Theory of Computing Report

Authors: Marie-Charlotte Brandenburg, Jesús A. De Loera, Chiara Meroni

We design exact algorithms for the ham-sandwich and centerpoint theorems for polytopal measures. Our key observation is that the cap-volume function of such a measure, i.e., the volume cut off by a halfspace, is piecewise rational on a natural decomposition of the space of oriented hyperplanes. This lets us recast prescribed-proportion cutting problems as semialgebraic feasibility problems. For fixed ambient dimension, this yields polynomial-time algorithms to decide the existence of cuts, describe the full solution set, and sample or enumerate solutions. We extend this framework to the center transversal theorem, showing that spaces of deep affine flats are semialgebraic, which holds for centerpoints. We further show that the set of centerpoints of a convex polytope coincides with its floating body at level $1/(d+1)$, a useful semialgebraic description.

Authors: Reymond Akpanya, Alice C. Niemeyer, Lukas Schnelle

A crystallographic group is a discrete subgroup of the Euclidean group $\operatorname{E}(n)$ that has a compact fundamental domain. Since such a crystallographic group $Γ$ is infinite, computing fundamental domains of $Γ$ is algorithmically challenging. We address this difficulty by targeting the computation of Dirichlet cells that can form fundamental domains of $Γ$. We show that the half-spaces defining such a Dirichlet cell can be derived from elements of $Γ$ acting on $\mathbb{R}^n$ that can be expressed as words of bounded length in a suitable generating set. Based on these results, we design an algorithm for the computation of fundamental domains of crystallographic groups and exploit it to study the construction of topological interlocking assemblies.

Authors: Foad Namjoo, Drew McClelland, Michael Matheny, Jeff M. Phillips

Anomaly detection in geospatial data is a crucial tool in geographic information science (GIS), with applications ranging from national security to public-health surveillance to the study of societal disparities. This work focuses on spatial scan statistics and addresses a key mismatch: spatial counts are typically aggregated into predefined regions (census tracts, zip codes, counties), whereas the most efficient scan algorithms operate on spatial point data. The standard remedy -- collapsing each region to its centroid, as in widely used tools such as SaTScan -- is convenient but, as we show, discards the region's spatial extent and causes a significant loss in statistical power. To resolve this, we propose a simple yet scalable fix: replace each spatial region with 20-50 points sampled uniformly from its geometry and spread the region's values evenly across them. This approach improves statistical power while maintaining computational tractability. A convergence analysis explains why so few samples per region suffice. We recommend this sampling-based conversion as the default way to apply point-based spatial scan statistics to region-aggregated data for anomaly detection.

Authors: Alexander Baumann, Therese Biedl, Mahmoud Elashmawi, Simon D. Fink, Maria Saumell, André Schulz

The reconstruction problem asks to construct a (convex) polygon that has a specified set of features, such as an ordered set of edge-lengths or an ordered set of polygon-angles. In this paper, we do a systematic exploration of the reconstruction problem in all scenarios where one or two sets of features have been specified. Some of these scenarios were well-studied already, for some we develop testing-algorithms and/or hardness results, and many give rise to interesting open problems for future study.

Authors: Srinivasan Arunachalam, Louis Schatzki

We study stabilizer state testing and learning with limited coherent quantum memory. Here an algorithm sequentially receives copies of an unknown $n$-qubit state, but may keep only $k$ qubits of coherent quantum memory between measurements. With unrestricted memory, seminal work of Gross, Nezami and Walter showed how to test $n$-qubit stabilizer states using $6$ copies, which is dimension independent, unlike the learning complexity of $Θ(n)$. We show that this testing-vs-learning separation is lost under memory constraints. More concretely we show that (1) The sample complexity of testing stabilizer states in the $k$-qubit memory framework is $Θ(n-k)$. Our upper bound goes via a novel connection to the hidden shift problem and the lower bound is proven using a novel approach to average case bounds on likelihood ratios via combinatorics of the stochastic orthogonal group. (2) The sample complexity of learning stabilizer states with $k$ qubits of memory, in the non-adaptive framework, is $Θ(n^2/k)$. As a further application of our techniques, we prove an exponential lower bound for purity testing even when the memory may be left coherent throughout the protocol. Our main results identify coherent quantum memory as the resource enabling the usual separation between stabilizer testing and learning. In particular, even with $k=0.99n$ qubits of memory, there is no constant-copy stabilizer tester; furthermore for $k=cn$ qubits of memory (for $0< c < 1$), stabilizer testing is as hard as learning, with both requiring $Θ(n)$ copies.

Authors: Cornelius Brand, Petteri Kaski, Jiaheng Wang

Strassen's theory of bilinear complexity provides a mathematical characterization of the arithmetic complexity of primitives such as matrix multiplication via the rank of tensors. However, the connection to tensor rank is known to break down in higher degrees of multilinearity. In this work, we highlight an unexplored connection between a generalized notion of tensor rank, which can be defined in Naslund's framework of partition ranks (JCTA 2020), and multiplicative complexity. These partition ranks allow us to control the multiplicative complexity, and thus arithmetic complexity, in any constant degree of multilinearity from below, while recovering Strassen's seminal characterization in the bilinear case. This enables novel potential applications of the rank-based approaches to problems in fine-grained algorithms and complexity, such as the hyperclique conjecture of Lincoln-Williams-Vassilevska Williams (SODA 2018). Moreover, we exhibit connections to established notions of rank, such as tensor slice rank (in the sense of Tao and Sawin), as well as its symmetric variant. For computing the latter symmetric variant, we point out a simple NP-hardness proof, contrasting the rather involved NP-hardness proof for ordinary, non-symmetric tensor slice rank by Bläser et al. (SODA 2021).

Authors: Wen Fang, Xianxian Li, Jun Liu, Jie Luo, Yongxin Tong, Ke Xu

Self-reference and solution independence are core properties underlying intractability. This paper establishes a finite combinatorial analogue of Gödel's incompleteness theorems within Boolean $K$-SAT. While standard random $K$-SAT has assignment correlations that disrupt solution independence, we resolve this via a logarithmic-width ensemble ($K = O(\log N)$). Here, satisfying assignments converge to a Poisson distribution, letting unsatisfiable and uniquely satisfiable formulas coexist. By executing a single-clause substitution conditioned on the unique solution, we construct structurally irreducible SAT/UNSAT pairs that are indistinguishable via local evaluation. Using algorithmic information theory and Shannon channels, we prove that deductive pipelines restricted to a sublinear window suffer from an informational blind spot, forcing a descriptive lower bound of $K(\mathcal{A}) \geq Ω(N^{1-δ})$. This deficit forces any Resolution refutation of the UNSAT instance to utilize wide clauses ($w(π) \geq Ω(N^{1-δ})$), triggering an exponential proof-tree explosion ($S(φ) \geq \exp(Ω(N^{1-2δ}))$). As $δ\rightarrow 0^+$, this bound converges to the worst-case $2^N$ threshold, reframing the Strong Exponential Time Hypothesis (SETH) as a direct projection of Gödel incompleteness onto finite computation. We diagnose the decades-long stagnation in complexity theory. Transitioning from Turing's class separation to a Gödelian paradigm of instance indistinguishability, we introduce a multi-dimensional comparative framework that contrasts these two historical lineages across distinct perspectives. The self-referential hardness exhibits physical invariance: it precludes quantum shortcuts due to the necessity of global semantic analysis and delineates a scaling bottleneck for machine learning architectures operating on lossy, local compression.

Authors: Daeho Lee, Shivam Nadimpalli, Mingda Qiao, Ronitt Rubinfeld

We initiate the study of *unate distributions* over $\{\pm1\}^n$ -- a natural analogue of unate Boolean functions -- by considering two basic testing problems that parallel well-studied questions for monotone distributions: - Uniformity Testing of Unate Distributions: We show that $\widetildeΘ(n^{3/2})$ samples are sufficient and necessary, in contrast to the $\widetildeΘ(n)$ sample complexity of the analogous problem for monotone distributions (Rubinfeld and Servedio, STOC 2005; Adamaszek, Czumaj, and Sohler, SODA 2010). - Unateness Testing of Arbitrary Distributions: We give a tester that uses $\widetilde{O}(n^{3/2})$ conditional samples in the subcube conditional model. On the other hand, every tester that draws conditional samples in a similar fashion, namely from $O(1)$-dimensional subcubes, must have an $\widetildeΩ(n^{2/3})$ complexity. In the same model, the complexity of monotonicity testing was recently shown to be $\widetildeΘ(n)$ (Chakrabarty et al., STOC 2025). Our algorithms for both problems significantly outperform the naive approach of reducing to the monotone case, which would incur $Ω(n^2)$ sample complexity. Our uniformity tester relies on a subroutine that "weakly" learns the hidden orientations of a unate distribution, together with a new correlation bound for these estimates. Both tools may be of independent interest in studying monotonicity and unateness over $\{\pm1\}^n$.

