Theory of Computing Report

The Faculty of Science at the University of Calgary invites applications for a Tenure-Track Assistant Professor in Computer Science position with a focus on Theoretical Foundations of Computer Science.

Website: https://careers.ucalgary.ca/jobs/17169318-tenure-track-assistant-professor-research-in-computer-science-faculty-of-science
Email: cpsc.hiring@ucalgary.ca

By shacharlovett

This is a live blog of Lecture 3 of my graduate seminar “Feedback, Learning, and Adaptation.” A table of contents is here.

I could teach an entire semester-long graduate class on gradient descent. First, I’d present gradient descent. Then I’d move to accelerated gradient descent. Then I could teach stochastic gradient descent, coordinate descent, projected gradient descent, proximal gradient descent… This would get us to Spring break. We could wrap up the semester with other assorted gradient potpourri. Indeed, Steve Wright and I packaged this course into a textbook: Optimization for Data Analysis.

Steve and I were inspired by the thousands of machine learning and optimization papers of the 2010s that made minor extensions in this gradient zoo. All of these papers proved their methods worked in the same way. They set up a Lyapunov function. They showed that it decreased as the algorithm evolved. QED.

Those Lyapunov functions were sometimes easy to come by. You’d always start with the function value itself. If it significantly decreased every iteration, then the algorithm would converge. You could also study the distance of the current iterate to the optimal solution. It took me a decade of beating my head against Nesterov’s inscrutable estimate sequences to realize that he was actually using a Lyapunov function too. In Nesterov’s accelerated method, this Lyapunov function has the form:

Showing functions like this were Lyapunov functions required pages of calculations, but all of these were manipulating exactly two inequalities. The most important assumption when analyzing gradient descent is that the gradients are Lipschitz. This means that the slope of the gradient function is bounded. Oftentimes, we also assume that the functions are strongly convex. This is equivalent to assuming the slope of the gradient function is bounded below.

Together, we had that the following two inequalities were true for any x and z.

Here, L is the Lipschitz constant of the gradient. m is the strong convexity parameter. Sometimes we’d use the second inequality with m=0. You might call those functions weakly convex. Convergence proofs cleverly sum up these two inequalities evaluated at different points in space to show that some Lyapunov function decreases. After enough Tetris-like puzzling, you surely prove that the Lyapunov function decreases.

These analyses appear to be assessing the stability of a dynamical system. That’s because they are. Gradient methods control a nonlinear system that takes a vector as input and outputs the gradient of a convex function evaluated at that vector.

The algorithm feeds “x” into the black box. The black box spits out “g.” The algorithm does some thinking and spits out another x. Eventually, the g emitted by the black box is always equal to zero.

In fact, all of the gradient-based algorithms are equivalent to PID controllers. Gradient descent is literally an integral controller. It is even after the same goals: gradient descent wants to find a point where the derivative is zero. Integral control seeks zero steady-state error. Accelerated methods are PID controllers. Projected gradient is a PI controller.

What if we just relabel that picture to align with control theory notation:

The slope bound assumption on the gradients is equivalent to assuming the black box has gain bounded between an upper and lower bound. This is the sort of thing control theorists have studied for a century. They call such nonlinear functions “sector bounded” and have a variety of tools to verify control algorithms when such uncertain nonlinearities are in the feedback loop.

In the paper “Analysis of Optimization Algorithms by Integral Quadratic Constraints,” Laurent Lessard, Andy Packard, and I translated these techniques to optimization algorithms. This let us search for Lyapunov-like proofs that your algorithm converges. With these tools, we could derive the same convergence rates and get novel robustness arguments. And the analyses were automatable, in the sense that we derived our proofs using other optimization algorithms.

A complementary approach to this problem developed by optimizers is the PEP framework, which uses a language more native to optimization. Both proof systems work because positive linear combinations of positive things are positive. Thus, you try to show a Lyapunov function decreases by writing this statement as an equality, and showing it’s the linear combination of a bunch of these inequalities. The computer can do this for you.

Lots of interesting insights come from this parallel between optimization and control. For instance, it shows there is no way to “fix” simple momentum methods like the Heavy Ball Method to prove they converge globally. The automated proof framework also helped us identify perturbations to the methods that could sometimes yield faster convergence rates or greater numerical stability.

But one thing I haven’t been able to do is turn this around: I want to teach a semester-long graduate course on PID control that would feel like my optimization class. I was hoping to get a start during this graduate seminar. I wanted to make it clear that most of the analysis could be done by Lyapunov functions. I wanted to move beyond sector-bounded nonlinear maps to more common dynamical system models in which people apply PID controllers. And I want to do all of this without ever taking a Laplace transform.

If there are any control theorists out there reading this with ideas for putting such a course together, please let me know! I know many would argue that PID controllers are solved, and the interesting stuff happens at a higher level. But to push the limits of what modern learning-control systems do, we have to understand the PID controls at the innermost loops of the complex system. Explaining this part of the architecture in a clean, modern way is a good pedagogical challenge for my control theory friends.

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By Ben Recht

The Algorithms group at IISc Bengaluru invites posdoc applications. Areas include Approximation/Online Algorithms, Game Theory, Computational Geometry, Optimization, Learning, and more. Fellowship: ₹80,000–1,30,000/month + travel/research grant.

Faculty: Siddharth Barman, Arindam Khan, Anand Louis, Rahul Saladi.

𝗔𝗽𝗽𝗹𝗶𝗰𝗮𝘁𝗶𝗼𝗻 𝗟𝗶𝗻𝗸: https://forms.gle/moz2vx7tiNFCbXmS6

Website: https://www.linkedin.com/posts/arindam-khan-445ab615_postdoc-jobs-algorithms-activity-7427285529886580736-6tUk?utm_source=share&utm_medium=member_desktop&rcm=ACoAAAMwLfUBJutL4b0gGJNLzdNG9Zwai-rt7_M
Email: ARINDAMKHAN@IISC.AC.IN

By shacharlovett

Authors: Ismael Rodriguez

We identify a few conditions $X$ such that $(P=NP \wedge X) \;\Rightarrow\; P=PSPACE$.

Authors: Gil Cohen, Dean Doron, Noam Goldgraber

Pseudorandom generators (PRGs) for low-degree polynomials are a central object in pseudorandomness, with applications to circuit lower bounds and derandomization. Viola's celebrated construction gives a PRG over the binary field, but with seed length exponential in the degree $d$. This exponential dependence can be avoided over sufficiently large fields. In particular, Dwivedi, Guo, and Volk constructed PRGs with optimal seed length over fields of size exponential in $d$. The latter builds on the framework of Derksen and Viola, who obtained optimal-seed constructions over fields of size polynomial in $d$, although growing with the number of variables $n$. In this work, we construct the first PRG with optimal seed length for degree-$d$ polynomials over fields of polynomial size, specifically $q \approx d^4$, assuming sufficiently large characteristic. Our construction follows the framework of prior work and reduces the required field size by replacing the hitting-set generator used in previous constructions with a new pseudorandom object. We also observe a threshold phenomenon in the field-size dependence. Specifically, we prove that constructing PRGs over fields of sublinear size, for example $q = d^{0.99}$ where $q$ is a power of two, would already yield PRGs for the binary field with comparable seed length via our reduction, provided that the construction imposes no restriction on the characteristic. While a breakdown of existing techniques has been noted before, we prove that this phenomenon is inherent to the problem itself, irrespective of the technique used.

