtain a near-linear-time algorithm. We also develop polynomial-time algorithms for weighted generalizations where the seminal work of Chen, Kleinberg, Lovász, Rajaraman, Sundaram, and Vetta (JACM'07) on confluent flows established only existential non-constructive results.

Tuesday, September 01 2026, 00:00

Test or Run? Scheduling Jobs of Unknown Length

Authors: Václav Rozhoň

A machine faces many jobs whose lengths are hidden. Spending one unit of time to inspect a job may reveal a short job that should be finished now, or it may reveal nothing useful while every other job waits. When should the machine keep looking, and when should it start working? We study natural variants of this question and provide optimal algorithms in both the worst-case and instance-optimal frameworks. The resulting algorithms are often quite simple, which may make them useful in practice.

Authors: Václav Rozhoň

A machine faces many jobs whose lengths are hidden. Spending one unit of time to inspect a job may reveal a short job that should be finished now, or it may reveal nothing useful while every other job waits. When should the machine keep looking, and when should it start working? We study natural variants of this question and provide optimal algorithms in both the worst-case and instance-optimal frameworks. The resulting algorithms are often quite simple, which may make them useful in practice.

Tuesday, September 01 2026, 00:00

The Cayley Completion of a Graph

Authors: Rigobert Fokam Souop, Laurent Bitjoka

A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $γ^{+}$ (additions only) and the Cayley edit distance $γ_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $γ^{+}(G)\ge nΔ^{*}/(2m)-1$, where $Δ^{*}$ is the least $d\geΔ$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $γ^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $γ^{+}$, while $γ_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $γ^{+}(P_n)=γ^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $γ_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.

Authors: Rigobert Fokam Souop, Laurent Bitjoka

A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $γ^{+}$ (additions only) and the Cayley edit distance $γ_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $γ^{+}(G)\ge nΔ^{*}/(2m)-1$, where $Δ^{*}$ is the least $d\geΔ$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $γ^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $γ^{+}$, while $γ_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $γ^{+}(P_n)=γ^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $γ_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.

Tuesday, September 01 2026, 00:00

On the maximum weight convex problem for some geometric graph-convexities

Authors: Fariza Aklouche, Pierre Bergé, Michel Habib

For a given geometric graph-convexity on a graph $G$ equipped with a weight function on the vertices with value in $\mathbb{Z}$, the Max Weight Convex Set problem consists in determining the convex set $S$ with maximum weight (sum of the weight of the vertices in $S$). Although the problem is NP-complete in general, it remains polynomial for particular cases. After a survey of known results, our main contribution uses a generalisation of the maximum subsequence problem to laminar trees. Then we derive a linear algorithm for proper interval graphs and a quadratic one for interval graphs. Both improve the state of the art.

Authors: Fariza Aklouche, Pierre Bergé, Michel Habib

For a given geometric graph-convexity on a graph $G$ equipped with a weight function on the vertices with value in $\mathbb{Z}$, the Max Weight Convex Set problem consists in determining the convex set $S$ with maximum weight (sum of the weight of the vertices in $S$). Although the problem is NP-complete in general, it remains polynomial for particular cases. After a survey of known results, our main contribution uses a generalisation of the maximum subsequence problem to laminar trees. Then we derive a linear algorithm for proper interval graphs and a quadratic one for interval graphs. Both improve the state of the art.

Tuesday, September 01 2026, 00:00

Gate-Efficient Implementation of the Query-Optimal Time-Dependent Hamiltonian Simulation

Authors: Boyang Chen, Minbo Gao, Zhengfeng Ji, Tongyang Li, Xinzhao Wang, Shuo Zhou

The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses $$ q = O\left( αT + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(αT) \right)} \right) $$ queries to $\mathrm{HAM\mbox{-}T}$ within $\varepsilon$ error for a Lipschitz-continuous time-dependent Hamiltonian $H(t)$ on $[0,T]$ satisfying $\left\lVert H(t)\right\rVert\leqα$. However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses $$ O\left[ q \left( a + \log\left(1 + \frac{T(α+ βT)}{\varepsilon} \right) \right) \right] $$ one- and two-qubit gates, where $a$ is the number of block-encoding ancilla qubits and $β$ is the Lipschitz constant of $H$. The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.