Authors: Avi Kadria, Liam Roditty, Virginia Vassilevska Williams

Let $G = (V, E)$ be a graph with $n = |V|$ nodes and $m = |E|$ edges. The $t$-Pairs Shortest Paths problem, introduced by Cohen [FOCS'93; SICOMP'99], asks to approximate the distances between $t$ prespecified pairs of vertices. Recently, this problem has received renewed attention, particularly in the case where $t = Θ(n)$: the $n$-Pairs Shortest Paths problem. In this setting, new algorithms and conditional lower bounds have been developed by Dalirrooyfard, Jin, Vassilevska Williams, and Wein [FOCS'22], and Chechik, Hoch, and Lifshitz [SODA'25]. In this paper, we present the first algorithm for the $n$-Pairs Shortest Paths problem in \textit{weighted} undirected graphs that achieves a $(2 - α)k$-approximation, for constant $α> 0$, that runs in $\tilde{O}(mn^{1/k} + n^{1 + 2/k})$ time. Specifically, we present a $1.622k$-approximation, improving upon the $(2k - 3)$-approximation of Chechik, Hoch, and Lifshitz [SODA'25] for graphs that are not super sparse, which answers in the affirmative the open question posed by them. We also develop improved approximation algorithms with better tradeoffs for unweighted graphs and dense weighted graphs that improve upon the results of Dalirrooyfard \etal~and Chechik, Hoch, and Lifshitz. Our main technical contribution is the new \textit{heavy-edge} technique. Using this technique, we transform an algorithm with an approximation guarantee that depends on $W_{uv}$, the weight of the heaviest edge on the shortest path between $u$ and $v$, into an algorithm with purely multiplicative approximation that does not depend on $W_{uv}$.

Authors: Qiao-Long Huang, Yichuan Cao, Ruichen Qiu, Xiao-Shan Gao

We study the problem of deterministically computing the exact root of a sparse polynomial in the multivariate setting. Let $f \in \F[x_1,\ldots,x_n]$ be a nonzero polynomial that is an exact $e$-th power, say $f = g^e$. Suppose $f$ is $s$-sparse, has an individual degree of at most $d$, and a total degree of $D = \tdeg(f)$. We prove a sparsity bound on the base polynomial $g$: \[ \|g\|_0 \le s^{D(2d+2)/e + 1}. \] Based on this bound, we develop a deterministic algorithm that computes the base $g$. % In contrast to the general deterministic factorization algorithm of Bhargava, Saraf, and Volkovich \cite{BhargavaSarafVolkovich2020}, which achieves only a quasi-polynomial dependence on the input parameters, our algorithm is \emph{polynomial-time} in the setting where the total degree $D$ is bounded. Specifically, the overall complexity is \[ \mathrm{poly}\left(s^{O(Dd)}, n, d, D\right) + s\cdot R(e), \] % where $R(e)$ denotes the cost of constructing a single $e$-th root of a scalar in the base field $\F$, and, when $\operatorname{char}(\F)\mid e$, the cost of computing a single Frobenius root of a scalar. % This term is field-dependent, and over finite fields, $\mathbb{Q}$, or number fields with a suitable representation, it is absorbed into the polynomial complexity bound. % Within the bounded total-degree regime, this yields a deterministic polynomial-time algorithm for exact-root computation.

Authors: Jiawei Zhang

This paper studies additive regret in the multi-secretary problem, defined as the gap between the expected offline prophet reward and the reward of the best online policy. Prior work established \(O(\log T)\) regret for bounded-density distributions with connected support and \(O((\log T)^2)\) upper bounds for bounded-density distributions with support gaps. It was unknown whether the extra logarithmic factor is necessary even in the one-resource model. We prove that it is necessary. For a mixture of two separated uniform distributions at the critical capacity, the optimal regret grows at least on the order of \((\log T)^2\). Thus the existing \(O((\log T)^2)\) upper bounds for bounded-density gapped instances, including those implied by network revenue management models with continuous rewards, are tight in this simplest specialization. The same framework also yields a matching lower bound for gapped distributions whose gap-facing densities vanish near the support edges; this companion result is given in the appendix. The proofs use Bellman certificates: feasible solutions to a relaxation of the exact Bellman recursion. This framework converts lower bounds into explicit certificate constructions and identifies why support gaps permit larger regret.

Authors: Riccardo Maso, Nicola Prezza, Carlo Tosoni

Pattern matching on strings is regarded as one of the core operations in computer science. Although researchers proposed several solutions to this problem, some of the most elegant and widely used approaches are based on the renowned Burrows-Wheeler transform (BWT). The success of the BWT lies in its pattern matching algorithm known as backward search, which is not only near-optimal in the RAM model, but also runs directly on a compressed representation of the input string. More recently, the backward search has been generalized to Wheeler deterministic finite automata (DFAs), a subclass of standard DFAs, without losing its near-optimal time efficiency. Similarly to the case of strings, this pattern matching algorithm for Wheeler DFAs has found applications in bioinformatics, where researchers have shown that specific pangenome graphs of human chromosomes can be transformed into Wheeler DFAs and consequently indexed using this strategy. However, this BWT-based index on Wheeler DFAs inherited a significant drawback from the original backward search, namely the high number of I/O operations triggered during the algorithm execution, which are in the worst-case lower-bounded by the length of the pattern. In this paper, we address this limitation by proposing the first cache-friendly algorithm specifically designed for Wheeler DFAs. Our new data structure reduces the number of I/O operations by employing a strategy analogous to the suffix array: it interleaves a binary search with fast sequential scans of the automaton. We empirically validate this new indexing strategy by running our algorithm on real-world Wheeler pangenome graphs. We show that while our data structure can use up to 15 times the space required by the backward search, it can also be 500 times faster and able to process a single character of the pattern in less than 3 ns.

Authors: Laura Merker, Lena Scherzer, Samuel Schneider

The geodesic treewidth of a graph $ G $ is the smallest $k$ for which there is a partition $\mathcal{P}$ into geodesics such that $G/\mathcal{P}$ has treewidth $k$, where $G/\mathcal{P}$ is obtained from $ G $ by contracting each part of $ \mathcal{P} $. Based on this notion, row treewidth was developed and is defined for a graph $ G $ as the smallest $ k $ such that $ G \subseteq H \boxtimes P $ for some graph $ H $ of treewidth $ k $ and a path $ P $. Equivalently, the row treewidth of a graph $ G $ is the smallest $ k $ for which there is a partition $ \mathcal{P} $ into disjoint unions of geodesics that are aligned with respect to some layering such that $ G/\mathcal{P} $ has treewidth $ k $. We separate the two notions by showing that bounded row treewidth does not imply bounded geodesic treewidth and by presenting a polynomial-time algorithm to decide whether a graph of treewidth 2 has geodesic treewidth 1, which is known to be NP-hard for row treewidth [Biedl, Eppstein, Ueckerdt, 2025]. More generally, we provide an algorithm to decide whether a given graph has geodesic treewidth at most $ d $ that is XP in the treewidth, whereas there is no such algorithm for row treewidth, unless P = NP [Biedl, Eppstein, Ueckerdt, 2025]. On the other hand, we show that computing the geodesic treewidth is NP-hard and that every graph with geodesic treewidth 1 has bounded row treewidth. Moreover, we improve the best known lower bound on the geodesic treewidth of planar graphs to 5.

Authors: Daniel Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, Meirav Zehavi

Courcelle's theorem states that there exists an algorithm that takes as input a graph $G$ of treewidth at most $t$ and a MSO formula $φ$, and determines whether $G$ satisfies $φ$ in time $f(φ,t) \cdot n$. It is folklore that the the function $f$ contains a tower of exponentials whose height depends as a linear function of the number of quantifier alternations of the input formula $φ$. A classic reduction of Frick and Grohe shows that, assuming the Exponential Time Hypothesis (ETH), the linear growth of the height of the tower is unavoidable. Nevertheless, there is still a huge gap between existing upper and lower bounds -- after all, there is quite a difference between a single exponential and a double exponential running time. In addition, this only gives us a very coarse understanding in the time complexity of Courcelle's theorem. In this paper, we prove a fine-grained version of Courcelle's theorem with nearly ETH-tight dependence on the treewidth parameter $t$ and the quantifier structure of $φ$ (specifically, the number of first order and second order variables in each quantifier alternation block).