Authors: Alexey Barsukov, Santiago Guzmán-Pro

We present a structural classification of constraint satisfaction problems (CSP) described by reflexive complete $2$-edge-coloured graphs. In particular, this classification extends the structural dichotomy for graph homomorphism problems known as the Hell--Nešetřil theorem (1990). Our classification is also efficient: we can check in polynomial time whether the CSP of a reflexive complete $2$-edge-coloured graph is in P or NP-complete, whereas for arbitrary $2$-edge-coloured graphs, this task is NP-complete. We then apply our main result in the context of matrix partition problems and sandwich problems. Firstly, we obtain one of the few algorithmic solutions to general classes of matrix partition problems. And secondly, we present a P vs. NP-complete classification of sandwich problems for matrix partitions.

Authors: Joseph Dorfer

We study the reconfiguration of odd matchings of combinatorial graphs. Odd matchings are matchings that cover all but one vertex of a graph. A reconfiguration step, or flip, is an operation that matches the isolated vertex and, consequently, isolates another vertex. The flip graph of odd matchings is a graph that has all odd matchings of a graph as vertices and an edge between two vertices if their corresponding matchings can be transformed into one another via a single flip. We show that computing the diameter of the flip graph of odd matchings is $Π_2^p$-hard. This complements a recent result by Wulf [FOCS25] that it is~$Π_2^p$-hard to compute the diameter of the flip graph of perfect matchings where a flip swaps matching edges along a single cycle of unbounded size. Further, we show that computing the radius of the flip graph of odd matchings is $Σ_3^p$-hard. The respective decision problems for the diameter and the radius are also complete in the respective level of the polynomial hierarchy. This shows that computing the radius of the flip graph of odd matchings is provably harder than computing its diameter, unless the polynomial hierarchy collapses. Finally, we reduce set cover to the problem of finding shortest flip sequences. As a consequence, we show $\log$-\APX-hardness and that the problem cannot be approximated by a sublogarithmic factor. By doing so, we answer a question asked by Aichholzer, Brenner, Dorfer, Hoang, Perz, Rieck, and Verciani [GD25].

Authors: John Bostanci, Andrew Huang, Vinod Vaikuntanathan

We show an unconditional classical oracle separation between the class of languages that can be verified using a quantum proof ($\mathsf{QMA}$) and the class of languages that can be verified with a classical proof ($\mathsf{QCMA}$). Compared to the recent work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), our proof is conceptually and technically simpler, and readily extends to other oracle separations. In particular, our techniques yield the first unconditional classical oracle separation between the class of languages that can be decided with quantum advice ($\mathsf{BQP}/\mathsf{qpoly}$) and the class of languages that can be decided with classical advice ($\mathsf{BQP}/\mathsf{poly}$), improving on the quantum oracle separation of Aaronson and Kuperberg (CCC 2007) and the classically-accessible classical oracle separation of Li, Liu, Pelecanos and Yamakawa (ITCS 2024). Our oracles are based on the code intersection problem introduced by Yamakawa and Zhandry (FOCS 2022), combined with codes that have extremely good list-recovery properties.

Authors: Lijie Chen, Jiatu Li, Igor C. Oliveira, Ryan Williams

In this work, we propose a new bounded arithmetic theory, denoted $APX_1$, designed to formalize a broad class of probabilistic arguments commonly used in theoretical computer science. Under plausible assumptions, $APX_1$ is strictly weaker than previously proposed frameworks, such as the theory $APC_1$ introduced in the seminal work of Jerabek (2007). From a computational standpoint, $APX_1$ is closely tied to approximate counting and to the central question in derandomization, the prBPP versus prP problem, whereas $APC_1$ is linked to the dual weak pigeonhole principle and to the existence of Boolean functions with exponential circuit complexity. A key motivation for introducing $APX_1$ is that its weaker axioms expose finer proof-theoretic structure, making it a natural setting for several lines of research, including unprovability of complexity conjectures and reverse mathematics of randomized lower bounds. In particular, the framework we develop for $APX_1$ enables the formulation of precise questions concerning the provability of prBPP=prP in deterministic feasible mathematics. Since the (un)provability of P versus NP in bounded arithmetic has long served as a central theme in the field, we expect this line of investigation to be of particular interest. Our technical contributions include developing a comprehensive foundation for probabilistic reasoning from weaker axioms, formalizing non-trivial results from theoretical computer science in $APX_1$, and establishing a tailored witnessing theorem for its provably total TFNP problems. As a byproduct of our analysis of the minimal proof-theoretic strength required to formalize statements arising in theoretical computer science, we resolve an open problem regarding the provability of $AC^0$ lower bounds in $PV_1$, which was considered in earlier works by Razborov (1995), Krajicek (1995), and Muller and Pich (2020).

Authors: Yang P. Liu, Shachar Lovett, Kunal Mittal

We prove that for any 3-player game $\mathcal G$, whose query distribution has the same support as the GHZ game (i.e., all $x,y,z\in \{0,1\}$ satisfying $x+y+z=0\pmod{2}$), the value of the $n$-fold parallel repetition of $\mathcal G$ decays exponentially fast: \[ \text{val}(\mathcal G^{\otimes n}) \leq \exp(-n^c)\] for all sufficiently large $n$, where $c>0$ is an absolute constant. We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant $ε>0$, the probability that the players win at least a $\left(\frac{3}{4}+ε\right)$ fraction of the $n$ coordinates is at most $\exp(-n^c)$, where $c=c(ε)>0$ is a constant. In both settings, our work exponentially improves upon the previous best known bounds which were only polynomially small, i.e., of the order $n^{-Ω(1)}$. Our key technical tool is the notion of \emph{algebraic spreadness} adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.

Authors: Alexander Firbas

A graph is geometric 1-planar if it admits a straight-line drawing where each edge is crossed at most once. We provide the first systematic study of the parameterized complexity of recognizing geometric 1-planar graphs. By substantially extending a technique of Bannister, Cabello, and Eppstein, combined with Thomassen's characterization of 1-planar embeddings that can be straightened, we show that the problem is fixed-parameter tractable when parameterized by treedepth. Furthermore, we obtain a kernel for Geometric 1-Planarity parameterized by the feedback edge number $\ell$. As a by-product, we improve the best known kernel size of $O((3\ell)!)$ for 1-Planarity and $k$-Planarity under the same parameterization to $O(\ell \cdot 8^{\ell})$. Our approach naturally extends to Geometric $k$-Planarity, yielding a kernelization under the same parameterization, albeit with a larger kernel. Complementing these results, we provide matching lower bounds: Geometric 1-Planarity remains \NP-complete even for graphs of bounded pathwidth, bounded feedback vertex number, and bounded bandwidth.

Authors: Lotte Blank

Given two polygonal curves $P$ and $Q$ defined by $n$ and $m$ vertices with $m\leq n$, we show that the discrete Fréchet distance in 1D cannot be approximated within a factor of $2-\varepsilon$ in $\mathcal{O}((nm)^{1-δ})$ time for any $\varepsilon, δ>0$ unless OVH fails. Using a similar construction, we extend this bound for curves in 2D under the continuous or discrete Fréchet distance and increase the approximation factor to $1+\sqrt{2}-\varepsilon$ (resp. $3-\varepsilon$) if the curves lie in the Euclidean space (resp. in the $L_\infty$-space). This strengthens the lower bound by Buchin, Ophelders, and Speckmann to the case where $m=n^α$ for $α\in(0,1)$ and increases the approximation factor of $1.001$ by Bringmann. For the discrete Fréchet distance in 1D, we provide an approximation algorithm with optimal approximation factor and almost optimal running time. Further, for curves in any dimension embedded in any $L_p$ space, we present a $(3+\varepsilon)$-approximation algorithm for the continuous and discrete Fréchet distance using $\mathcal{O}((n+m^2)\log n)$ time, which almost matches the approximation factor of the lower bound for the $L_\infty$ metric.

