Theory of Computing Report

Chain Quality (CQ) is a core blockchain property. Roughly speaking, it says: Owning $3\%$ of the stake gives you inclusion rights over valid inputs of your choice in roughly $3\%$ of the blockspace over time. For Nakamoto style chains, this is called Ideal CQ (see here). Chain quality was sufficient for early generations of blockchains that had low throughput, but modern blockchains have much higher bandwidth and can commit many...

By Ittai Abraham, Pranav Garimidi, Joachim Neu

Authors: Konstantinos Tsirkas, Leda Wang, Ilias Zadik

Over the last decades, two distinct approaches have been instrumental to our understanding of the computational complexity of statistical estimation. The statistical physics literature predicts algorithmic hardness through local stability and monotonicity properties of the Franz--Parisi (FP) potential \cite{franz1995recipes,franz1997phase}, while the mathematically rigorous literature characterizes hardness via the limitations of restricted algorithmic classes, most notably low-degree polynomial estimators \cite{hopkins2017efficient}. For many inference models, these two perspectives yield strikingly consistent predictions, giving rise to a long-standing open problem of establishing a precise mathematical relationship between them. In this work, we show that for estimation problems the power of low-degree polynomials is equivalent to the monotonicity of the annealed FP potential for a broad family of Gaussian additive models (GAMs) with signal-to-noise ratio $λ$. In particular, subject to a low-degree conjecture for GAMs, our results imply that the polynomial-time limits of these models are directly implied by the monotonicity of the annealed FP potential, in conceptual agreement with predictions from the physics literature dating back to the 1990s.

Authors: Samuel González-Castillo, Joon Hyung Lee, Alfons Laarman

We study PRODSAT-QSAT($k$): given rank-one $k$-local projectors, determine whether a quantum $k$-SAT instance admits a satisfying product state. We present a CDCL-style refutation framework that searches a finite partition of each qubit's Bloch sphere while a sound theory solver checks region feasibility using a geometric overapproximation of the projection amplitudes for each constraint. When the theory solver proves that no state in a region can satisfy a constraint, it produces a sound conflict clause that blocks that region; accumulated blocking clauses can yield a global result of product-state unsatisfiability (UN-PRODSAT). We formalise the problem, prove the soundness of the clause-learning rule, and describe a practical algorithm and implementation.

Authors: Angshul Majumdar

We study the computational complexity of constrained nonnegative Gram feasibility. Given a partially specified symmetric matrix together with affine relations among selected entries, the problem asks whether there exists a nonnegative matrix $H \in \mathbb{R}_+^{n\times r}$ such that $W = HH^\top$ satisfies all specified entries and affine constraints. Such factorizations arise naturally in structured low-rank matrix representations and geometric embedding problems. We prove that this feasibility problem is $\exists\mathbb{R}$-complete already for rank $r=2$. The hardness result is obtained via a polynomial-time reduction from the arithmetic feasibility problem \textsc{ETR-AMI}. The reduction exploits a geometric encoding of arithmetic constraints within rank-$2$ nonnegative Gram representations: by fixing anchor directions in $\mathbb{R}_+^2$ and representing variables through vectors of the form $(x,1)$, addition and multiplication constraints can be realized through inner-product relations. Combined with the semialgebraic formulation of the feasibility conditions, this establishes $\exists\mathbb{R}$-completeness. We further show that the hardness extends to every fixed rank $r\ge 2$. Our results place constrained symmetric nonnegative Gram factorization among the growing family of geometric feasibility problems that are complete for the existential theory of the reals. Finally, we discuss limitations of the result and highlight the open problem of determining the complexity of unconstrained symmetric nonnegative factorization feasibility.

Authors: Zach Hunter, Aleksa Milojević, Benny Sudakov, Istvan Tomon

Determining the randomized (or distributional) communication complexity of disjointness is a central problem in communication complexity, having roots in the foundational work of Babai, Frankl, and Simon in the 1980s and culminating in the famous works of Kalyanasundaram-Schnitger and Razborov in 1992. However, the question of obtaining tight bounds for product distributions persisted until the more recent work of Bottesch, Gavinsky, and Klauck resolved it. In this note we revisit this classical problem and give a short, streamlined proof of the best bounds, with improved quantitative dependence on the error parameter. Our approach is based on a simple combinatorial lemma that may be of independent interest: if two sets drawn independently from two distributions are disjoint with non-negligible probability, then one can extract two subfamilies of reasonably large measure that are fully cross-disjoint (equivalently, a large monochromatic rectangle for disjointness).

Authors: Vincent Delecroix, Oscar Fontaine, Arnaud de Mesmay

A triangulation of a surface is k-irreducible if every non-contractible curve has length at least k and any edge contraction breaks this property. Equivalently, every edge belongs to a non-contractible curve of length k and there are no shorter non-contractible curves. We prove that a k-irreducible triangulation of a surface of genus g has $O(k^2g)$ triangles, which is optimal. This is an improvement over the previous best bound $k^{O(k)} g^2$ of Gao, Richter and Seymour [Journal of Combinatorial Theory, Series B, 1996].