Authors: Ilie Sarpe, Federico Altieri, Andrea Pietracaprina, Geppino Pucci, Fabio Vandin

The silhouette is one of the most widely used measures to assess the quality of a $k$-clustering of a dataset of $n$ elements. Its evaluation requires no information beyond the clustering assignment. In addition, the silhouette is extremely easy to interpret, providing a score to measure the quality of a clustering as a whole or for each element. The exact computation of the: (i) silhouette of each element of a dataset; and (ii) the global silhouette of the clustering; require $Θ(n^2)$ distance calculations, under general metrics. The quadratic complexity $Θ(n^2)$ is extremely prohibitive, especially on massive modern datasets. Surprisingly, existing approximate methods using $O(n^2)$ distance calculations are heuristics not offering provable and controllable guarantees on the quality of their results. We introduce the first rigorous and efficient algorithms to estimate: (i) the (local) silhouette of each element of a dataset; and (ii) the (global) silhouette; of any metric $k$-clustering. Our methods, based on sampling, perform $O(nk\varepsilon^{-2}\ln (nk/δ))$ distance computations, and provide estimates with additive error $O(\varepsilon)$ with probability at least $1-δ$. That is, parameters $\varepsilon$ and $δ$ in $(0,1)$ control the trade-off between accuracy and efficiency. We also introduce a scalable and distributed design of our methods for the MapReduce and Massively Parallel Computing (MPC) frameworks. Our distributed algorithms use a constant number of rounds and sublinear local memory. Finally, we perform extensive experiments against state-of-the-art approaches. The results show that our new techniques yield the best trade-off between accuracy and efficiency for both local and global silhouette estimation. In addition, our methods scale efficiently to massive datasets for which an exact computation of the silhouette is not practical.

Authors: Da Wei Zheng

We present the first truly subquadratic time algorithm to compute diameter and eccentricity in real-weighted directed graphs with constant distance VC-dimension and strongly sublinear-sized balanced separators. This runs in $O(n^{2-1/(2h-2)}\textrm{polylog}(n))$ time for real-weighted $K_h$-minor-free digraphs. Prior to this work, truly subquadratic time computation of diameter was only known for real-weighted planar graphs, while extensions to broader classes like minor-free graphs were restricted to unweighted settings. In particular, existing algorithms that use VC-dimension [Ducoffe, Habib, Viennot; SICOMP 2022][Le, Wulff-Nilsen; SODA 2024][Chan, Chang, Gao, Le, Kisfaludi-Bak, Zheng; FOCS 2025] work with small integer weights, but do not naturally generalize to real weights. We overcome this barrier by introducing a randomized search-to-decision reduction, demonstrating that VC-dimension is a sufficiently powerful tool in the real-weighted regime.

Authors: Édouard Bonnet, Carl Feghali, Manolis Vasilakis

Given a connected graph $G$ and a source $s \in V(G)$, what is the smallest number of rounds necessary for all vertices of $G$ to receive a message initially only held by $s$, where at each round every informed vertex passes the message to one of its neighbors? This problem is called Telephone Broadcast and is suprisingly hard: it remains NP-hard on cycles intersecting at a single shared vertex, in particular, graphs of pathwidth 2 with a linear feedback vertex set of size 1, as well as on graphs with treedepth at most 6 [Egami et al.; MFCS '25]. Vertex cover number, vertex integrity, and distance to clique are among the few parameters for which Telephone Broadcast is fixed-parameter tractable. There is a $2^{\mathcal{O}(\mathrm{vc}^3)} n^{\mathcal{O}(1)}$-time algorithm parameterized by vertex cover number $\mathrm{vc}$ [Fomin, Fraigniaud, Golovach; TCS '24], a double-exponential algorithm parameterized by vertex integrity $\mathrm{vi}$, and a $2^{\mathcal{O}(k^2)} n^{\mathcal{O}(1)}$-time algorithm parameterized by distance to clique $k$ [Egami et al.; MFCS '25]. In this paper, we give improved parameterized algorithms for Telephone Broadcast with running times $2^{\mathcal{O}(\mathrm{vc} \log \mathrm{vc})} n^{\mathcal{O}(1)}$, $2^{\mathcal{O}(\mathrm{vi}^2 \log \mathrm{vi})} n^{\mathcal{O}(1)}$, and $2^{\mathcal{O}(k \log k)} n^{\mathcal{O}(1)}$. The main ingredient that makes our algorithms faster is a Turing reduction to edge-weighted $b$-Matching.

Authors: Shintaro Matsushita, Takayoshi Shoudai, Yusuke Suzuki

Unordered trees appear in applications where the order among child vertices is insignificant, such as abstract syntax trees and chemical structures. To describe patterns in such trees, we propose unordered term tree patterns, which employ height-constrained variables that restrict trunk length and subtree height. We formalize the pattern matching problem between an unordered term tree pattern and an unordered tree, and present an $O(N \cdot \max\{nD^{3/2}, \mathcal{S}\})$-time algorithm, where $n$ and $N$ are the numbers of vertices in the pattern and tree, $D$ is the maximum vertex degree, and $\mathcal{S}$ is the sum of trunk constraints. Computational results show that the algorithm runs efficiently in practice.

Authors: Yuta Tsuruzono, Hiroki Arimura, Shunsuke Inenaga

The compact directed acyclic word graph (CDAWG) of a string can be viewed in two equivalent ways: as the edge-compacted DAWG of the string, and as the DAG obtained from the suffix tree by merging the nodes whose subtrees are isomorphic. By exploiting these two views in opposite directions, we show how to build, for the (reversed) input string of length $n$, the CDAWG with $e_L$ edges in $O(e_L\log n\log(n/r))$ time with $O(r\log(n/r)+e_L)$ words of working space, provided that the fully functional compressed suffix tree of Gagie, Navarro, and Prezza of size $O(r\log(n/r))$ is available. Here, $r$ denotes the number of BWT-runs of the input string.

Authors: Taiki Kaneda, Hiroki Arimura, Shunsuke Inenaga

We study fully persistent dynamic strings with equality and longest common extension (LCE) queries. Straightforward full persistence is problematic for the splay-based FeST structure, since the same unbalanced past version can be reused indefinitely and the usual amortized analysis no longer applies. We give a fully persistent dynamic LCE structure, called FeAVL, based on path copying over AVL trees. For an operation involving string(s) of total length $n$, it supports split, concatenate, and single-character updates in worst-case $O(\log n)$ time, equality in worst-case $O(\log n)$ time w.h.p., and LCE in worst-case $O(\log n+\log^2\ell)$ time w.h.p., where $\ell$ is the answer; each update creates only $O(\log n)$ new permanent nodes. We also give a grammar-compressed instantiation via AVL grammars: starting from an initial grammar of size $g_0$, after $U$ updates, the total number of permanent grammar nodes is $O(g_0+I+U\log n_{\max})$, where $I$ is the number of inserted fresh characters and $n_{\max}$ is the maximum string length appearing during the update sequence.

Authors: Billy Jin, Nathan Klein, David P. Williamson

One of the most famous conjectures in combinatorial optimization is the four-thirds conjecture, which states that the integrality gap of the Subtour LP relaxation of the TSP is equal to $\frac{4}{3}$. For 40 years, the best known upper bound was $1.5$, due to Wolsey. Recently, Karlin, Klein, and Oveis Gharan showed that the max entropy algorithm for the TSP gives an improved bound of $1.5 - 10^{-36}$. In this paper, we show that the maximum entropy algorithm is a $\frac{10}{7}$-approximation for half-integral cycle cut instances of the TSP. This class of instances contains examples which demonstrate the subtour LP has an integrality gap of at least $\frac{4}{3}$, as well as examples showing that the performance of the max entropy algorithm is no better than $\frac{11}{8}$. We note that in the authors recently gave an algorithm upper bounding the integrality gap of this class of instances by $\frac{4}{3}$, so this work does not (and could not) provide an improved bound on the integrality gap. However, since there is no reason to believe that the analysis of the maximum entropy algorithm on general instances is tight, our work provides hope (and potentially direction) for improved analysis on other instance classes.

Authors: Gagan Aggarwal, Anupam Gupta, Yifan Wang, Mingfei Zhao

The "Trading Prophet" problem challenges an online trader to maximize its profit by buying and selling assets under stochastic prices and capacity constraints, competing against an offline prophet with full foresight. In previous work, each arriving asset was assumed to have a single price $p_t$, and the trader was allowed to either buy a copy at this price (subject to having available capacity), or sell a copy (if it already held at least one copy in hand). However, this abstraction can fail to capture the structural asymmetry of decentralized dealer-based markets, where buying and selling opportunities could be distinct, and driven by individual preferences. To address this, we introduce the Asymmetric Trading Prophets problem, where at each timestep the trader observes a price tuple $(b_t, s_t)$ -- representing the cost to buy, and the revenue from selling at this timestep. Importantly, the $(b_t,s_t)$ tuple could be potentially arbitrarily correlated. We provide the first competitive analysis for this asymmetric trading prophets problem, characterizing the achievable profit based on the trader's capacity $B$ and initial inventory $B_0$. For the unit-capacity case of $B=1$, we design online algorithms that achieve constant competitive ratios for both i.i.d. and non-i.i.d. distributions on the price tuples, when the trader has one initial copy ($B_0=1$). For the general capacity case where $B$ can be large, we give algorithms for i.i.d. distributions that achieve a competitive ratio of $1 - Θ(\log B_0/\sqrt{B_0})$. Finally, for the symmetric case (where the price tuple satisfies $b_t=s_t$), we improve this to get a competitive ratio of $1 - O(\log B/\sqrt{B})$, demonstrating that the performance approaches optimality as the capacity increases. We show that both ratios are tight up to a logarithmic factor.