Authors: Zhengding Hu, Jingwen Sun, Le Jiang, Yuhao Wang, Junqing Lin, Yi Zong, Guangzhong Sun

As one of the most fundamental problems in graph processing, the Single-Source Shortest Path (SSSP) problem plays a critical role in numerous application scenarios. However, existing GPU-based solutions remain inefficient, as they typically rely on a single, fixed queue design that incurs severe synchronization overhead, high memory latency, and poor adaptivity to diverse inputs. To address these inefficiencies, we propose MultiLevelMultiQueue (MLMQ), a novel data structure that distributes multiple queues across the GPU's multi-level parallelism and memory hierarchy. To realize MLMQ, we introduce a cache-like collaboration mechanism for efficient inter-queue coordination, and develop a modular queue design based on unified Read and Write primitives. Within this framework, we expand the optimization space by designing a set of GPU-friendly queues, composing them across multiple levels, and further providing an input-adaptive MLMQ configuration scheme. Our MLMQ design achieves average speedups of 1.87x to 17.13x over state-of-the-art implementations. Our code is open-sourced at https://github.com/Leo9660/MLMQ.git.

Authors: Diptarka Chakraborty, Rudrayan Kundu, Nidhi Purohit, Aravinda Kanchana Ruwanpathirana

Given a set of strings over a specified alphabet, identifying a median or consensus string that minimizes the total distance to all input strings is a fundamental data aggregation problem. When the Hamming distance is considered as the underlying metric, this problem has extensive applications, ranging from bioinformatics to pattern recognition. However, modern applications often require the generation of multiple (near-)optimal yet diverse median strings to enhance flexibility and robustness in decision-making. In this study, we address this need by focusing on two prominent diversity measures: sum dispersion and min dispersion. We first introduce an exact algorithm for the diameter variant of the problem, which identifies pairs of near-optimal medians that are maximally diverse. Subsequently, we propose a $(1-ε)$-approximation algorithm (for any $ε>0$) for sum dispersion, as well as a bi-criteria approximation algorithm for the more challenging min dispersion case, allowing the generation of multiple (more than two) diverse near-optimal Hamming medians. Our approach primarily leverages structural insights into the Hamming median space and also draws on techniques from error-correcting code construction to establish these results.

Authors: T. R. Avila, Lars Rohwedder, Leo Wennmann

Recent concurrent work by Dupré la Tour and Fujii and by Hollender, Manurangsi, Meka, and Suksompong [ITCS'26] introduced a generalization of classical discrepancy theory to non-additive functions, motivated by applications in fair division. As many classical techniques from discrepancy theory seem to fail in this setting, including linear algebraic methods like the Beck-Fiala Theorem [Discrete Appl. Math '81], it remains widely open whether comparable non-additive bounds can be achieved. Towards a better understanding of non-additive discrepancy, we study coverage functions in a sparse setting comparable to the classical Beck-Fiala Theorem. Our setting generalizes the additive Beck-Fiala setting, rank functions of partition matroids, and edge coverage in graphs. More precisely, assuming each of the $n$ items covers only $t$ elements across all functions, we prove a constructive discrepancy bound that is polynomial in $t$, the number of colors $k$, and $\log n$.

Authors: Yaniv Sadeh, Haim Kaplan

We study the maintenance of a $(Δ+C)$-edge-coloring ($C\ge 1$) in a fully dynamic graph $G$ with maximum degree $Δ$. We focus on minimizing \emph{recourse} which equals the number of recolored edges per edge updates. We present a new technique based on an object which we call a \emph{shift-tree}. This object tracks multiple possible recolorings of $G$ and enables us to maintain a proper coloring with small recourse in polynomial time. We shift colors over a path of edges, but unlike many other algorithms, we do not use \emph{fans} and \emph{alternating bicolored paths}. We combine the shift-tree with additional techniques to obtain an algorithm with a \emph{tight} recourse of $O\big( \frac{\log n}{\log \frac{Δ+C}{Δ-C}}\big)$ for all $C \ge 0.62Δ$ where $Δ-C = O(n^{1-δ})$. Our algorithm is the first deterministic algorithm to establish tight bounds for large palettes, and the first to do so when $Δ-C=o(Δ)$. This result settles the theoretical complexity of the recourse for large palettes. Furthermore, we believe that viewing the possible shifts as a tree can lead to similar tree-based techniques that extend to lower values of $C$, and to improved update times. A second application is to graphs with low arboricity $α$. Previous works [BCPS24, CRV24] achieve $O(ε^{-1}\log n)$ recourse per update with $C\ge (4+ε)α$, and we improve by achieving the same recourse while only requiring $C \ge (2+ε)α- 1$. This result is $Δ$-adaptive, i.e., it uses $Δ_t+C$ colors where $Δ_t$ is the current maximum degree. Trying to understand the limitations of our technique, and shift-based algorithms in general, we show a separation between the recourse achievable by algorithms that only shift colors along a path, and more general algorithms such as ones using the Nibbling Method [BGW21, BCPS24].

Authors: Shinsaku Sakaue, Yuichi Yoshida

We study online learning in the random-order model, where the multiset of loss functions is chosen adversarially but revealed in a uniformly random order. Building on the batch-to-online conversion by Dong and Yoshida (2023), we show that if an offline algorithm admits a $(1+\varepsilon)$-approximation guarantee and the effect of $\varepsilon$ on its average sensitivity is characterized by a function $\varphi(\varepsilon)$, then an adaptive choice of $\varepsilon$ yields a small-loss regret bound of $\tilde O(\varphi^{\star}(\mathrm{OPT}_T))$, where $\varphi^{\star}$ is the concave conjugate of $\varphi$, $\mathrm{OPT}_T$ is the offline optimum over $T$ rounds, and $\tilde O$ hides polylogarithmic factors in $T$. Our method requires no regularity assumptions on loss functions, such as smoothness, and can be viewed as a generalization of the AdaGrad-style tuning applied to the approximation parameter $\varepsilon$. Our result recovers and strengthens the $(1+\varepsilon)$-approximate regret bounds of Dong and Yoshida (2023) and yields small-loss regret bounds for online $k$-means clustering, low-rank approximation, and regression. We further apply our framework to online submodular function minimization using $(1\pm\varepsilon)$-cut sparsifiers of submodular hypergraphs, obtaining a small-loss regret bound of $\tilde O(n^{3/4}(1 + \mathrm{OPT}_T^{3/4}))$, where $n$ is the ground-set size. Our approach sheds light on the power of sparsification and related techniques in establishing small-loss regret bounds in the random-order model.

Authors: Prathamesh Dharangutte, Jingcheng Liu, Pasin Manurangsi, Akbar Rafiey, Phanu Vajanopath, Zongrui Zou

We study approximation algorithms for Maximum Constraint Satisfaction Problems (Max-CSPs) under differential privacy (DP) where the constraints are considered sensitive data. Information-theoretically, we aim to classify the best approximation ratios possible for a given privacy budget $\varepsilon$. In the high-privacy regime ($\varepsilon \ll 1$), we show that any $\varepsilon$-DP algorithm cannot beat a random assignment by more than $O(\varepsilon)$ in the approximation ratio. We devise a polynomial-time algorithm which matches this barrier under the assumptions that the instances are bounded-degree and triangle-free. Finally, we show that one or both of these assumptions can be removed for specific CSPs--such as Max-Cut or Max $k$-XOR--albeit at the cost of computational efficiency.