Authors: Boyang Chen, Minbo Gao, Zhengfeng Ji, Tongyang Li, Xinzhao Wang, Shuo Zhou

The query-optimal algorithm of [CGWZ26] for general time-dependent Hamiltonian simulation uses $$ q = O\left( αT + \frac{\log(1/\varepsilon)}{\log\left(e + \log(1/\varepsilon)/(αT) \right)} \right) $$ queries to $\mathrm{HAM\mbox{-}T}$ within $\varepsilon$ error for a Lipschitz-continuous time-dependent Hamiltonian $H(t)$ on $[0,T]$ satisfying $\left\lVert H(t)\right\rVert\leqα$. However, its direct circuit implementation incurs a substantially larger gate overhead. In this note, we give an implementation of the same algorithm that retains its optimal query complexity and uses $$ O\left[ q \left( a + \log\left(1 + \frac{T(α+ βT)}{\varepsilon} \right) \right) \right] $$ one- and two-qubit gates, where $a$ is the number of block-encoding ancilla qubits and $β$ is the Lipschitz constant of $H$. The main ingredient is an exact dyadic factorization of the ordered update product in the underlying one-query transducer.

Tuesday, September 01 2026, 00:00

Beating Quadratic Time--Message Trade-off in Distributed Minimum Spanning Tree Construction

Authors: Taisuke Izumi, Naoki Kitamura, Toshimitsu Masuzawa

We present a new distributed algorithm for computing a minimum spanning tree (MST) in the \textsf{CONGEST-KT$_{1}$} model, where messages are limited to $O(\log n)$ bits and each vertex initially knows the identifiers of its neighbors. Our algorithm exposes a two-parameter time--message trade-off: for any $0 \leq λ\leq κ\leq 1/2$, it runs in $\tilde{O}(n^λD_G + n^{1 - κ- λ} + n^{1 - 2κ+ λ} + n^{1/2})$ rounds and uses $\tilde{O}(\min\{m, n^{1 + κ}\})$ messages, where $n$, $m$, and $D_G$ are the number of vertices, edges, and thenetwork diameter, respectively. In particular, setting $(κ, λ) = (1/3, 1/6)$ yields an MST algorithm running in $\tilde{O}(n^{1/2} + n^{1/6}D_G)$ rounds with only $\tilde{O}(n^{4/3})$ messages. Under the mild assumption $D_G = O(n^{1/3})$, this is round-optimal while improving the best known message bound of $\tilde{O}(n^{3/2})$. More broadly, our algorithm breaks the quadratic time--message trade-off barrier $\mathrm{\# rounds} \cdot \mathrm{\# messages} = \tildeΩ(n^2)$, which no previous MST algorithm in the \textsf{CONGEST-KT$_{1}$} model has been able to overcome, and it does so for almost the entire range of the diameter $D_G$. As a byproduct, we also obtain new low-message broadcast, spanning-tree, and leader-election algorithms.

Authors: Taisuke Izumi, Naoki Kitamura, Toshimitsu Masuzawa

We present a new distributed algorithm for computing a minimum spanning tree (MST) in the \textsf{CONGEST-KT$_{1}$} model, where messages are limited to $O(\log n)$ bits and each vertex initially knows the identifiers of its neighbors. Our algorithm exposes a two-parameter time--message trade-off: for any $0 \leq λ\leq κ\leq 1/2$, it runs in $\tilde{O}(n^λD_G + n^{1 - κ- λ} + n^{1 - 2κ+ λ} + n^{1/2})$ rounds and uses $\tilde{O}(\min\{m, n^{1 + κ}\})$ messages, where $n$, $m$, and $D_G$ are the number of vertices, edges, and thenetwork diameter, respectively. In particular, setting $(κ, λ) = (1/3, 1/6)$ yields an MST algorithm running in $\tilde{O}(n^{1/2} + n^{1/6}D_G)$ rounds with only $\tilde{O}(n^{4/3})$ messages. Under the mild assumption $D_G = O(n^{1/3})$, this is round-optimal while improving the best known message bound of $\tilde{O}(n^{3/2})$. More broadly, our algorithm breaks the quadratic time--message trade-off barrier $\mathrm{\# rounds} \cdot \mathrm{\# messages} = \tildeΩ(n^2)$, which no previous MST algorithm in the \textsf{CONGEST-KT$_{1}$} model has been able to overcome, and it does so for almost the entire range of the diameter $D_G$. As a byproduct, we also obtain new low-message broadcast, spanning-tree, and leader-election algorithms.

Tuesday, September 01 2026, 00:00

Socially Fair Clustering: Parameterized Approximation and Local Search