Authors: Karl Bringmann, Nick Fischer, Vasileios Nakos

In the seminal sparse matrix multiplication problem the goal is to compute the product of two $n \times n$ matrices when the matrices are sparse, i.e., when the number of nonzeros in the input matrices $m_{in}$ and/or the number of nonzeros in the output matrix $m_{out}$ are much smaller than $n^2$. In this paper, we explore the generalized problem of (approximately) computing the $k$ largest output entries, with an approximation error dependent solely on the smaller entries -- from the viewpoint of sparse recovery, this can be seen as a robust variant of sparse matrix multiplication. Despite the substantial research dedicated to sparse matrix multiplication, almost no existing algorithms are robust in this sense. The one exception is Pagh's algorithm in time $\widetilde O(m_{in} + nk)$ [ITCS'12], and it remained open whether other algorithms can be similarly made robust. Our principal contribution is a black-box reduction from robust sparse matrix multiplication to conventional sparse matrix multiplication with only polylogarithmic overhead. Specifically, we show that any sparse matrix multiplication algorithm with running time $T(n, m_{in}, m_{out})$ can be transformed into a robust algorithm running in time $\widetilde O(T(n, m_{in}, k))$. This reduction leverages an extensive toolkit from sparse recovery, and intriguingly, also involves solving a knapsack-type problem. By plugging in the state-of-the-art algorithm for sparse matrix multiplication by Abboud, Bringmann, Fischer, and Künnemann [SODA'24], we achieve significantly improved bounds such as $O((m_{in} + k)^{1.346})$. Notably, in the regime where $k \geq m_{in}^{1.762}$, our reduction culminates in an almost-optimal $k^{1+o(1)}$-time algorithm.

The Ramanujan challenge

Ido Kaminer shared with me the following information  about The Ramanujan Challenge for AI, and I am happy to share it with the readers of this blog. The challenge page is at  ramanujanmachine.com/ramanujan-challenge; Here is the The full challenge paper, and a quote from Ido’s email.

“The challenge launched today and will run until August 1, 2026. It consists of [ten] research-level problems on explicit formulas for mathematical constants, designed to test whether AI systems can move from a concrete formula to a valid proof or symbolic derivation.

We designed the rules to make the challenge compatible with formal and code-based systems. Accepted submissions may be formal proofs, CAS-based derivations, or human-readable proofs accompanied by reproducible code. The goal is not only to test whether AI can find answers, but whether it can produce derivations that can be checked in a structured way.”

 

The second problem

By Gil Kalai

Scott’s foreword: Cynthia Dwork is Gordon McKay Professor of Computer Science at Harvard, and a pioneer in the fields of differential privacy and algorithmic fairness. On my recent travels to the SigmaWest science camp and then STOC, there was much talk about a recent Trump administration action that would ban not only differential privacy, but essentially all modern techniques for preserving privacy in large datasets, for example in the 2030 US Census. I realize that many of us have “outrage fatigue,” but this particular outrage hits extremely close to home for the CS theory community. So when Cynthia approached me at STOC to propose a guest post on the issue, of course I said yes. The post that she sent me, below, is cosigned by many other leaders in the field.

On June 4, 2026, the U.S. Secretary of Commerce issued a directive (DAO 216-26) relegating confidentiality protection in all Bureau of Economic Analysis (BEA) and U.S. Census Bureau publications to techniques dating back to the early 1970s, turning its back on over half a century of progress and protections for data subjects. Advances in confidentiality provision had enabled the Census Bureau to share increasing quantities of data at more granular detail. The order will result in less useful (or fewer available) statistics, weaker protection, or both. We write to illustrate the danger posed by the order and to mobilize the scientific community to speak out against it.

The acting force behind this order is political interest, not scientific merit. DAO 216-26 bypassed legally required administrative procedures. It fulfills a promise made by the architects of the Heritage Foundation’s Project 2025, and reflects both the rhetoric and misunderstandings of representatives of the Center for Renewing America (CRA), an organization founded by OMB Director Russell Vought. CRA’s explainer on the use of differential privacy in the 2020 Census is up-front about the stakes: “Even if the citizenship question is added to the Census, it will be impossible to ascertain the status of individuals so long as differential privacy is used.” But masking this sort of personal characteristics data is legally required by the Census Act (13 U.S. Code Section 9), which makes it a crime to “make any publication whereby the data furnished by any particular [individual] can be identified.” Confidentiality is also widely understood as critical to ensuring that people respond to the census. 

DAO-216-26 bans differential privacy and other modern (and not so modern) techniques. It restricts disclosure avoidance techniques to “coarsening,” which it describes as “reducing the level of detail or specificity of published statistics, such as through rounding, aggregating (grouping), and/or the use of ranges.” “Suppression” (“expressly redacting certain values”) may also be used, but only as a “last resort.” DAO-216-26 forbids “noise infusion”, described as “methods that involve modifying a dataset by adding random values, or noise.” 

Noise infusion was invented precisely to address the increasing demand for granular data in the face of confidentiality laws that forbid publishing reidentifiable data. Coarsening and suppression were satisfactory for most national, aggregate statistical series, like the Principal Federal Economic Indicators. However, these techniques failed when applied to business and demographic data at fine geographic or industrial detail. By forbidding noise infusion, the directive bans the disclosure avoidance techniques at the core of dozens of data releases over the last three decades. It bans input noise infusion, used in the Quarterly Workforce Indicators since 2002 and, until now, planned for the Bureau of Economic Analysis statistics [1].  It bans swapping, used for decennial census publications since 1990. It also bans differential privacy,  the best currently known approach for obtaining the most data utility for any given level of privacy. Differential privacy was used for sharing data on commuting patterns (OnTheMap) since 2008 and for publications based on the 2020 Census. Until the recent directive, differential privacy was planned for the 2030 Census too.  Many other products and procedures are implicated as well.

1.      Illustrations

DAO-216-26 is incompatible with the Census Bureau’s dual mandate to provide confidentiality and fitness for use. To illustrate this, we recall and expand on an example due to Nathan Goldschlag, inspired by the County Business Patterns (CBP) data, which provides statistics on business activity broken down by industry and geography. Goldschlag describes three scenarios, illustrating the tension between providing useful information and maintaining confidentiality of responses as required by the Census Act. 

·       “There is only one brewery in a small county. If the CBP published the exact count of brewery employees in that county, it would be disclosing the information of one business (how many workers it employs), a clear violation of the law.²

·        “There are two breweries in a small county, and the CBP again publishes the exact count of brewery employees. If I own one of those breweries, I could learn how many employees my competitor has, again violating the law.

·        “There are more than two breweries in a small county, but the CBP chooses not to publish the total number of brewery employees out of concern that it might compromise the privacy of the businesses. If I’m a prospective brewery owner, I may deem the project too risky to pursue without information about the market I’m entering.”

In Goldschlag’s example, coarsening makes the published statistics useless.  We now add a fourth scenario, showing that it also fails to maintain confidentiality.  To keep things simple, assume none of us owns any of the businesses in the new example. The County has two towns with one brewery each, North Bend and South Bend. Furthermore, North Bend has a mobile bottling company and South Bend has a stationary bottling company. That’s a total of four beer-related business entities in the County.  Two of these businesses, the North-Bend brewery and the South Bend bottling company, are publicly-owned.

• The CBP publishes five statistics:
  • (A) The total number of employees in beer-related businesses in North Bend: Because there is only one brewing company in North Bend and only one bottling company in North Bend, the category is coarsened to “beer-related”.
    
  • (B) The total number of employees in beer-related businesses in South Bend: Because there is only one brewing company in South Bend and only one bottling company in South Bend, the category is coarsened to “beer-related”.
    
  • (C) The total number of employees in brewing only:  Because there is only one brewing company in each of North Bend and South Bend, the statistic is coarsened to the total number of employees in brewing only in the County.
    
  • (D) The total number of employees in bottling only: Because there is only one bottling company in each of North Bend and South Bend, the statistic is coarsened to the total number of employees in bottling only in the County.
    
  • (E) The total number of employees at publicly owned companies: Because there is only one publicly owned company in each of North Bend and South Bend, the statistic is coarsened to the total number of employees in publicly owned companies in the County. 

We now have 5 equations in 4 unknowns. Using only 4 of these (A, B, C, and E), we can solve for the exact number of employees at each of the four companies with high school algebra.

In the above (fictional but realistic) scenario, the County Business Patterns were released with good-faith coarsenings for the geographical, business, and ownership categories.  Nonetheless, even without inside knowledge of one of the companies’ number of employees, we can completely reconstruct all four numbers.  What happened?  The coarsenings interacted poorly.  Noise infusion perturbs that set of equations, preventing exact reconstruction.