Pseudorandom generators (PRGs) for low-degree polynomials are a central object in pseudorandomness, with applications to circuit lower bounds and derandomization. Viola’s celebrated construction (CC 2009) gives a PRG over the binary field, but with seed length exponential in the degree $d$. This exponential dependence can be avoided over sufficiently large fields. In particular, Dwivedi, Guo, and Volk (RANDOM 2024) constructed PRGs with optimal seed length over fields of size exponential in $d$. The latter builds on the framework of Derksen and Viola (FOCS 2022), who obtained optimal-seed constructions over fields of size polynomial in $d$, although growing with the number of variables $n$. In this work, we construct the first PRG with optimal seed length for degree-$d$ polynomials over fields of polynomial size, specifically $q \approx d^4$, assuming, as in [DGV], sufficiently large characteristic. Our construction follows the framework of [DV, DGV] and reduces the required field size by replacing the hitting-set generator used in prior work with a new pseudorandom object. We also observe a threshold phenomenon in the field-size dependence. Specifically, we prove that constructing PRGs over fields of sublinear size, for example $q = d^{0.99}$ where $q$ is a power of two, would already yield PRGs for the binary field with comparable seed length via our reduction, provided that the construction imposes no restriction on the characteristic. While a breakdown of existing techniques has been noted before, we prove that this phenomenon is inherent to the problem itself, irrespective of the technique used.

This is just a quick announcement that I’ll be hosting Nate Soares—who coauthored the self-explanatorily titled If Anyone Builds It, Everyone Dies with Eliezer Yudkowsky—tomorrow (Tuesday) at 5PM at UT Austin, for a brief talk followed by what I’m sure will be an extremely lively Q&A about his book. Anyone in the Austin area is welcome to join us.

By Scott

Authors: Jonah Brown-Cohen, Geoffrey Irving, Simon C. Marshall, Ilan Newman, Georgios Piliouras, Mario Szegedy

AI safety via debate uses two competing models to help a human judge verify complex computational tasks. Previous work has established what problems debate can solve in principle, but has not analysed the practical cost of human oversight: how many queries must the judge make to the debate transcript? We introduce Debate Query Complexity}(DQC), the minimum number of bits a verifier must inspect to correctly decide a debate. Surprisingly, we find that PSPACE/poly (the class of problems which debate can efficiently decide) is precisely the class of functions decidable with O(log n) queries. This characterisation shows that debate is remarkably query-efficient: even for highly complex problems, logarithmic oversight suffices. We also establish that functions depending on all their input bits require Omega(log n) queries, and that any function computable by a circuit of size s satisfies DQC(f) <= log(s) + 3. Interestingly, this last result implies that proving DQC lower bounds of log(n) + 6 for languages in P would yield new circuit lower bounds, connecting debate query complexity to central questions in circuit complexity.

Authors: Matthias Christandl, Aram W. Harrow, Greta Panova, Pietro M. Posta, Michael Walter

Some representation-theoretic multiplicities, such as the Kostka and the Littlewood-Richardson coefficients, admit a combinatorial interpretation that places their computation in the complexity class #P. Whether this holds more generally is considered an important open problem in mathematics and computer science, with relevance for geometric complexity theory and quantum information. Recent work has investigated the quantum complexity of particular multiplicities, such as the Kronecker coefficients and certain special cases of the plethysm coefficients. Here, we show that a broad class of representation-theoretic multiplicities is in #BQP. In particular, our result implies that the plethysm coefficients are in #BQP, which was only known in special cases. It also implies all known results on the quantum complexity of previously studied coefficients as special cases, unifying, simplifying, and extending prior work. We obtain our result by multiple applications of the Schur transform. Recent work has improved its dependence on the local dimension, which is crucial for our work. We further describe a general approach for showing that representation-theoretic multiplicities are in #BQP that captures our approach as well as the approaches of prior work. We complement the above by showing that the same multiplicities are also naturally in GapP and obtain polynomial-time classical algorithms when certain parameters are fixed.

Authors: Sandip Das, Sk Samim Islam, Daniel Lokshtanov

A multipacking in an undirected graph $G=(V,E)$ is a set $M\subseteq V$ such that for every vertex $v\in V$ and for every integer $r\geq 1$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$, that is, there are at most $r$ vertices in $M$ at a distance at most $r$ from $v$ in $G$. The Multipacking problem asks whether a graph contains a multipacking of size at least $k$. For more than a decade, it remained an open question whether the Multipacking problem is NP-complete or solvable in polynomial time. Whereas the problem is known to be polynomial-time solvable for certain graph classes (e.g., strongly chordal graphs, grids, etc). Foucaud, Gras, Perez, and Sikora [Algorithmica 2021] made a step towards solving the open question by showing that the Multipacking problem is NP-complete for directed graphs and it is W[1]-hard when parameterized by the solution size. In this paper, we prove that the Multipacking problem is NP-complete for undirected graphs, which answers the open question. Moreover, the problem is W[2]-hard for undirected graphs when parameterized by the solution size. Furthermore, we have shown that the problem is NP-complete and W[2]-hard (when parameterized by the solution size) even for various subclasses: chordal, bipartite, and claw-free graphs. Whereas, it is NP-complete for regular, and CONV graphs (intersection graphs of convex sets in the plane). Additionally, the problem is NP-complete and W[2]-hard (when parameterized by the solution size) for chordal $\cap$ $\frac{1}{2}$-hyperbolic graphs, which is a superclass of strongly chordal graphs where the problem is polynomial-time solvable. On the positive side, we present an exact exponential-time algorithm for the Multipacking problem on $n$-vertex general graphs, which breaks the $2^n$ barrier by achieving a running time of $O^*(1.58^n)$.

Authors: Yaé U. Gaba

The complexity quasi-metric, introduced by Schellekens, provides a topological framework where the asymmetric nature of computational comparisons -- stating that one algorithm is faster than another carries different information than stating the second is slower than the first -- finds precise mathematical expression. In this paper we develop a comprehensive theory of expansive homeomorphisms on complexity quasi-metric spaces. Our central result establishes that the scaling transformation $ψ_α(f)(n)=αf(n)$ is expansive on the complexity space $(\C,d_\C)$ if and only if $α\neq 1$. The $δ$-stable sets arising from this dynamics correspond exactly to asymptotic complexity classes, providing a dynamical characterisation of fundamental objects in complexity theory. We prove that the canonical coordinates associated with $ψ_α$ are hyperbolic with contraction rate $λ=1/α$ and establish a precise connection between orbit separation in the dynamical system and the classical time hierarchy theorem of Hartmanis and Stearns. We further investigate unstable sets, conjugate dynamics, and topological entropy estimates for the scaling map. Throughout, concrete algorithms and Python implementations accompany the proofs, making every result computationally reproducible. SageMath verification snippets are inlined alongside the examples, and the full code is available in the companion repository.