Authors: Evanthia Papadopoulou, Zeyu Wang

We consider the Voronoi diagram of lines in $\mathbb{R}^3$ under the Euclidean metric, and give a full classification of its structure in the base case of four lines in general position. We first show that the number of vertices in the Voronoi diagram of four lines in general position is always even, between 0 and 8, and all such numbers can be realized. We identify a key structure for the diagram formation, called a \emph{twist}, which is a pair of consecutive intersections among trisector branches; only two types of twists are possible, so-called \emph{full} and \emph{partial} twists. A full twist is a purely local structure, which can be inserted or removed without affecting the rest of the diagram. Assuming no full twists, the nearest and the farthest Voronoi diagrams of four lines, each have 15 distinct topologies, which are in one-to-one correspondence; the two-dimensional faces are all unbounded, and the total number of vertices is at most six. The unbounded features of the farthest diagram, encoded in a two-dimensional spherical map, are also in one-to-one correspondence. The identified topologies are all realizable. Any Voronoi diagram of four lines in general position in $\mathbb{R}^3$ can be obtained from one of these topologies by inserting full twists; each twist induces a bounded face of exactly two vertices in both the nearest and farthest diagrams. We obtain the classification by an exhaustive search algorithm using some new structural and combinatorial observations of line Voronoi diagrams.

Authors: Jonathan Richard Shewchuk

The restricted Delaunay triangulation of a closed surface $Σ$ and a finite point set $V \subset Σ$ is a subcomplex of the Delaunay tetrahedralization of $V$ whose triangles approximate $Σ$. It is well known that if $V$ is a sufficiently dense sample of a smooth $Σ$, then the union of the restricted Delaunay triangles is homeomorphic to $Σ$. We show that an $ε$-sample with $ε\leq 0.3245$ suffices. By comparison, Dey proves it for a $0.18$-sample; our improved sampling bound reduces the number of sample points required by a factor of $3.25$. More importantly, we improve a related sampling bound of Cheng et al. for Delaunay surface meshing, reducing the number of sample points required by a factor of $21$. The first step of our homeomorphism proof is particularly interesting: we show that for a $0.44$-sample, the restricted Voronoi cell of each site $v \in V$ is homeomorphic to a disk, and the orthogonal projection of the cell onto $T_vΣ$ (the plane tangent to $Σ$ at $v$) is star-shaped.

Authors: Chengyuan Deng, Jie Gao, Kevin Lu, Feng Luo, Cheng Xin

For a metric space $(X, d)$, a family $\mathcal{H}$ of locality sensitive hash functions is called $(r, cr, p_1, p_2)$ sensitive if a randomly chosen function $h\in \mathcal{H}$ has probability at least $p_1$ (at most $p_2$) to map any $a, b\in X$ in the same hash bucket if $d(a, b)\leq r$ (or $d(a, b)\geq cr$). Locality Sensitive Hashing (LSH) is one of the most popular techniques for approximate nearest-neighbor search in high-dimensional spaces, and has been studied extensively for Hamming, Euclidean, and spherical geometries. An $(r, cr, p_1, p_2)$-sensitive hash function enables approximate nearest neighbor search (i.e., returning a point within distance $cr$ from a query $q$ if there exists a point within distance $r$ from $q$) with space $O(n^{1+ρ})$ and query time $O(n^ρ)$ where $ρ=\frac{\log 1/p_1}{\log 1/p_2}$. But LSH for hyperbolic spaces $\mathbb{H}^d$ remains largely unexplored. In this work, we present the first LSH construction native to hyperbolic space. For the hyperbolic plane $(d=2)$, we show a construction achieving $ρ\leq 1/c$, based on the hyperplane rounding scheme. For general hyperbolic spaces $(d \geq 3)$, we use dimension reduction from $\mathbb{H}^d$ to $\mathbb{H}^2$ and the 2D hyperbolic LSH to get $ρ\leq 1.59/c$. On the lower bound side, we show that the lower bound on $ρ$ of Euclidean LSH extends to the hyperbolic setting via local isometry, therefore giving $ρ\geq 1/c^2$.

Authors: Tsuri Farhana, Omrit Filtser, Shalev Goldshtein

We study unlabeled multi-robot motion planning for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Banyassady et al. (SoCG'22) guarantee feasibility in simple polygons under start--start and target--target distances of at least $4$, and start--target distances of at least $3$, but without optimality guarantees. Solovey et al. (RSS'15) provide a near-optimal solution in general polygonal domains, under stricter conditions: start/target positions must have pairwise distance at least $4$, and at least $\sqrt{5}\approx2.236$ from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments. In this paper we present a generalized algorithm that achieve different trade-offs on the robots-separation and obstacles-separation bounds, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) the robots-separation is $2\tfrac{2}{3}$ and the obstacles-separation is $1\tfrac{2}{3}$, or (ii) the robots-separation is $\approx3.291$ and the obstacles-separation $\approx1.354$. Additionally, we introduce a different strategy yielding a polynomial-time solution when the robots-separation is only $2$, and the obstacles-separation is $3$. Finally, we show that without any robots-separation assumption, obstacles-separation of at least $1.5$ may be necessary for a solution to exist.