2.      Impediments to Implementation

The Commerce Department now claims the directive’s return to the outdated “tradstat” traditional statistical techniques of the 70s is good for data consumers: “This update to our disclosure limitation method protects respondents and provides the public with more essential economic information.”  (Emphasis added.)  As we saw from Goldschlag’s example, coarsening does just the opposite.

And it can’t be fixed. Coarsening by definition reduces access to fine-grained information.  Our example of three poorly interacting coarsenings shows that this sacrifice is for naught: without noise infusion, confidentiality is destroyed by elementary calculations.  For population surveys, this is precisely what formal noise infusion methods, like differential privacy, protect against; this is the “fancy math” that Goldschlag mentions in his post and that holds personal characteristics, like citizenship status, in confidence.  

3.      Confidentiality is Critical for Federal Statistics 

The scientific community continues to debate the best techniques for protecting the confidentiality of respondents’ data, but DAO-216-26 is not driven by science. It is driven by political interests. Those issuing this order are willing to risk the public’s trust in the process. We think that this is wrong-headed and dangerous. 

Civil servants will do their best to comply with this order while still following the laws that require them to protect the confidentiality of respondents’ data. To balance these competing mandates, they may seek to produce less data or coarsen data so much that it is unusable. Or they might be pushed by political actors to publish data that can be easily unmasked, like in the brewery examples above. Regardless of their choices, they will be hard-pressed to guarantee respondents’ confidentiality, which will prompt many businesses and individuals to simply not answer. This is devastating for an agency that delivers democracy’s data.

Conclusion

Rather than political actors overruling the government’s own statisticians, we need deep investment in our nation’s statistical agencies, ensuring that agencies have the staff and support to improve their methods using the best available tools. Regardless of how the scientific community feels about any specific privacy-enhancing technique, we must collectively reject this anti-scientific approach to governing federal statistics. Too much is at stake. 

How to Take Action

Share this post with others in your professional network and community.

Contact your Congressional representative and voice your concerns. Calling or writing to your representative is one of the most effective and easiest things a constituent can do that should only take a couple minutes of your time.

1. Find your representative contact information here.
2. State your concern. Here is a sample script: “My name is [Name], and I am a constituent from [City] in your district [ZIP CODE]. I am calling because I am concerned about the the U.S. Secretary of Commerce issued a directive (DAO 216-26) that wants to relegate confidentiality protection in all Bureau of Economic Analysis and U.S. Census Bureau data products and statistics to outdated and ineffective statistical techniques. If followed, this order will destroy the Commerce public data our nation relies on for important decisions, such as where to build necessary services for our community’s well-being. I want the DAO to be rescinded. I want proper administrative procedure to be followed. I want technical decisions such as the choice of method used to balance utility and confidentiality to be informed by professionals in the federal statistical agencies, not made unilaterally by political operatives.”
   1. Optional is stating what kind of constituent, such as a retired teacher or a working professional.

Volunteer to help preserve Census working papers and documentation. Pages explaining “noise infusion” and “differential privacy” are already going offline. Archive relevant methodology pages and technical documentation. You can also do this via the Internet Archive’s Wayback Machine (“Save Page Now”).

John Abowd 
Aloni Cohen
Cynthia Dwork
Jae June Lee
Jayshree Sarathy
Adam Smith
Salil Vadhan

[1] BEA Working Paper WP2026-9, now purged by the Department of Commerce.  As of 6/22 Google returns:

Bureau of Economic Analysis (BEA) (.gov)
https://bea.gov › files › papers › BEA-WP2026-9

By Scott

We show that one-way functions suffice for constructing very efficient argument systems for proving the correctness of bounded-space computations. Taking $\kappa$ to be a cryptographic security parameter and $n$ to be the input length, our argument system applies to general computations running in time $T$ and space $S$. The protocol has ${\mathit{polylog}}(T)$ rounds, the communication complexity is $S\cdot {\mathit{poly}}({\mathit{log}}(T),\kappa)$ and the verifier runtime is $(S + n) \cdot {\mathit{poly}}({\mathit{log}}(T),\kappa)$. In particular, the complexity of the verifier is only poly-logarithmic in the computation length. The honest prover runs in time ${\mathit{poly}}(T,\kappa)$, so the protocol is doubly-efficient. If the one-way function is secure against $\lambda(\kappa)$-size adversaries, then the argument system is sound against cheating provers of size $\lambda(\kappa)/{\mathit{poly}}(T)$. Prior to this work, doubly-efficient argument systems with poly-logarithmic dependence on $T$ required assuming (at least) the existence of collision-resistant hash functions [Kilian, STOC 1992]. For unconditionally sound doubly-efficient protocols, the RRR protocol for bounded-space computations [Reingold, Rothblum and Rothblum, STOC 2016] has communication and verification complexities that grow with $T^{\sigma}$ for an arbitrarily small constant $\sigma > 0$.

Celebrating 100 Years: Avi 70 + CSDM 30
Dates: June 14-18, 2027

The Institute for Advanced Study’s School of Mathematics is pleased to announce a conference celebrating Avi Wigderson’s 70th birthday and retirement, together with 30 years of the Computer Science and Discrete Mathematics program (CSDM) at IAS. The conference will honor both Avi’s extraordinary scientific legacy and his transformative role in shaping the intellectual community around theoretical computer science and discrete mathematics at IAS.

Avi Wigderson has made foundational contributions across an exceptional range of areas in theoretical computer science and mathematics. His work has had a profound impact on complexity theory broadly, and especially on cryptography, circuit and proof complexity, communication complexity, pseudorandomness, and expander graphs; in recent years, he has also opened deep and unexpected connections to invariant theory. These contributions have shaped major research directions and influenced generations of mathematicians and theoretical computer scientists.

Posted on behalf of the organizers, Irit Dinur and Amir Shpilka. Additional information is available at https://www.ias.edu/math/events/celebrating-100-years-avi-70-csdm-30.

By shuchic

Authors: Abrahim Ladha, Yiran Luo, Alan Tian

While the Church-Turing thesis asserts that effective calculability explicates to sets decidable by a Turing machine, the Cobham-Edmonds thesis asserts that feasible computation explicates to the complexity class $\mathsf{P}$, those decidable by a polynomial-time bounded Turing machine. The Church-Turing thesis has been placed under rigorous scrutiny and has several convincing arguments in its favor, but the Cobham-Edmonds thesis has not undergone a similar examination. Many of the arguments in its favor simply suggest that $\mathsf{P}$ is a useful assumption, rather than a necessary target. This paper presents analogous arguments in favor of the Cobham-Edmonds thesis.

Authors: Mark Giesbrecht, Armin Jamshidpey, Éric Schost

Let $\mathsf{E}=\mathbb F_q[x]/(Γ)$ be an algebraic extension of degree $n$ over the finite field $\mathbb F_q$, given by a $Γ\in\mathbb F_q[x]$ monic and irreducible. It is classical that any such $\mathsf{E}$ contains an element $β\in\mathsf{E}$ that is normal over $\mathbb F_q$, i.e., the conjugates $β,β^q,\ldots,β^{q^{n-1}}$ form an $\mathbb F_q$-basis of $\mathsf{E}$. In this paper we give a deterministic algorithm which finds such a normal element using $O_ε((n^2\log q)^{1+ε})+O\,\tilde{}\,(n\log^2 q)$ bit operations, for any $ε>0$. The algorithm works by showing that, for a parameter $t\in\mathbb F_q$, the element $β_t=(θ-t)^{-1}$ is normal except for at most $n(n-1)$ values of $t$. This is established by constructing a "cleared Moore" circulant matrix over $\mathbb F_{q^n}[\mathcal T]$, whose determinant degree at most $n(n-1)$, such that $β_t$ is normal if and only the determinant is non-zero at $t\in\mathbb F_q$. For faster computation over the base field, we replace this by an equivalent trace Gram circulant matrix over $\mathbb F_q[\mathcal T]$. A main algorithmic contribution is a fast determinant algorithm for circulant matrices of polynomials, which uses triangular set projection and modular composition techniques to achieve a near-linear cost. Given an $n\times n$ circulant matrix over $\mathbb F_q[t]$ whose entries have degree at most $m>0$, we show how to compute its determinant deterministically with $O_ε((nm\log q)^{1+ε})$ bit operations. We complete the solution by showing how to extend this to finite fields of size less than $n(n-1)$, through an embedding in a low-degree extension field, at poly-logarithmic additional cost.