Authors: Pin-Hsian Lee, Te-Cheng Liu, Meng-Tsung Tsai

We give a short, self-contained, and easily verifiable proof that determining the outerthickness of a general graph is NP-hard. This resolves a long-standing open problem on the computational complexity of outerthickness. Moreover, our hardness result applies to a more general covering problem $P_F$, defined as follows. Fix a proper graph class $F$ whose membership is decidable. Given an undirected simple graph $G$ and an integer $k$, the task is to cover the edge set $E(G)$ by at most $k$ subsets $E_1,\ldots,E_k$ such that each subgraph $(V(G),E_i)$ belongs to $F$. Note that if $F$ is monotone (in particular, when $F$ is the class of all outerplanar graphs), any such cover can be converted into an edge partition by deleting overlaps; hence, in this case, covering and partitioning are equivalent. Our result shows that for every proper graph class $F$ whose membership is decidable and that satisfies all of the following conditions: (a) $F$ is closed under topological minors, (b) $F$ is closed under $1$-sums, and (c) $F$ contains a cycle of length $3$, the problem $P_F$ is NP-hard for every fixed integer $k\ge 3$. In particular: For $F$ equal to the class of all outerplanar graphs, our result settles the long-standing open problem on the complexity of determining outerthickness. For $F$ equal to the class of all planar graphs, our result complements Mansfield's NP-hardness result for the thickness, which applies only to the case $k=2$. It is also worth noting that each of the three conditions above is necessary. If $F$ is the class of all eulerian graphs, then cond. (a) fails. If $F$ is the class of all pseudoforests, then cond. (b) fails. If $F$ is the class of all forests, then cond. (c) fails. For each of these three classes $F$, the problem $P_F$ is solvable in polynomial time for every fixed integer $k\ge 3$, showing that none of the three conditions can be dropped.

Authors: Kaleem Ullah Qasim, Jiashu Zhang, Hao Li, Muhammad Kafeel Shaheen

Training language models to solve complex mathematical problems benefits from curriculum learning progressively training on simpler subproblems. However, existing decomposition methods are often heuristic, offering no guarantees that subproblems are simpler, that solving them aids the parent task, or that their relationships are mathematically grounded. We observe that symbolic differentiation provides a natural structure for verified decomposition: calculus rules explicitly define how expressions reduce to simpler components with provable properties. We introduce Verify-RL, a framework where every parent-child decomposition satisfies three verifiable conditions: strictly decreasing structural complexity, solution containment, and formal rule derivation. Unlike heuristic methods where a significant fraction of decompositions are invalid our properties admit automatic verification through symbolic computation, achieving "verification by construction" Experiments demonstrate that eliminating invalid decompositions yields sizable gains, accuracy on the hardest problems more than doubles from 32% to 68%, with a 40% relative improvement overall.

Authors: Clément L. Canonne, Kenny Chen, Julián Mestre

We study the extremal Forrelation problem, where, provided with oracle access to Boolean functions $f$ and $g$ promised to satisfy either $\textrm{forr}(f,g)=1$ or $\textrm{forr}(f,g)=-1$, one must determine (with high probability) which of the two cases holds while performing as few oracle queries as possible. It is well known that this problem can be solved with \emph{one} quantum query; yet, Girish and Servedio (TQC 2025) recently showed this problem requires $\widetildeΩ(2^{n/4})$ classical queries, and conjectured that the optimal lower bound is $\widetildeΩ(2^{n/2})$. Through a completely different construction, we improve on their result and prove a lower bound of $Ω(2^{0.4999n})$, which matches the conjectured lower bound up to an arbitrarily small constant in the exponent.

Authors: Ruijie Zhu, Jiahao Lu, Wenbo Hu, Xiaoguang Han, Jianfei Cai, Ying Shan, Chuanxia Zheng

We introduce MotionCrafter, a video diffusion-based framework that jointly reconstructs 4D geometry and estimates dense motion from a monocular video. The core of our method is a novel joint representation of dense 3D point maps and 3D scene flows in a shared coordinate system, and a novel 4D VAE to effectively learn this representation. Unlike prior work that forces the 3D value and latents to align strictly with RGB VAE latents-despite their fundamentally different distributions-we show that such alignment is unnecessary and leads to suboptimal performance. Instead, we introduce a new data normalization and VAE training strategy that better transfers diffusion priors and greatly improves reconstruction quality. Extensive experiments across multiple datasets demonstrate that MotionCrafter achieves state-of-the-art performance in both geometry reconstruction and dense scene flow estimation, delivering 38.64% and 25.0% improvements in geometry and motion reconstruction, respectively, all without any post-optimization. Project page: https://ruijiezhu94.github.io/MotionCrafter_Page

Authors: Ivor van der Hoog, Eva Rotenberg, Jack Spalding-Jamieson, Lasse Wulf

Many fundamental problems in computational geometry admit no algorithm running in $o(n \log n)$ time for $n$ planar input points, via classical reductions from sorting. Prominent examples include the computation of convex hulls, quadtrees, onion layer decompositions, Euclidean minimum spanning trees, KD-trees, Voronoi diagrams, and decremental closest-pair. A classical result shows that, given $n$ points sorted along a single direction, the convex hull can be constructed in linear time. Subsequent works established that for all of the other above problems, this information does not suffice. In 1989, Aggarwal, Guibas, Saxe, and Shor asked: Under which conditions can a Voronoi diagram be computed in $o(n \log n)$ time? Since then, the question of whether sorting along TWO directions enables a $o(n \log n)$-time algorithm for such problems has remained open and has been repeatedly mentioned in the literature. In this paper, we introduce the Presort Hierarchy: A problem is 1-Presortable if, given a sorting along one axis, it permits a (possibly randomised) $o(n \log n)$-time algorithm. It is 2-Presortable if sortings along both axes suffice. It is Presort-Hard otherwise. Our main result is that quadtrees, and by extension Delaunay triangulations, Voronoi diagrams, and Euclidean minimum spanning trees, are 2-Presortable: we present an algorithm with expected running time $O(n \sqrt{\log n})$. This addresses the longstanding open problem posed by Aggarwal, Guibas, Saxe, and Shor (albeit randomised). We complement this result by showing that some of the other above geometric problems are also 2-Presortable or Presort-Hard.

Authors: Vineet Kumar Rakesh, Ahana Bhattacharjee, Soumya Mazumdar, Tapas Samanta, Hemendra Kumar Pandey, Amitabha Das, Sarbajit Pal

Talking-head avatars are increasingly adopted in educational technology to deliver content with social presence and improved engagement. However, many recent talking-head generation (THG) methods rely on GPU-centric neural rendering, large training sets, or high-capacity diffusion models, which limits deployment in offline or resource-constrained learning environments. A deterministic and CPU-oriented THG framework is described, termed Symbolic Vedic Computation, that converts speech to a time-aligned phoneme stream, maps phonemes to a compact viseme inventory, and produces smooth viseme trajectories through symbolic coarticulation inspired by Vedic sutra Urdhva Tiryakbhyam. A lightweight 2D renderer performs region-of-interest (ROI) warping and mouth compositing with stabilization to support real-time synthesis on commodity CPUs. Experiments report synchronization accuracy, temporal stability, and identity consistency under CPU-only execution, alongside benchmarking against representative CPU-feasible baselines. Results indicate that acceptable lip-sync quality can be achieved while substantially reducing computational load and latency, supporting practical educational avatars on low-end hardware. GitHub: https://vineetkumarrakesh.github.io/vedicthg

Authors: Mark Yan Lok Yip, Gary P. T. Choi

Rod-based structures are commonly used in practical applications in science and engineering. However, in many design, analysis, and manufacturing tasks, handling the rod-based structures in three dimensions directly is generally challenging. To simplify the tasks, it is usually more desirable to achieve a two-dimensional representation of the rod-based structures via some suitable geometric mappings. In this work, we develop a novel method for computing a low-distortion planar embedding of rod-based structures. Specifically, we identify geometrical constraints that aim to preserve key length and angle quantities of the 3D rod-based structures and prevent the occurrence of overlapping rods in the planar embedding. Experimental results with a variety of rod-based structures are presented to demonstrate the effectiveness of our approach. Moreover, our method can be naturally extended to the design and mapping of hybrid structures consisting of both rods and surface elements. Altogether, our approach paves a new way for the efficient design and fabrication of novel three-dimensional geometric structures for practical applications.