Authors: Tianyu Bell Pan, Damon L. Woodard

Large language models (LLMs) demonstrate strong performance, but they often lack transparency. We introduce GeoLAN, a training framework that treats token representations as geometric trajectories and applies stickiness conditions inspired by recent developments related to the Kakeya Conjecture. We have developed two differentiable regularizers, Katz-Tao Convex Wolff (KT-CW) and Katz-Tao Attention (KT-Attn), that promote isotropy and encourage diverse attention. Our experiments with Gemma-3 (1B, 4B, 12B) and Llama-3-8B show that GeoLAN frequently maintains task accuracy while improving geometric metrics and reducing certain fairness biases. These benefits are most significant in mid-sized models. Our findings reveal scale-dependent trade-offs between geometric precision and performance, suggesting that geometry-aware training is a promising approach to enhance mechanistic interpretability.

Authors: Zhao Song

In this paper, we show how to generalize the lazy update regime from dynamic matrix product [Cohen, Lee, Song STOC 2019, JACM 2021] to dynamic kronecker product. We provide an algorithm that uses $n^{ω( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a}$ amortized update time and $ n^{ω( \lceil(k-s)/2 \rceil, \lfloor (k-s)/2 \rfloor,a )}$ worst case query time for dynamic kronecker product problem. Unless tensor MV conjecture is false, there is no algorithm that can use both $n^{ω( \lceil k/2 \rceil, \lfloor k/2 \rfloor, a )-a-Ω(1)}$ amortized update time, and $ n^{ω( \lceil(k-s)/2 \rceil, \lfloor (k-s)/2 \rfloor,a )-Ω(1)}$ worst case query time.

Authors: Fedor V. Fomin, Petr A. Golovach, M. S. Ramanujan, Saket Saurabh

In the d-Euclidean Distance Matrix Completion (d-EDMC) problem, one aims to determine whether a given partial matrix of pairwise distances can be extended to a full Euclidean distance matrix in d dimensions. This problem is a cornerstone of computational geometry with numerous applications. While classical work on this problem often focuses on exploiting connections to semidefinite programming typically leading to approximation algorithms, we focus on exact algorithms and propose a novel distance-from-triviality parameterization framework to obtain tractability results for d-EDMC. We identify key structural patterns in the input that capture entry density, including chordal substructures and coverability of specified entries by fully specified principal submatrices. We obtain: (1) The first fixed-parameter algorithm (FPT algorithm) for d-EDMC parameterized by d and the maximum number of unspecified entries per row/column. This is achieved through a novel compression algorithm that reduces a given instance to a submatrix on O(1) rows (for fixed values of the parameters). (2) The first FPT algorithm for d-EDMC parameterized by d and the minimum number of fully specified principal submatrices whose entries cover all specified entries of the given matrix. This result is also achieved through a compression algorithm. (3) A polynomial-time algorithm for d-EDMC when both d and the minimum fill-in of a natural graph representing the specified entries are fixed constants. This result is achieved by combining tools from distance geometry and algorithms from real algebraic geometry. Our work identifies interesting parallels between EDM completion and graph problems, with our algorithms exploiting techniques from both domains.

Authors: Amanda Redlich

This paper analyzes a variation on the well-known "power of two choices" allocation algorithms. Classically, the smallest of $d$ randomly-chosen options is selected. We investigate what happens when the largest of $d$ randomly-chosen options is selected. This process generates a power-law-like distribution: the $i^{th}$-smallest value scales with $i^{d-1}$, where $d$ is the number of randomly-chosen options, with high probability. We give a formula for the expectation and show the distribution is concentrated around the expectation

Authors: Rudra Prakash, S. Janardhanan, Shaunak Sen

This paper analyses the computational complexity of validated interval methods for uncertain nonlinear systems. Interval analysis produces guaranteed enclosures that account for uncertainty and round-off, but its adoption is often limited by computational cost in high dimensions. We develop an algorithm-level worst-case framework that makes the dependence on the initial search volume $\mathrm{Vol}(X_0)$, the target tolerance $\varepsilon$, and the costs of validated primitives explicit (inclusion-function evaluation, Jacobian evaluation, and interval linear algebra). Within this framework, we derive worst-case time and space bounds for interval bisection, subdivision$+$filter, interval constraint propagation, interval Newton, and interval Krawczyk. The bounds quantify the scaling with $\mathrm{Vol}(X_0)$ and $\varepsilon$ for validated steady-state enclosure and highlight dominant cost drivers. We also show that determinant and inverse computation for interval matrices via naive Laplace expansion is factorial in the matrix dimension, motivating specialised interval linear algebra. Finally, interval Newton and interval Krawczyk have comparable leading-order costs; Krawczyk is typically cheaper in practice because it inverts a real midpoint matrix rather than an interval matrix. These results support the practical design of solvers for validated steady-state analysis in applications such as biochemical reaction network modelling, robust parameter estimation, and other uncertainty-aware computations in systems and synthetic biology.