Authors: Madhulatha Mandarapu, Sandeep Kunkunuru

Filter-and-refine spatial joins have always avoided touching exact geometry for certified candidate pairs, but the field never modeled the decompression cost of the pairs that survive the filter. When geometry is stored in a compressed, progressively-decodable multiresolution codec, the join's true cost is bytes decoded. We study provably-exact polygon intersection joins over a Douglas-Peucker level-of-detail (LOD) ladder, certified by a two-sided Hausdorff-margin test, and make two contributions. First, a reproducible mechanism and harness: on real U.S. Census TIGER water polygons, our progressive certificate join returns the exact join result while decoding 3.4-16.8x (median 5.9x) fewer vertices than naive decompress-then-refine, and about 4.9x fewer than the single-approximation multi-step baseline of Brinkhoff et al. (1994), with zero correctness violations (set-equality against a full-precision oracle) across 31 workloads. Second, a characterization we call the decode-work law: decode work is governed by each pair's signed-clearance margin -- how close it is to the predicate-flip boundary -- independent of object size, because the certificate descends the ladder only until its resolution beats the margin. The law is clean on controlled geometry (held-out R2=0.87, size-independent) and directional on real data (R2 ~= 0.55). We are explicit about what does not hold: a near-boundary-vertex predictor is the wrong model (we pre-registered one and rejected it), a selectivity regime forecaster did not materialize, and the worst case is the trivial Omega(v) read bound on adversarially interleaved boundaries. We contribute the mechanism, budget-honest decode accounting, and an open harness; we do not claim a new index.

Authors: Erin W. Chambers, Christopher Fillmore, Shankha Shubhra Mukherjee, Rohit Roy, Elizabeth Stephenson, Mathijs Wintraecken

Vineyards, or time-varying families of persistence diagrams, are widely used in topological data analysis (TDA) pipelines to track how topological features change and evolve as a parameter varies. When the parameter traces a closed loop, a vineyard can exhibit monodromy: diagram points permute over the course of a full traversal, which obstructs feature tracking and can complicate downstream analysis of such data. Chambers et al. considered the periodic vineyards that arise from the radial persistence transform, which maps the manifold to a family of persistence diagrams, where each diagram fixes a base point and considers the filtration that is based on Euclidean distance to that point, and showed that monodromy and knotting can occur. Other recent work by Arya et al. considers geometric conditions that exclude monodromy in two dimensions, in an effort to better understand when this effect happens. That said, understanding when and why monodromy occurs is a fundamental open problem with direct practical consequences for many data analysis pipelines. In this work, we study this question for 1-manifolds in $\mathbb{R}^2$, using a surprising connection with tools from singularity theory, and provide a classification for the causes of monodromy in vineyards. More precisely, we prove that the vineyard of a sufficiently small loop $γ$ cannot exhibit monodromy unless it contains a specific singularity of the distance function. The central geometric object in our analysis is the symmetry set, which is the locus of centers of spheres tangent in more than one point to the manifold; this object classifies singularities of the distance function, and in our setting, dictates precisely when monodromy occurs. This characterization opens the door to the development of algorithmic criteria for detecting and utilizing (or avoiding) monodromy in TDA pipelines.

Authors: José Fernández Goycoolea, Andrea de las Heras-Parrilla, Luis H. Herrera, Clemens Huemer, Carlos Seara

Recently, Andrade and Dahl introduced combinatorial Fiedler theory by studying a parameter $b(G)$ defined as the $\ell_1$-analog of the Rayleigh quotient minimization characterization of the algebraic connectivity of a graph $G=(V,E)$. In this work, we study the corresponding maximization problem, which plays the role of the $\ell_1$-analog of the largest Laplacian eigenvalue. We show that the new parameter $B(G)$ associated with this maximization problem admits a simple exact description: it is the average of the two largest vertex degrees of $G$. A unified combinatorial treatment of the minimization and maximization problems is presented first. Later, both optimization problems are reinterpreted in a geometrical setting. The feasible set is identified with a $(n-2)$-dimensional cuboctahedron shell where $n=|V|$. Additional structure is presented for this polyhedron, including the fact that maximizing solutions arise at its vertices and minimizing solutions arise at the centers of its facets. Finally, we analyze the number of optimal vectors for $b(G)$ and $B(G)$ for several graph families. Although the value of $B(G)$ is determined by the two largest degrees, we prove that counting the vectors that attain this value is actually $\#\mathrm{P}$-complete.

Authors: Ahmad Kamaludeen, Somnath Kundu, Yeganeh Bahoo

We study the symmetric bouncing of a point robot within orthogonally-joined rectangles with equal width, which we refer to as pipes. We provide an exhaustive case analysis of every trajectory pattern inside a single rectangular pipe segment, identifying the conditions under which the robot exits. We then extend the analysis to L-shaped pipes and, more generally, to linear chains of $k$ orthogonally connected pipe segments. We prove exit guarantees for the special angle $α= π/4$. Furthermore, these results extend to pipes with curved joints.

Authors: Clément Dallard, Daniël Paulusma, Erik Jan van Leeuwen

The Chromatic Sum problem asks, given a graph $G$ and an integer $k$, whether $G$ admits a colouring $c$ with sum $\sum_{v\in V}c(v) \leq k$. We study the complexity of Chromatic Sum on graph classes defined by some set of forbidden graphs. First, we show that three known frameworks fully classify the complexity of Chromatic Sum on $HH$-minor-free graphs and $HH$-topological-minor-free graphs for any set of graphs $HH$, and on $HH$-subgraph-free graphs for any finite set of graphs $HH$. To show this, we prove a new NP-completeness result for Chromatic Sum on certain subdivisions of planar subcubic graphs. Next, we consider other containment relations. We formalise a novel framework of problems that are NP-complete for planar graphs as well as for graphs of bounded independence number. For every problem in this framework, we obtain an almost complete complexity classification on $H$-induced-minor-free graphs, $H$-induced-topological-minor-free graphs, and $H$-free graphs for every graph $H$. We show that Chromatic Sum belongs to this framework, as do several other problems. We also define a more fine-grained framework for the induced subgraph relation. We apply this to obtain a complete complexity classification for Chromatic Sum on $H$-free graphs, as well as for several other problems. We justify the choice of this framework by proving that Chromatic Sum is NP-complete for graphs of clique-width at most $3$. This result complements a known polynomial-time result for graphs of clique-width at most $2$.

Authors: Édouard Bonnet, Maël Dumas, Julien Duron

For every $\varepsilon > 0$, Max Independent Set admits a polynomial-time $n^\varepsilon$-approximation algorithm on $n$-vertex graphs of effectively bounded twin-width [Bergé et al., STACS '23]. The approximation factor actually obtained is more precisely $n^{O(1/ \log \log n)}$. Prior to the current paper, no approximation hardness was known for this problem, and the existence of a polynomial-time approximation scheme (PTAS) was repeatedly raised as an open question. We answer this question in a strong sense: We show that there is a constant $γ> 0$ such that a polynomial-time $n^{γ/ (\log \log n)^2}$-approximation algorithm for Max Independent Set on graphs of twin-width at most 4 would refute the Exponential-Time Hypothesis (ETH). This lower bound further holds if a 4-sequence is provided as part of the input. We show the same hardness of approximation for Min Coloring, which also has a nearly matching $n^{O(1/ \log \log n)}$-approximation algorithm on graphs of effectively bounded twin-width. We also clarify the parameterized complexity of $k$-Independent Set on graphs of bounded radius-$r$ merge-width when the range of $r$ is limited. There is a fixed-parameter tractable algorithm for $k$-Independent Set on graphs given with radius-$2^{O(k^2)}$ merge sequences of bounded width [Dreier and Toruńczyk, STOC '25]. We complement this result by showing that $k$-Independent Set is W[1]-hard on graphs given with radius-$o(k)$ merge sequences of bounded width. We further show that this result also holds for $k$-Dominating Set.

Authors: Deeparnab Chakrabarty, Hang Liao

We investigate the power of CUT queries to reveal the structure of unknown hypergraphs. While simple graphs allow for optimal $O(n)$-query connectivity algorithms, hypergraphs face a fundamental identifiability barrier in that distinct hypergraphs can share identical cut-profiles, making exact edge learning impossible in general, a primitive crucial in the graph connectivity algorithms. We first present a zero-error randomized algorithm that identifies the connected components of any weighted hypergraph using $O(n)$ expected queries, matching the $Ω(n)$ lower bound. This approach bypasses the reconstruction barrier by introducing the notion of ``independent families'' -- vertex subpartitions that do not share hyperedges -- and iteratively coarsening them using auxiliary weighted graph connectivity techniques [Liao-Chakrabarty, 2024]. Second, we demonstrate that the impossibility of exact learning depends on hyperedge parity. For even-parity hypergraphs, we show that the structure is reconstructible using a Möbius transform on the CUT function to implement binary-search-style vertex identification. This yields deterministic algorithms for obtaining $k$-connectivity certificates for $r$-bounded even hypergraphs in $\tilde{O}_r(kn)$ queries. Finally, we bypass parity and rank constraints for linear hypergraphs, achieving a subquadratic $\tilde{O}(kn^{1.5})$ query complexity for $k$-connectivity. This significantly improves upon the general $\tilde{O}(n^2)$ bound derived via symmetric submodular function minimization.