Authors: Pavel Paták, Zuzana Patáková

We discuss recent progress on topological Helly-type theorems and their variants. We provide an overview of two different proof techniques, one based on the nerve lemma, while the other on non-embeddability.

Authors: Mohammad Shahverdikondori, Sepehr Elahi, Patrick Thiran, Negar Kiyavash

Motivated by optimization oracles in bandits with network interference, we study the Neighborhood-Aware Graph Labeling (NAGL) problem. Given a graph $G = (V,E)$, a label set of size $L$, and local reward functions $f_v$ accessed via evaluation oracles, the objective is to assign labels to maximize $\sum_{v \in V} f_v(x_{N[v]})$, where each term depends on the closed neighborhood of $v$. Two vertices co-occur in some neighborhood term exactly when their distance in $G$ is at most $2$, so the dependency graph is the squared graph $G^2$ and $\mathrm{tw}(G^2)$ governs exact algorithms and matching fine-grained lower bounds. Accordingly, we show that this dependence is inherent: NAGL is NP-hard even on star graphs with binary labels and, assuming SETH, admits no $(L-\varepsilon)^{\mathrm{tw}(G^2)}\cdot n^{O(1)}$-time algorithm for any $\varepsilon>0$. We match this with an exact dynamic program on a tree decomposition of $G^2$ running in $O\!\left(n\cdot \mathrm{tw}(G^2)\cdot L^{\mathrm{tw}(G^2)+1}\right)$ time. For approximation, unless $\mathsf{P}=\mathsf{NP}$, for every $\varepsilon>0$ there is no polynomial-time $n^{1-\varepsilon}$-approximation on general graphs even under the promise $\mathrm{OPT}>0$; without the promise $\mathrm{OPT}>0$, no finite multiplicative approximation ratio is possible. In the nonnegative-reward regime, we give polynomial-time approximation algorithms for NAGL in two settings: (i) given a proper $q$-coloring of $G^2$, we obtain a $1/q$-approximation; and (ii) on planar graphs of bounded maximum degree, we develop a Baker-type polynomial-time approximation scheme (PTAS), which becomes an efficient PTAS (EPTAS) when $L$ is constant.

Authors: Yu-Sheng Shih, Meng-Tsung Tsai, Yen-Chu Tsai, Ying-Sian Wu

We study the space complexity of four variants of the standard subgraph finding problem in the streaming model. Specifically, given an $n$-vertex input graph and a fixed-size pattern graph, we consider two settings: undirected simple graphs, denoted by $G$ and $H$, and oriented graphs, denoted by $\vec{G}$ and $\vec{H}$. Depending on the setting, the task is to decide whether $G$ contains $H$ as a subgraph or as an induced subgraph, or whether $\vec{G}$ contains $\vec{H}$ as a subgraph or as an induced subgraph. Let Sub$(H)$, IndSub$(H)$, Sub$(\vec{H})$, and IndSub$(\vec{H})$ denote these four variants, respectively. An oriented graph is well-oriented if it admits a bipartition in which every arc is oriented from one part to the other, and a vertex is non-well-oriented if both its in-degree and out-degree are non-zero. For each variant, we obtain a complete dichotomy theorem, briefly summarized as follows. (1) Sub$(H)$ can be solved by an $\tilde{O}(1)$-pass $n^{2-Ω(1)}$-space algorithm if and only if $H$ is bipartite. (2) IndSub$(H)$ can be solved by an $\tilde{O}(1)$-pass $n^{2-Ω(1)}$-space algorithm if and only if $H \in \{P_3, P_4, co\mbox{-}P_3\}$. (3) Sub$(\vec{H})$ can be solved by a single-pass $n^{2-Ω(1)}$-space algorithm if and only if every connected component of $\vec H$ is either a well-oriented bipartite graph or a tree containing at most one non-well-oriented vertex. (4) IndSub$(\vec{H})$ can be solved by an $\tilde{O}(1)$-pass $n^{2-Ω(1)}$-space algorithm if and only if the underlying undirected simple graph $H$ is a $co\mbox{-}P_3$.

Authors: Jan Dreier, Nikolas Mählmann, Sebastian Siebertz

A theorem of Ding, Oporowski, Oxley, and Vertigan implies that any sufficiently large twin-free graph contains a large matching, a co-matching, or a half-graph as a semi-induced subgraph. The sizes of these unavoidable patterns are measured by the matching index, co-matching index, and half-graph index of a graph. Consequently, graph classes can be organized into the eight classes determined by which of the three indices are bounded. We completely classify the parameterized complexity of Independent Set, Clique, and Dominating Set across all eight of these classes. For this purpose, we first derive multiple tractability and hardness results from the existing literature, and then proceed to fill the identified gaps. Among our novel results, we show that Independent Set is fixed-parameter tractable on every graph class where the half-graph and co-matching indices are simultaneously bounded. Conversely, we construct a graph class with bounded half-graph index (but unbounded co-matching index), for which the problem is W[1]-hard. For the W[1]-hard cases of our classification, we review the state of approximation algorithms. Here, we contribute an approximation algorithm for Independent Set on classes of bounded half-graph index.

Authors: Zhao Song

Brand, Nanongkai, and Saranurak introduced a conjecture known as the Hinted Mv Conjecture. Although it was originally formulated for the matrix case, we generalize it here to the tensor setting.

Authors: Siddharth Barman, Nirjhar Das, Shivam Gupta, Kirankumar Shiragur

Nearest Neighbor Search (NNS) is a fundamental problem in data structures with wide-ranging applications, such as web search, recommendation systems, and, more recently, retrieval-augmented generations (RAG). In such recent applications, in addition to the relevance (similarity) of the returned neighbors, diversity among the neighbors is a central requirement. In this paper, we develop principled welfare-based formulations in NNS for realizing diversity across attributes. Our formulations are based on welfare functions -- from mathematical economics -- that satisfy central diversity (fairness) and relevance (economic efficiency) axioms. With a particular focus on Nash social welfare, we note that our welfare-based formulations provide objective functions that adaptively balance relevance and diversity in a query-dependent manner. Notably, such a balance was not present in the prior constraint-based approach, which forced a fixed level of diversity and optimized for relevance. In addition, our formulation provides a parametric way to control the trade-off between relevance and diversity, providing practitioners with flexibility to tailor search results to task-specific requirements. We develop efficient nearest neighbor algorithms with provable guarantees for the welfare-based objectives. Notably, our algorithm can be applied on top of any standard ANN method (i.e., use standard ANN method as a subroutine) to efficiently find neighbors that approximately maximize our welfare-based objectives. Experimental results demonstrate that our approach is practical and substantially improves diversity while maintaining high relevance of the retrieved neighbors.