Authors: Elvio G. Amparore

Range-Based Set Reconciliation (RBSR) synchronizes ordered sets by recursively comparing summaries of contiguous ranges and refining only the mismatching parts. While its communication complexity is well understood, its local computational cost fundamentally depends on the storage backend that must answer repeated range-summary, rank, and enumeration queries during refinement. We argue that a natural storage abstraction for RBSR implementations based on composable range aggregates is a \emph{range-summarizable order-statistics store} (RSOS): a dynamic ordered-set structure supporting composable summaries of contiguous ranges together with rank/select navigation. This identifies and formalizes the backend contract needed for efficient recursive refinement, combining range-summary support with order-statistics navigation for balanced partitioning. We then show that a specific augmentation of B\textsuperscript{+}-trees with subtree counts and composable summaries realizes a RSOS, and we derive corresponding bounds on local reconciliation work in this abstract storage model. Finally, we introduce AELMDB, an extension of LMDB that realizes this design inside a persistent memory-mapped engine, and evaluate it through an integration with Negentropy. The results show that placing the reconciliation oracle inside the storage tree substantially reduces local reconciliation cost on the evaluated reconciliation-heavy workloads compared with an open-source persistent baseline based on auxiliary tree caches, while the window-subrange ablation further confirms the usefulness of the systems optimizations built on top of the core aggregate representation.

Authors: Zeyu Ding, Katja Ickstadt, Nadja Klein, Alexander Munteanu, Simon Omlor

Efficient and scalable non-parametric or semi-parametric regression analysis and density estimation are of crucial importance to the fields of statistics and machine learning. However, available methods are limited in their ability to handle large-scale data. We address this issue by developing a novel coreset construction for multivariate conditional transformation models (MCTMs) to enhance their scalability and training efficiency. To the best of our knowledge, these are the first coresets for semi-parametric distributional models. Our approach yields substantial data reduction via importance sampling. It ensures with high probability that the log-likelihood remains within multiplicative error bounds of $(1\pm\varepsilon)$ and thereby maintains statistical model accuracy. Compared to conventional full-parametric models, where coresets have been incorporated before, our semi-parametric approach exhibits enhanced adaptability, particularly in scenarios where complex distributions and non-linear relationships are present, but not fully understood. To address numerical problems associated with normalizing logarithmic terms, we follow a geometric approximation based on the convex hull of input data. This ensures feasible, stable, and accurate inference in scenarios involving large amounts of data. Numerical experiments demonstrate substantially improved computational efficiency when handling large and complex datasets, thus laying the foundation for a broad range of applications within the statistics and machine learning communities.

Authors: Daisuke Shibatani, Yutaro Yamaguchi

We study an application of fair division theory to school redistricting. Procaccia, Robinson, and Tucker-Foltz (SODA 2024) recently proposed a mathematical model to generate redistricting plans that provide theoretically guaranteed fairness among demographic groups of students. They showed that an almost proportional allocation can be found by adding $O(g \log g)$ extra seats in total, where $g$ is the number of groups. In contrast, for three or more groups, adding $o(n)$ extra seats is not sufficient to obtain an almost envy-free allocation in general, where $n$ is the total number of students. In this paper, we focus on the case of two groups. We introduce a relevant relaxation of envy-freeness, termed 1-relaxed envy-freeness, which limits the capacity violation not in total but at each school to at most one. We show that there always exists a 1-relaxed envy-free allocation, which can be found in polynomial time.

 If I get an email offering me a $1000 for I DON"T KNOW SINCE  I ignore it and don't even bother looking for other signs it is a scam. 

If I get an email offering me $100  I may look more carefully and often they are legit (most common is to give a per-publication review of a math book---sometimes just questions, but more often a written report). 

Most offers I get are either $1000 or $100. Today I got one for $750 which inspired this post (I ignored the offer without checking). 

Which nets more people $100 or $1000?

1) If people are like me then $100 fools more people. But people like me will still CHECK CAREFULLY. I sometimes feed the email into ChatGPT for an opinion to see if it's a scam. (Spellcheck still things ChatGPT should be spelled catgut.) 

2) Are there people who would fill out the survey (or whatever) for $1000 but not for $100? I ask non rhetorically as always. Are such people more gullible?

Would scammers make more money if they offered $100 instead of $1000 ?

1) More people would fall for the $100 scam. Or maybe not---do some people not bother if it's only $100?

2) Depending on how they are scamming you, will they get less out of it if they only offer $100?

Here are types of scams:

1) They send you a check for $1000 + x and say WHOOPS- please email us a check for $x. I've heard of this in the Can you tutor our daughter in math? scam. For this one, offering $1000 nets the scammer more money since for $100+x, x will be smaller than $1000+x.

2) They want to harvest your personal information. For these I don't think they will gain more if they do 1000 vs 100. 

One more thought:

1) I said that for $100 I take it seriously but for $1000 I don't

2) I said that $750 I do not take it seriously.

3) What's the cutoff?  Obviously $289.

By gasarch

Sunday, 22 March, 2026 – 11:00 to 13:00

Today the seminar will take place via zoom. After Pesach we hope to make it in Ross 70 seminar room.  

Repeats every week every Sunday until the end of June 2026

Kazhdan Seminar: Boolean Functions, Hypercontractivity, and Applications

Instructors: Noam Lifshitz and Gil Kalai 

Abstract:

Boolean functions are central objects in combinatorics, complexity theory, probability theory, and other areas of mathematics and computer science. Fourier methods have come to play an important role in the analysis of Boolean functions, and so are hypercontractive inequalities. New hypercontractive inequalities for global functions were developed in the last decade and have led to the resolution of some long-standing problems and to further applications to group theory and representation theory.