Authors: Georgios Amanatidis, Giulio Giaconi, Evangelos Markakis, Nicos Protopapas

We study the online fair division of indivisible mixed manna among agents with additive valuation functions. Under the standard online model, at each time step an indivisible item arrives; each agent may assign it a positive, negative, or zero value, and it must be irrevocably allocated, before the arrival of the next item. At the same time, we also wish to maintain some fairness guarantee, and in this work we focus on envy-freeness (EF) and one of its most prominent relaxations, envy-freeness up to one item (EF1). Given the strong negative and the scarce positive results for this problem without additional assumptions, we augment our algorithms with buffers that can store and rearrange a limited number of items. This setting interpolates naturally between the fully online case (no buffer) and the fully offline case (a buffer large enough to hold all items). We show that algorithms equipped with reasonably sized buffers can achieve strong guarantees for personalized $k$-value instances, i.e., instances in which each agent assigns at most $k$ distinct values to items. In particular, we construct allocations that are EF1 at every time step and EF at most time steps, using a buffer of size linear in $k$ and in the number of agents. Our approach relies on novel combinatorial arguments and on constructing a sequence of envy-free matchings that allocates most items. Finally, we extend our results to general additive valuation functions, with a dependence on the largest per-agent ratio between two values of the same sign, and we also identify limitations of our approach via impossibility results on the use of buffers with smaller size.

Authors: Ishay Haviv

In the fair allocation problem under conflict constraints, the goal is to partition the vertices of a graph among agents in a fair manner, such that no two adjacent vertices are assigned to the same agent. We study this problem for agents with common preferences through the lens of three fairness criteria: stochastic-dominance envy-freeness up to one item for preference orders (SD-EF1), envy-freeness up to one item for monotone additive valuations (EF1), and envy-freeness up to one item from each side for general additive valuations (EF[1,1]). To do so, we introduce a hierarchy of variants of the strong chromatic number, a graph quantity introduced independently by Alon and Fellows in the early nineties. Our results reveal a close connection between fair allocation under conflict constraints and the first two levels of this hierarchy, providing a unified route to both existential and algorithmic results. For SD-EF1, we fully characterize the number of agents needed to guarantee a fair allocation of a given graph for every common preference order. For EF1 and EF[1,1], we provide analogous sufficient conditions, extending a result on path graphs due to Equbal, Gurjar, Igarashi, Kumar, Manurangsi, Nath, Saxena, Vaish, and Yoneda. We also show that, unlike in the SD-EF1 setting, the sufficient conditions for EF1 and EF[1,1] are not necessary in general. Our framework yields existential and algorithmic consequences in terms of the maximum degree. We obtain that every graph with maximum degree $Δ$ admits SD-EF1, EF1, and EF[1,1] allocations for common preferences whenever the number of agents is at least $3Δ-1$. We further provide, for any $\varepsilon>0$, deterministic polynomial-time algorithms that find such allocations whenever the number of agents is at least $(3+\varepsilon)Δ$. These guarantees strengthen earlier work by Barman and Viswanathan on equitable colorings.

Authors: Philip Bille, Inge Li Gørtz, Máximo Pérez-López, Simon R. Tarnow

Given a Wheeler NFA $\mathcal{A}$, the Wheeler determinization problem is to construct a Wheeler DFA $\mathcal{D}$ that accepts the same language as $\mathcal{A}$. We use the notation $n_{\mathcal{A}},m_{\mathcal{A}}$ for the number of vertices and edges of $\mathcal{A}$, and equivalently $n_{\mathcal{D}},m_{\mathcal{D}}$ for $\mathcal{D}$. Alanko et al. [SODA 2020, Inf. Comp. 2021] show that we can solve this problem in $O(n_{\mathcal{A}}^3)$ time. In this paper, we show how to improve the running time to $O(n_{\mathcal{A}} + m_{\mathcal{A}} + n_{\mathcal{D}} + m_{\mathcal{D}})$ when given the Wheeler order of $\mathcal{A}$ (which can be computed in $O(m_{\mathcal{A}}\log n_{\mathcal{A}})$ with an algorithm by Becker et al. [ESA 2023]). Our running time is a factor $n_{\mathcal{A}}^2/σ$ faster than the state of the art, where $σ$ is the size of the alphabet. Furthermore, for $σ=O(1)$ we have the first linear time algorithm for this problem. We show that our bound is tight for sorted inputs with any combination of $n$ and $σ$, by giving a family of inputs for which our output $\mathcal{D}$ is minimum, and of maximum size $Θ(nσ)$.

Authors: Dean Kraizberg, Ron Peretz

We study a dynamic averaging process on the cycle \(C_n\). At each discrete time, an edge is chosen uniformly at random, one unit of load is introduced, and the two endpoint loads are replaced by their common average after the new unit has been added. Starting from the zero configuration, we prove that the expected gap between the largest and smallest loads is \(O(\sqrt n)\), uniformly in time. Building on the lower-bound argument of Alistarh, Nadiradze, and Sabour for the expected square of the gap, we further show that the expected gap is \(Ω(\sqrt n)\) in the long run. This confirms their conjecture that the expected gap is of order \(\sqrt n\).

Authors: Avi Kadria, Liam Roditty, Virginia Vassilevska Williams

We study the problem of approximating the length of a shortest cycle in a given graph, known as the girth of the graph. The state-of-the-art approximation algorithms for unweighted graphs by Kadria et al. [SODA'22] and Roditty and Trabelsi [arXiv'25] achieve the following trade-off: for every integer $k\geq 2$, there is an $\tilde{O}(n^{1+2/k})$ time algorithm that achieves a $(2k/3)$-approximation for the girth in unweighted $n$-node graphs. The first result of this paper is to achieve the same trade-off for $m$-edge, $n$-node graphs with non-negative real edge weights: a $2k/3$-approximation algorithm running in $\tilde{O}(m+n^{1+2/k})$ time. The dependence on $m$ is unavoidable in weighted graphs. Our result improves on the work of Kadria et al.~[SODA'23] and Ducoffe [ICALP'19 and SIDMA'21], who were only able to achieve such a trade-off for some values of $k$. We also prove new fine-grained lower bounds for girth approximation and related problems in unweighted graphs.

Authors: Koustav Bhanja, Merav Parter, Asaf Petruschka

We study Single-Source Mincut Sensitivity Oracles: compact data structures that, when queried with an edge e, report those affected vertices whose mincut value to source $s$ changes upon the insertion or failure of e. Insertion queries were treated by Baswana, Gupta, and Knollmann [Algorithmica '22], who showed an extremely compact oracle with only O(n) space. In this work, we consider edge failure queries, which are of even greater interest, but far more challenging. The current-best approaches give O(n^2) space: either using n-1 fixed-pair oracles of O(n) space each, based on the Picard-Queyranne representation [MPS '80], or using the O(n^2) space all-pairs oracle by Baswana and Pandey [SODA '22]. -Our key result is an optimal O(n) space single-source mincut sensitivity oracle for edge failure queries. It reports the set of affected vertices in O(n) time, thus matching the state-of-the-art bounds for the insertion case. -Additionally, we provide oracles with near-optimal query times at the cost of increasing the space to O(n^{1.5}). They can determine if any given vertex is affected by an insertion/failure of an edge in O(log n) time, or reports all affected vertices in amortized O(\log^3 n) time per vertex. Such oracles of subquadratic space were previously unknown, even for insertion. Our main technical contribution is in establishing novel and intricate connections between two seemingly distant objects, representing two different families of mincuts. The first is the DAG representation of farthest mincuts to the source, which was the central tool introduced by Baswana, Gupta, and Knollmann. The second is the Connectivity Carcass for Steiner mincuts of Dinitz and Vainshtein [STOC '94], which generalizes well-known cactus representations of global mincuts. Our work demonstrates the relatively unexplored potential of the carcass beyond its obvious Steiner mincuts scope.

Authors: Bennet Edler, Klaus Jansen, Felix Ohnesorge, Lis Pirotton

In the problem of Parallel Task Scheduling (PTS), we are asked to schedule $n$ jobs, each with a fixed processing time and machine requirement, such that the completion time of the last job is minimized. Jansen and Rau (2019) presented an algorithm for PTS that achieves an approximation ratio of $(3/2)\text{OPT} + p_{\max}$. They additionally posed the open question whether an approximation ratio of $(4/3)\text{OPT} + p_{\max}$ is possible. In this work, we present such an algorithm with a running time of $O(n\log n)$. The problem of Multiple Cluster Scheduling (MCS) is a natural extension of PTS where we are given $N$ clusters each of $m$ machines to schedule jobs. Jansen and Rau (2019) adapted their PTS algorithm to MCS with the following results: (1) a 2 approximation, and (2) a near-linear 9/4 approximation if $N$ is divisible by 3. We improve the running time of their 2-approximation and generalize the 9/4 approximation to the general case. The 2-approximation for MCS is tight, since one cannot hope for an approximation ratio better than 2, unless P=NP [Zhuk, 2006]. In addition to our theoretical results, we implement our algorithm and show its practical applicability.