Authors: Ziyi Cai, D. D. Gao, Prasanna Ramakrishnan, Kangning Wang

We study the design of voting rules in the metric distortion framework. It is known that any deterministic rule suffers distortion of at least $3$, and that randomized rules can achieve distortion strictly less than $3$, often at the cost of reduced transparency and interpretability. In this work, we explore the trade-off between these paradigms by asking whether it is possible to break the distortion barrier of $3$ using only "bounded" randomness. We answer in the affirmative by presenting a voting rule that (1) achieves distortion of at most $3 - \varepsilon$ for some absolute constant $\varepsilon > 0$, and (2) selects a winner uniformly at random from a deterministically identified list of constant size. Our analysis builds on new structural results for the distortion and approximation of Maximal Lotteries and Stable Lotteries.

Authors: Lunjia Hu, Jon Schneider, Yifan Wu

We give a randomized online algorithm that guarantees near-optimal $\widetilde O(\sqrt T)$ expected swap regret against any sequence of $T$ adaptively chosen Lipschitz convex losses on the unit interval. This improves the previous best bound of $\widetilde O(T^{2/3})$ and answers an open question of Fishelson et al. [2025b]. In addition, our algorithm is efficient: it runs in $\mathsf{poly}(T)$ time. A key technical idea we develop to obtain this result is to discretize the unit interval into bins at multiple scales of granularity and simultaneously use all scales to make randomized predictions, which we call multi-scale binning and may be of independent interest. A direct corollary of our result is an efficient online algorithm for minimizing the calibration error for general elicitable properties. This result does not require the Lipschitzness assumption of the identification function needed in prior work, making it applicable to median calibration, for which we achieve the first $\widetilde O(\sqrt T)$ calibration error guarantee.

Authors: Yubin Zhu, Zitan Chen

We derive Singleton-type bounds on the free distance and column distances of trellis codes. Our results show that, at a given time instant, the maximum attainable column distance of trellis codes can exceed that of convolutional codes. Moreover, using expander graphs, we construct trellis codes over constant-size alphabets that achieve a rate-distance trade-off arbitrarily close to that of convolutional codes with a maximum distance profile. By comparison, all known constructions of convolutional codes with a maximum distance profile require working over alphabets whose size grows at least exponentially with the number of output symbols per time instant.

Authors: Panagiotis Charalampopoulos, Jonas Ellert, Manal Mohamed

The Cartesian tree of a sequence captures the relative order of the sequence's elements. In recent years, Cartesian tree matching has attracted considerable attention, particularly due to its applications in time series analysis. Consider a text $T$ of length $n$ and a pattern $P$ of length $m$. In the exact Cartesian tree matching problem, the task is to find all length-$m$ fragments of $T$ whose Cartesian tree coincides with the Cartesian tree $CT(P)$ of the pattern. Although the exact version of the problem can be solved in linear time [Park et al., TCS 2020], it remains rather restrictive; for example, it is not robust to outliers in the pattern. To overcome this limitation, we consider the approximate setting, where the goal is to identify all fragments of $T$ that are close to some string whose Cartesian tree matches $CT(P)$. In this work, we quantify closeness via the widely used Hamming distance metric. For a given integer parameter $k>0$, we present an algorithm that computes all fragments of $T$ that are at Hamming distance at most $k$ from a string whose Cartesian tree matches $CT(P)$. Our algorithm runs in time $\mathcal O(n \sqrt{m} \cdot k^{2.5})$ for $k \leq m^{1/5}$ and in time $\mathcal O(nk^5)$ for $k \geq m^{1/5}$, thereby improving upon the state-of-the-art $\mathcal O(nmk)$-time algorithm of Kim and Han [TCS 2025] in the regime $k = o(m^{1/4})$. On the way to our solution, we develop a toolbox of independent interest. First, we introduce a new notion of periodicity in Cartesian trees. Then, we lift multiple well-known combinatorial and algorithmic results for string matching and periodicity in strings to Cartesian tree matching and periodicity in Cartesian trees.

Authors: Emilio Cruciani, Sebastian Forster, Antonis Skarlatos

Given a weighted undirected graph, a number of clusters $k$, and an exponent $z$, the goal in the $(k, z)$-clustering problem on graphs is to select $k$ vertices as centers that minimize the sum of the distances raised to the power $z$ of each vertex to its closest center. In the dynamic setting, the graph is subject to adversarial edge updates, and the goal is to maintain explicitly an exact $(k, z)$-clustering solution in the induced shortest-path metric. While efficient dynamic $k$-center approximation algorithms on graphs exist [Cruciani et al. SODA 2024], to the best of our knowledge, no prior work provides similar results for the dynamic $(k,z)$-clustering problem. As the main result of this paper, we develop a randomized incremental $(k, z)$-clustering algorithm that maintains with high probability a constant-factor approximation in a graph undergoing edge insertions with a total update time of $\tilde O(k m^{1+o(1)}+ k^{1+\frac{1}λ} m)$, where $λ\geq 1$ is an arbitrary fixed constant. Our incremental algorithm consists of two stages. In the first stage, we maintain a constant-factor bicriteria approximate solution of size $\tilde{O}(k)$ with a total update time of $m^{1+o(1)}$ over all adversarial edge insertions. This first stage is an intricate adaptation of the bicriteria approximation algorithm by Mettu and Plaxton [Machine Learning 2004] to incremental graphs. One of our key technical results is that the radii in their algorithm can be assumed to be non-decreasing while the approximation ratio remains constant, a property that may be of independent interest. In the second stage, we maintain a constant-factor approximate $(k,z)$-clustering solution on a dynamic weighted instance induced by the bicriteria approximate solution. For this subproblem, we employ a dynamic spanner algorithm together with a static $(k,z)$-clustering algorithm.

Authors: Moran Feldman, Justin Ward

We study the problem of maximizing a non-negative monotone submodular objective $f$ subject to the intersection of $k$ arbitrary matroid constraints. The natural greedy algorithm guarantees $(k+1)$-approximation for this problem, and the state-of-the-art algorithm only improves this approximation ratio to $k$. We give a $\frac{2k\ln2}{1+\ln2}+O(\sqrt{k})<0.819k+O(\sqrt{k})$ approximation for this problem. Our result is the first multiplicative improvement over the approximation ratio of the greedy algorithm for general $k$. We further show that our algorithm can be used to obtain roughly the same approximation ratio also for the more general problem in which the objective is not guaranteed to be monotone (the sublinear term in the approximation ratio becomes $O(k^{2/3})$ rather than $O(\sqrt{k})$ in this case). All of our results hold also when the $k$-matroid intersection constraint is replaced with a more general matroid $k$-parity constraint. Furthermore, unlike the case in many of the previous works, our algorithms run in time that is independent of $k$ and polynomial in the size of the ground set. Our algorithms are based on a hybrid greedy local search approach recently introduced by Singer and Thiery (STOC 2025) for the weighted matroid $k$-intersection problem, which is a special case of the problem we consider. Leveraging their approach in the submodular setting requires several non-trivial insights and algorithmic modifications since the marginals of a submodular function $f$, which correspond to the weights in the weighted case, are not independent of the algorithm's internal randomness. In the special weighted case studied by Singer and Thiery, our algorithms reduce to a variant of their algorithm with an improved approximation ratio of $k\ln2+1-\ln2<0.694k+0.307$, compared to an approximation ratio of $\frac{k+1}{2\ln2}\approx0.722k+0.722$ guaranteed by Singer and Thiery.

Authors: Wojciech Gabryelski, Zbigniew Gołȩbiewski, Martin Pépin

We propose two efficient algorithms for generating uniform random directed acyclic graphs, including an asymptotically optimal exact-size sampler that performs $\frac{n^2}{2} + o(n^2)$ operations and requests to a random generator. This was achieved by extending the Boltzmann model for graphical generating functions and by using various decompositions of directed acyclic graphs. The presented samplers improve upon the state-of-the-art algorithms in terms of theoretical complexity and offer a significant speed-up in practice.