Prerequisites

Basic knowledge of probability, linear algebra, and group theory.

Related blog posts: Joram’s seminar 2025: Hypercontractivity, Groups and Representations;  Noam Lifshitz: A new hypercontractivity inequality — The proof!; To cheer you up in difficult times 3: A guest post by Noam Lifshitz on the new hypercontractivity inequality of Peter Keevash, Noam Lifshitz, Eoin Long and Dor Minzer; Nati’s Influence.

---

Tentative Weekly Schedule

Part I: Classical Tools on the Hypercube

1. 1. Foundations: Fourier expansion on the Boolean cube , the Noise Operator, and Total Influence. Basic isoperimetric inequalities. 

• The Classical Hypercontractive Inequality: The Bonami-Beckner-Gross inequality. Proof and immediate consequences.

1. 1. Fundamental Theorems: The KKL Theorem (Kahn-Kalai-Linial) and Friedgut’s Junta Theorem. Variance bounds for first passage percolation. 

By Gil Kalai

Determining the randomized (or distributional) communication complexity of disjointness is a central problem in communication complexity, having roots in the foundational work of Babai, Frankl, and Simon in the 1980s and culminating in the famous works of Kalyanasundaram-Schnitger and Razborov in 1992. However, the question of obtaining tight bounds for product distributions persisted until the more recent work of Bottesch, Gavinsky, and Klauck resolved it. In this note we revisit this classical problem and give a short, streamlined proof of the best bounds, with improved quantitative dependence on the error parameter. Our approach is based on a simple combinatorial lemma that may be of independent interest: if two sets drawn independently from two distributions are disjoint with non-negligible probability, then one can extract two subfamilies of reasonably large measure that are fully cross-disjoint (equivalently, a large monochromatic rectangle for disjointness).

Glóð.ai, one of the two EU-cofunded projects in which my colleague Anna Liebel is involved, is opening a number of research opportunities in the area of AI and HPC. What follows is largely a copy-paste of text Anna wrote.

Reykjavik University has 5 PhD/Postdoc positions open for applications. Some of them are purely at the Department of Computer Science, several will be co-supervised between our department and our engineering colleagues, and one is even at the Department of Law.

If you or any of your (recent) students want to help my colleagues make the European AI infrastructure stronger and more competitive, work in research and support local businesses and the public sector with AI adoption and development, all while living in stunning Iceland, please apply!

Here are the links for the position announcements.

• https://jobs.50skills.com/ru/en/41631
• https://jobs.50skills.com/ru/en/41348
• https://jobs.50skills.com/ru/en/41225
• https://jobs.50skills.com/ru/en/41239
• https://jobs.50skills.com/ru/is/41088 (PhD in Law, in Icelandic)

By Luca Aceto

Fully Funded Joint PhD – BITS Pilani × RMIT (Trustworthy AI, LLMs, RAG) 2 positions on:

Unlearning & AI behavior
RAG + Right to be Forgotten

CS/AI/ML/Data/Math/EE
₹42.8k–45.8k + AUD stipend
India + Melbourne
Deadline: 28 Mar 2026

Website: https://www.linkedin.com/feed/update/urn:li:activity:7436398877475483648/
Email: yash.sinha@pilani.bits-pilani.ac.in

By shacharlovett

I’ve been wanting to write a summary of the Cultural AI conference I attended at NYU last week, but I’ve been struggling to succinctly capture my thoughts. That’s indicative of the depth and complexity of how AI meets culture, and the different perspectives and disciplines might not lend themselves to a tidy summary.1 As I often do when trying to wrap my head around complex things, I will stop worrying and just blog through it.

The talk that serves as my hub in the complex network of cultural AI is Cosma Shalizi’s “Aware of All Internet Traditions: Large Language Models as Information Retrieval and Synthesis.” That language models simultaneously retrieve information and synthesize new content isn’t controversial. Nor is the fact that this synthesis is formulaic. The current synthesis is next-token prediction trained on all written information, whose output is warped by some selective post-training. By design, language models mechanistically reproduce the recurring regularities in their training data. That training data consists of all the text files on the internet and what is easily available in printed books. Hence, the regularities are the tropes, stereotypes, templates, conventions, and genres of language and code.

The formulaic generation of discourse looks like discourse in ways we could never have imagined. But with hindsight, we shouldn’t be surprised. Human culture is very formulaic! There are long-standing formulas for oral tradition, for generating small talk, or for generating scientific papers. As Cosma put it, in the single sentence that summarizes the entire Cultural AI conference:2

Not having to think is often a good thing! Tradition lets us externalize certain processes so we can focus on other tasks. Formalities strengthen cultural connections. Traditions in communication help us understand each other better and come to consensus faster.