Authors: Adam Polak, Jonas Schmidt

One of the three key hypotheses of fine-grained complexity asserts that computing All-Pairs Shortest Paths (APSP) requires cubic time, up to subpolynomial factors, in the worst case. We initiate the study of APSP in the paradigm of algorithms with predictions, also known as learning-augmented algorithms. We propose an APSP algorithm that takes as additional input a \emph{prediction} (e.g., given by a model learned from similar instances seen in the past) consisting of sets of vertices causing the shortest \emph{detour} for each pair of vertices. The algorithm runs in time $\mathcal{O}(n^{2.83} + ηn)$, where $η$ denotes the \emph{prediction error} defined as the number of pairs of vertices for which, informally speaking, the prediction was not sufficient to compute and certify optimality of the shortest path length. This is already subcubic when the prediction error is (polynomially) smaller than its maximum possible values $n^2$, i.e., whenever the prediction is at least slightly better than terrible. We build on the co-nondeterministic algorithm for the Exact Triangle problem by Chan, Vassilevska Williams, and Xu (STOC 2023), essentially enabling this algorithm to detect mistakes in the nondeterministic certificate and recover from them. Our result constitutes the first necessary step towards designing learning-augmented algorithms for problems with known fine-grained lower bounds conditioned on the APSP Hypothesis.

Authors: Neta Singer, Theophile Thiery

Given a monotone submodular function, we consider the problem of finding a maximum-valued set in the intersection of $k$ matroids. Our main result is a polynomial time local search based algorithm achieving a $\frac{k}{2} + o(k)$ approximation guarantee. This asymptotically matches the best-known guarantee of $\frac{k}{2} + ε$ in the unweighted setting by Lee, Sviridenko, and Vondrák (2009). Prior to this work, the state-of-the-art was a $\frac{\ln(4)k}{1+\ln(2)} + o(k)$-approximation algorithm obtained by Feldman and Ward (2026). Our approach extends to Matroid $k$-Parity yielding the same approximation guarantee. In contrast to the weight bucketing approach underlying the recent advances of Singer and Thiery (2025) and Feldman and Ward (2026), our algorithm processes elements greedily in decreasing order of marginal value and searches for sufficiently profitable swaps, whose gain exceeds a parameter $α$ given as a function of $k$. We further combine this idea with the weight bucketing approach to obtain improved guarantees for weighted $k$-Set Packing. Our second main result is a $\frac{\ln(4)k}{3} + o(k)$-approximation algorithm for weighted $k$-Set Packing, improving on the state of the art $\frac{k}{2.00561} + O(1)$-approximation by Neuwohner (2023).

Authors: Yu Yao, Huanjian Zhou, Andi Han, Wei Huang, Masashi Sugiyama

Discrete diffusion models are widely used for learning and generating discrete distributions. As the generation process is inherently sequential, the acceleration of sampling is of significant importance. In this work, we parallelize the mainstream $τ$-leaping algorithm for absorbing discrete diffusion in a Continuous-Time Markov Chain (CTMC) framework. By leveraging the continuous-time stochastic integral form of the $τ$-leaping algorithm and the Picard iteration method, we achieve parallel-in-time sampling acceleration and provide a proof of exponential-factorial convergence for our algorithm. We improve the overall time complexity of $τ$-leaping under absorbing settings from ${\mathcal{O}}(d \log S)$ to ${\mathcal{O}}(\log (d\log S)\cdot \log d)$ with respect to NFE. Empirically, our method shows consistent acceleration across synthetic and real-data settings. The new sampler achieves at most $7$--$9\times$ runtime speedup for synthetic distribution, and maintains the same quality with $50\%$ fewer NFE and $1.45$--$1.86\times$ runtime speedups in image/text tasks on a single GPU. Our research expands the potential of discrete diffusion models for efficient parallel inference, with broader implications for applications such as molecular structure and language generation.

Authors: Hiroki Shibata, Haruki Umezaki, Takuya Mieno, Yuto Nakashima, Shunsuke Inenaga

A non-empty string is closed if its length is one or its longest border appears exactly twice in the string. An occurrence of a closed substring is a maximal closed substring (MCS) if it cannot be extended to the left or to the right while preserving closedness. MCSs can be regarded as a general class of maximal repetitive structures including runs. In this paper, we study the computation of MCSs of a string given in an online manner, where one character is appended to the string at a time. Our algorithm detects newly formed MCSs after each append operation by using the rightmost previous occurrences of suffixes. To support this efficiently, we introduce the link-cut suffix tree (LCST), a novel data structure combining an online suffix tree with a link-cut tree. The LCST maintains rightmost occurrence information for substrings represented in the suffix tree in $O(n \log n)$ total time and $O(n)$ space, where $n$ is the length of the input string. Using the LCST, we obtain an $O(n \log n)$-time online algorithm for computing all MCSs, which is worst-case optimal. As further direct applications of the LCST, we obtain online algorithms for rightmost LZ77 factorizations and most recent match queries.

Authors: Junhao Gan, Xiao Sun, Seeun William Umboh

We study the online min-cost perfect matching with delay (MPMD) problem where $m$ requests arrive in a metric space of $n$ points. In MPMD, an algorithm can choose to match a request or to delay, and the objective is to minimise the sum of connection and delay costs. The connection cost of a match is the distance between the locations of two matched requests in the metric, and the increase of the delay cost is a function of the set of unmatched requests at every moment. In this paper, we study two different types of delay functions, size-based (MPMD-Size) and convex delays (MPMD-Convex). The study of MPMD-Size was initiated by Deryckere and Umboh (APPROX/RANDOM 2023) where the instantaneous delay increment is a non-negative monotone function of the number of unmatched requests. Our bounds are in terms of $n$, as opposed to Deryckere and Umboh's bounds that depend on $m$. Our results settle the deterministic competitive ratio (up to constants). At the heart of these results is a succinct encoding scheme of MPMD-Size on a given $n$-point metric as a metrical task system problem on a $2^{n-1}$-point metric. We also consider MPMD-Convex proposed by Liu et al. (ISAAC 2018) where the delay cost incurred by each request is a uniform convex delay function of the time difference between its arrival time and the moment that it is matched by the algorithm. They focused on delay functions $f$ that are unbounded, non-decreasing, continuous, and satisfy $f(0)=f'(0)=0$, and showed that the deterministic competitive ratio is $Ω(n)$ for $n$-point uniform metrics. We show that, surprisingly, when $f$ is a non-negative, monotone polynomial with $f'(0)>0$, there is an $O(1)$-competitive deterministic algorithm for uniform metrics. Our result completes our understanding of MPMD-Convex on uniform metrics for a broad class of functions.

Authors: Toshiharu Minematsu, Shunsuke Inenaga

We study longest-common-extension (LCE) queries and lexicographic minimizer maintenance on the suffix tree of a sliding window. The main difficulty is that a sliding suffix tree is maintained in an implicit Ukkonen-style form: some suffixes of the current window are not represented by leaves. We show that the longest implicit (i.e. non-leaf) suffix induces a periodic representative map that folds every implicit suffix to an explicit suffix leaf in constant time. Combined with leaf pointers [Leonard et al., PSC 2026] and a dynamic LCA data structure [Cole & Hariharan, SICOMP 2005], this yields a linear-space data structure with amortized constant-time window shifts and worst-case constant-time LCE queries over a constant-size alphabet. For minimizers, the LCE structure gives a direct exact solution, but it uses more machinery than fixed-depth comparisons require. We therefore give an alternative LCE-free algorithm that reports minimizers in constant time per window shift, which is built on BP-linked suffix trees [Sumiyoshi et al, SPIRE 2024] and a standard order maintenance data structure (e.g. [Bender et al., ESA 2002]).

Authors: Mikhail Fadin, Huseyin Topaloglu

One of the traditional approaches for constructing approximate policies for dynamic assortment optimization problems is to use sampling-based inventory-agnostic policies. Such policies are called sampling-based, as they sample an assortment of products from a fixed distribution at each time period to offer to a customer of each type. Such policies are called inventory-agnostic, as the sampled assortments may include products without remaining inventories, so if a customer chooses a product without remaining inventories, then she leaves without a purchase. Inventory-agnostic nature of a policy is not a concern, because it is known that if the policy samples an assortment that includes products without remaining inventories, then dropping the products without remaining inventories does not degrade the performance. However, sampling-based nature of a policy is a concern, because sampling brings another source of uncertainty in the performance. In this paper, we give an algorithm to de-randomize any sampling-based inventory-agnostic policy, so the de-randomized policy offers a deterministic sequence of assortments within the support of the original policy without degrading the performance. Furthermore, we give a variation of our de-randomization algorithm that searches for a deterministic sequence of assortments beyond the support of the original policy. We show that we can implement the latter variation efficiently as long as we can solve the static assortment optimization problem under the choice model governing the choice process of the customers. As our crowning technical contribution, we study locally-optimal deterministic policies, where changing any single one of the assortments in the policy does not improve the total expected revenue. We show that any locally-optimal policy has a performance guarantee of 1/2 - epsilon when compared with the best sampling-based policy.