Authors: Tian Zhang, Ashwin Padaki, Jiaming Liang, Zack Ives, Erik Waingarten

Vector similarity search is an essential primitive in modern AI and ML applications. Most vector databases adopt graph-based approximate nearest neighbor (ANN) search algorithms, such as DiskANN (Subramanya et al., 2019), which have demonstrated state-of-the-art empirical performance. DiskANN's graph construction is governed by a reachability parameter $α$, which gives a trade-off between construction time, query time, and accuracy. However, adaptively tuning this trade-off typically requires rebuilding the index for different $α$ values, which is prohibitive at scale. In this work, we propose RP-Tuning, an efficient post-hoc routine, based on DiskANN's pruning step, to adjust the $α$ parameter without reconstructing the full index. Within the $α$-reachability framework of prior theoretical works (Indyk and Xu, 2023; Gollapudi et al., 2025), we prove that pruning an initially $α$-reachable graph with RP-Tuning preserves worst-case reachability guarantees in general metrics and improved guarantees in Euclidean metrics. Empirically, we show that RP-Tuning accelerates DiskANN tuning on four public datasets by up to $43\times$ with negligible overhead.

Authors: Nicola Cotumaccio

Recently, a new paradigm was introduced in automata theory. The main idea is to classify regular languages according to their propensity to be sorted, establishing a deep connection between automata theory and data compression [J. ACM 2023]. This parameterization leads to two hierarchies of regular languages: a deterministic hierarchy and a non-deterministic hierarchy. While the deterministic hierarchy is well understood, the non-deterministic hierarchy appears much more complex. This is true even for the richest and most studied level of the hierarchies, corresponding to the class of Wheeler languages. In this paper, we study Wheeler language through the lens of bisimulations. We first show that the standard notion of bisimulation is not appropriate. Then, we introduce Wheeler bisimulations, that is, bisimulations that respect the convex structure of the considered Wheeler automata. Although there are some differences between the properties of bisimulations and the properties of Wheeler bisimulations, we show that Wheeler bisimulations induce a unique minimal Wheeler NFA (analogously to standard bisimulations). In particular, in the deterministic case, we retrieve the minimum Wheeler deterministic automaton of a given language. We also show that the minimal Wheeler NFA induced by Wheeler bisimulations can be built in linear time. This is in contrast with standard bisimulations, for which the corresponding minimal NFA can be built in $ O(m \log n) $ time (where $ m $ is the number of edges and $ n $ is the number of states) by adapting Paige-Tarjan's partition refinement algorithm.

Authors: Joon Suk Huh

Modern data workflows are inherently adaptive, repeatedly querying the same dataset to refine and validate sequential decisions, but such adaptivity can lead to overfitting and invalid statistical inference. Adaptive Data Analysis (ADA) mechanisms address this challenge; however, there is a fundamental tension between computational efficiency and sample complexity. For $T$ rounds of adaptive analysis, computationally efficient algorithms typically incur suboptimal $O(\sqrt{T})$ sample complexity, whereas statistically optimal $O(\log T)$ algorithms are computationally intractable under standard cryptographic assumptions. In this work, we shed light on this trade-off by identifying a natural class of data distributions under which both computational efficiency and optimal sample complexity are achievable. We propose a computationally efficient ADA mechanism that attains optimal $O(\log T)$ sample complexity when the data distribution is dense with respect to a known prior. This setting includes, in particular, feature--label data distributions arising in distribution-specific learning. As a consequence, our mechanism also yields a sample-efficient (i.e., $O(\log T)$ samples) statistical query oracle in the distribution-specific setting. Moreover, although our algorithm is not based on differential privacy, it satisfies a relaxed privacy notion known as Predicate Singling Out (PSO) security (Cohen and Nissim, 2020). Our results thus reveal an inherent connection between adaptive data analysis and privacy beyond differential privacy.

Authors: Sharareh Alipour

Many multiagent tasks -- such as reviewer assignment, coalition formation, or fair resource allocation -- require selecting a group of agents such that collaboration remains effective even in the worst case. The \emph{weighted max-min $T$-join problem} formalizes this challenge by seeking a subset of vertices whose minimum-weight matching is maximized, thereby ensuring robust outcomes against unfavorable pairings. We advance the study of this problem in several directions. First, we design an algorithm that computes an upper bound for the \emph{weighted max-min $2k$-matching problem}, where the chosen set must contain exactly $2k$ vertices. Building on this bound, we develop a general algorithm with a \emph{$2 \ln n$-approximation guarantee} that runs in $O(n^4)$ time. Second, using ear decompositions, we propose another upper bound for the weighted max-min $T$-join cost. We also show that the problem can be solved exactly when edge weights belong to $\{1,2\}$. Finally, we evaluate our methods on real collaboration datasets. Experiments show that the lower bounds from our approximation algorithm and the upper bounds from the ear decomposition method are consistently close, yielding empirically small constant-factor approximations. Overall, our results highlight both the theoretical significance and practical value of weighted max-min $T$-joins as a framework for fair and robust group formation in multiagent systems.

Authors: Owen Shen, Haoran Xu, Yinyu Ye, Peter Glynn, Patrick Jaillet

We study online configuration selection with admission control problem, which arises in LLM serving, GPU scheduling, and revenue management. In a planning horizon with $T$ periods, we consider a two-layer framework for the decisions made within each time period. In the first layer, the decision maker selects one of the $K$ configurations (ex. quantization, parallelism, fare class) which induces distribution over the reward-resource pair of the incoming request. In the second layer, the decision maker observes the request and then decides whether to accept it or not. Benchmarking this framework requires care. We introduce a \textbf{switching-aware fluid oracle} that accounts for the value of mixing configurations over time, provably upper-bounding any online policy. We derive a max-min formulation for evaluating the benchmark, and we characterize saddle points of the max-min problem via primal-dual optimality conditions linking equilibrium, feasibility, and complementarity. This guides the design of \textbf{SP-UCB--OLP} algorithm, which solves an optimistic saddle point problem and achieves $\tilde{O}(\sqrt{KT})$ regret.

Authors: Sreenivas Gollapudi, Kostas Kollias, Kamesh Munagala, Aravindan Vijayaraghavan

Conformal prediction provides rigorous, distribution-free uncertainty guarantees, but often yields prohibitively large prediction sets in structured domains such as routing, planning, or sequential recommendation. We introduce "graph-based conformal compression", a framework for constructing compact subgraphs that preserve statistical validity while reducing structural complexity. We formulate compression as selecting a smallest subgraph capturing a prescribed fraction of the probability mass, and reduce to a weighted version of densest $k$-subgraphs in hypergraphs, in the regime where the subgraph has a large fraction of edges. We design efficient approximation algorithms that achieve constant factor coverage and size trade-offs. Crucially, we prove that our relaxation satisfies a monotonicity property, derived from a connection to parametric minimum cuts, which guarantees the nestedness required for valid conformal guarantees. Our results on the one hand bridge efficient conformal prediction with combinatorial graph compression via monotonicity, to provide rigorous guarantees on both statistical validity, and compression or size. On the other hand, they also highlight an algorithmic regime, distinct from classical densest-$k$-subgraph hardness settings, where the problem can be approximated efficiently. We finally validate our algorithmic approach via simulations for trip planning and navigation, and compare to natural baselines.