Indeed, our vast externalized cultural intelligence is the jewel of human tradition. Cosma cites Jacques Barzun’s conception of the House of Intellect: intellect is the communal form of society’s intelligence. “[I]t is intelligence stored up and made into habits of discipline, signs and symbols of meaning, chains of reasoning and spurs to emotion — a shorthand and a wireless by which the mind can skip connectives, recognize ability, and communicate truth.” According to Barzun, intellect lets society share and externalize knowledge. It belongs to society, not any individual. It connects individual intelligences. It lives after any single intelligence dies.

GenAI is the mechanization of this intellect. It is the mechanization of all of our traditions.

With James Evans, Henry Farrell, and Alison Gopnik, Cosma has been preaching the gospel that AI is a cultural technology for several years. He’s gone through several iterations of what that means and what it implies, but mechanized tradition is the characterization that resonates most with me. Mechanized tradition of Barzun’s artificial intellect is a far better description of GenAI technology than “artificial intelligence.” This frame helps us get away from the silly C-suite sci-fi navel-gazing about the personalities inside the data centers. Claude is not a person. It is a mechanized intellect. A Lore Laundering Machine. The frame of mechanized tradition helps me build a social metascience of our LLM condition.

Let me give you a fun example.

In the same session as Cosma, Wouter Haverals gave a rhizomatic inspection of the tradition of literary style. What is style anyway? We love to ask LLMs to write in new styles. It’s funny to have it generate poetry. One of my most common queries is how to rewrite emails to sound less angry.

But humans are also great at mimicking style. It can be a fun, creative game to do the sort of rewriting we now task AI with. And our audience can all tell when something hits or misses the mark when we very ape a particular tradition.

Wouter introduced Raymond Queneau’s Exercices de Style, a book consisting of 99 rewritings of the same story in different styles. The main story is simple enough. Here’s Barbara Wright’s 1958 translation:

In the S bus, in the rush hour. A chap of about 26, felt hat with a cord instead of a ribbon, neck too long, as if someone’s been having a tug-of-war with it. People getting off. The chap in question gets annoyed with one of the men standing next to him. He accuses him of jostling him every time anyone goes past. A snivelling tone which is meant to be aggressive. When he sees a vacant seat he throws himself on to it.

Two hours later, I meet him in the Cour de Rome, in front of the gare Saint-Lazare. He’s with a friend who’s saying: You ought to get an extra button put on your overcoat.” He shows him where (at the lapels) and why.

Queneau rewrites this story in the past and the present. In reported speech. In the passive voice. In haiku.

Summer S long neck

plait hat toes abuse retreat

station button friend

Obviously, you can feed an LLM Queneau’s original story and prompt it to write in each of the prescribed styles. Can LLM capture the style? How could you know that LLM did a good job?

The only way to answer such questions is to lean on the tradition of vulgar positivism. In a delightfully recursive metanarrative,3 Wouter and his co-author Meredith Martin ran a survey experiment. On the platform Prolific, they asked “real people” a series of questions about Wright’s translations of Queneau’s original stories and AI-generated versions. They ran several variants. In one, mimicking the style of Kevin Roose’s mechanistically obnoxious New York Times quiz last week, they didn’t tell the participants how the two stories were generated and simply asked which better captured the style. In the second variant, they asked which captured the style better when the participants knew whether the story was Queneau’s or AI’s. In the third experiment, they asked for preferences with the true labels switched.

What happened next was entirely predictable. Without attribution of authorship, the “people” on Prolific slightly preferred the AI version, choosing the AI 55% of the time. There were 186 participants and 930 pairwise judgments, so statistical tradition would spew out a confidence interval somewhere between 3 and 7 percentage points wide, depending on the pedantry of Reviewer 2. Make of that what you will. On the other hand, with the correct labels, “people” only chose the AI 48% of the time. Most hilariously, when the labels were swapped, “people” chose what they thought was human 62% of the time.

To situate these numbers within our broader house of intellectual tradition, Haverals and Martin adopted a recently instituted social-scientific tradition: silicon sampling. They ran a survey experiment where the participants were LLMs. When prompted with the same survey, LLMs chose AI-writing 50% of the time without labels. But with the correct labels, the machines judged Queneau superior 70% of the time. And with the swapped labels, AI chose what was presented as Queneau 64% of the time. As the title of Wouter and Meredith’s paper says, “Everyone prefers human writers, even AI.”

There’s nothing surprising in these survey results, and that shouldn’t be surprising. Survey experiments are a woefully limited way to understand the social condition. They are completely mechanical. Of course, this sort of impoverished social science can be done by mechanical literary analysis. Silicon-sampled survey experiments enable us to mechanically generate stories from illusory correlations. These stories are interpreted traditionally as either informative or absurd, depending on the academic tradition in which you were raised. The recursion continues indefinitely. There are so many patterns and regularities in human behavior, and by simulating common text strings, we get text conforming to these regularities. To rephrase Nelson Goodman, regularities are where you find them, and in human tradition, you find them everywhere.

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By Ben Recht

Authors: Yuhan Ye, Saurabh Amin, Asuman Ozdaglar

We study how to construct compressed datasets that suffice to recover optimal decisions in linear programs with an unknown cost vector $c$ lying in a prior set $\mathcal{C}$. Recent work by Bennouna et al. provides an exact geometric characterization of sufficient decision datasets (SDDs) via an intrinsic decision-relevant dimension $d^\star$. However, their algorithm for constructing minimum-size SDDs requires solving mixed-integer programs. In this paper, we establish hardness results showing that computing $d^\star$ is NP-hard and deciding whether a dataset is globally sufficient is coNP-hard, thereby resolving a recent open problem posed by Bennouna et al. To address this worst-case intractability, we introduce pointwise sufficiency, a relaxation that requires sufficiency for an individual cost vector. Under nondegeneracy, we provide a polynomial-time cutting-plane algorithm for constructing pointwise-sufficient decision datasets. In a data-driven regime with i.i.d.\ costs, we further propose a cumulative algorithm that aggregates decision-relevant directions across samples, yielding a stable compression scheme of size at most $d^\star$. This leads to a distribution-free PAC guarantee: with high probability over the training sample, the pointwise sufficiency failure probability on a fresh draw is at most $\tilde{O}(d^\star/n)$, and this rate is tight up to logarithmic factors. Finally, we apply decision-sufficient representations to contextual linear optimization, obtaining compressed predictors with generalization bounds scaling as $\tilde{O}(\sqrt{d^\star/n})$ rather than $\tilde{O}(\sqrt{d/n})$, where $d$ is the ambient cost dimension.

Authors: Thomas Bläsius, Emil Dohse, Deborah Haun, Laura Merker

Hyperbolic uniform disk graphs (HUDGs) are intersection graphs of disks with some radius $r$ in the hyperbolic plane, where $r$ may be constant or depend on the number of vertices in a family of HUDGs. We show that HUDGs with constant clique number do not admit \emph{product structure}, i.e., that there is no constant $c$ such that every such graph is a subgraph of $H \boxtimes P$ for some graph $H$ of treewidth at most $c$. This justifies that HUDGs are described as not having a grid-like structure in the literature, and is in contrast to unit disk graphs in the Euclidean plane, whose grid-like structure is evident from the fact that they are subgraphs of the strong product of two paths and a clique of constant size [Dvořák et al., '21, MATRIX Annals]. By allowing $H$ to be any graph of constant treewidth instead of a path-like graph, we reject the possibility of a grid-like structure not merely by the maximum degree (which is unbounded for HUDGs) but due to their global structure. We complement this by showing that for every (sub-)constant $r$, HUDGs admit product structure, whereas the typical hyperbolic behavior is observed if $r$ grows with the number of vertices. Our proof involves a family of $n$-vertex HUDGs with radius $\log n$ that has bounded clique number but unbounded treewidth, and one for which the ratio of treewidth and clique number is $\log n / \log \log n$. Up to a $\log \log n$ factor, this negatively answers a question raised by Bläsius et al. [SoCG '25] asking whether balanced separators of HUDGs with radius $\log n$ can be covered by less than $\log n$ cliques. Our results also imply that the local and layered tree-independence number of HUDGs are both unbounded, answering an open question of Dallard et al. [arXiv '25].

Authors: Sunil Arya, David M. Mount

We prove polarity duality for covering problems in Hilbert geometry. Let $G$ and $K$ be convex bodies in $\mathbb{R}^d$ where $G \subset \operatorname{int}(K)$ and $\operatorname{int}(G)$ contains the origin. Let $N^H_K(G,α)$ and $S^H_K(G,α)$ denote, respectively, the minimum numbers of radius-$α$ Hilbert balls in the geometry induced by $K$ needed to cover $G$ and $\partial G$. Our main result is a Hilbert-geometric analogue of the König-Milman covering duality: there exists an absolute constant $c \geq 1$ such that for any $α\in (0,1]$, \[ c^{-d}\,N^H_{G^{\circ}}(K^{\circ},α) ~ \leq ~ N^H_K(G,α) ~ \leq ~ c^{d}\,N^H_{G^{\circ}}(K^{\circ},α), \] and likewise, \[ c^{-d}\,S^H_{G^{\circ}}(K^{\circ},α) ~ \leq ~ S^H_K(G,α) ~ \leq ~ c^{d}\,S^H_{G^{\circ}}(K^{\circ},α). \] We also recover the classical volumetric duality for translative coverings of centered convex bodies, and obtain a new boundary-covering duality in that setting. The Hilbert setting is subtler than the translative one because the metric is not translation invariant, and the local Finsler unit ball depends on the base point. The proof involves several ideas, including $α$-expansions, a stability lemma that controls the interaction between polarity and expansion, and, in the boundary case, a localized relative isoperimetric argument combined with Holmes--Thompson area estimates. In addition, we provide an alternative proof of Faifman's polarity bounds for Holmes--Thompson volume and area in the Funk and Hilbert geometries.

Authors: Haisheng Zhu, Taotao He, Mohit Tawarmalani

We present a novel relaxation framework for general mixed-integer nonlinear programming (MINLP) grounded in computational geometry. Our approach constructs polyhedral relaxations by convexifying finite sets of strategically chosen points, iteratively refining the approximation to converge toward the simultaneous convex hull of factorable function graphs. The framework is underpinned by three key contributions: (i) a new class of explicit inequalities for products of functions that strictly improve upon standard factorable and composite relaxation schemes; (ii) a proof establishing that the simultaneous convex hull of multilinear functions over axis-aligned regions is fully determined by their values at corner points, thereby generalizing existing results from hypercubes to arbitrary axis-aligned domains; and (iii) the integration of computational geometry tools, specifically voxelization and QuickHull, to efficiently approximate feasible regions and function graphs. We implement this framework and evaluate it on randomly generated polynomial optimization problems and a suite of 619 instances from \texttt{MINLPLib}. Numerical results demonstrate significant improvements over state-of-the-art benchmarks: on polynomial instances, our relaxation closes an additional 20--25\% of the optimality gap relative to standard methods on half the instances. Furthermore, compared against an enhanced factorable programming baseline and Gurobi's root-node bounds, our approach yields superior dual bounds on approximately 30\% of \texttt{MINLPLib} instances, with roughly 10\% of cases exhibiting a gap reduction exceeding 50\%.

Authors: Patrick Loiseau, Simon Mauras, Minrui Xu

We study auction design in the celebrated interdependence model introduced by Milgrom and Weber [1982], where a mechanism designer allocates a good, maximizing the value of the agent who receives it, while inducing truthfulness using payments. In the lesser-studied procurement auctions, one allocates a chore, minimizing the cost incurred by the agent selected to perform it. Most of the past literature in theoretical computer science considers designing truthful mechanisms with constant approximation for the value setting, with restricted domains and monotone valuation functions. In this work, we study the general computational problems of optimizing the approximation ratio of truthful mechanism, for both value and cost, in the deterministic and randomized settings. Unlike most previous works, we remove the domain restriction and the monotonicity assumption imposed on value functions. We provide theoretical explanations for why some previously considered special cases are tractable, reducing them to classical combinatorial problems, and providing efficient algorithms and characterizations. We complement our positive results with hardness results for the general case, providing query complexity lower bounds, and proving the NP-Hardness of the general case.

Authors: Sitan Chen, Jingqiu Ding, Mahbod Majid, Walter McKelvie

Bayesian methods lie at the heart of modern data science and provide a powerful scaffolding for estimation in data-constrained settings and principled quantification and propagation of uncertainty. Yet in many real-world use cases where these methods are deployed, there is a natural need to preserve the privacy of the individuals whose data is being scrutinized. While a number of works have attempted to approach the problem of differentially private Bayesian estimation through either reasoning about the inherent privacy of the posterior distribution or privatizing off-the-shelf Bayesian methods, these works generally do not come with rigorous utility guarantees beyond low-dimensional settings. In fact, even for the prototypical tasks of Gaussian mean estimation and linear regression, it was unknown how close one could get to the Bayes-optimal error with a private algorithm, even in the simplest case where the unknown parameter comes from a Gaussian prior. In this work, we give the first efficient algorithms for both of these problems that achieve mean-squared error $(1+o(1))\mathrm{OPT}$ and additionally show that both tasks exhibit an intriguing computational-statistical gap. For Bayesian mean estimation, we prove that the excess risk achieved by our method is optimal among all efficient algorithms within the low-degree framework, yet is provably worse than what is achievable by an exponential-time algorithm. For linear regression, we prove a qualitatively similar lower bound. Our algorithms draw upon the privacy-to-robustness framework of arXiv:2212.05015, but with the curious twist that to achieve private Bayes-optimal estimation, we need to design sum-of-squares-based robust estimators for inherently non-robust objects like the empirical mean and OLS estimator. Along the way we also add to the sum-of-squares toolkit a new kind of constraint based on short-flat decompositions.

Authors: Alexander Munteanu, Simon Omlor, Jeff M. Phillips

We establish new exponential in dimension lower bounds for the Maximum Halfspace Discrepancy problem, which models linear classification. Both are fundamental problems in computational geometry and machine learning in their exact and approximate forms. However, only $O(n^d)$ and respectively $\tilde O(1/\varepsilon^d)$ upper bounds are known and complemented by polynomial lower bounds that do not support the exponential in dimension dependence. We close this gap up to polylogarithmic terms by reduction from widely-believed hardness conjectures for Affine Degeneracy testing and $k$-Sum problems. Our reductions yield matching lower bounds of $\tildeΩ(n^d)$ and respectively $\tildeΩ(1/\varepsilon^d)$ based on Affine Degeneracy testing, and $\tildeΩ(n^{d/2})$ and respectively $\tildeΩ(1/\varepsilon^{d/2})$ conditioned on $k$-Sum. The first bound also holds unconditionally if the computational model is restricted to make sidedness queries, which corresponds to a widely spread setting implemented and optimized in many contemporary algorithms and computing paradigms.

Authors: Oswin Aichholzer, Joseph Dorfer, Sándor P. Fekete, Phillip Keldenich, Peter Kramer, Stefan Schirra

We give an overview of the 2026 Computational Geometry Challenge targeting the problem of finding a Central Triangulation under Parallel Flip Operations in triangulations of point sets. A flip is the parallel exchange of a set of edges in a triangulation with opposing diagonals of the convex quadrilaterals containing them. The challenge objective was, given a set of triangulations of a fixed point set, to determine a central triangulation with respect to parallel flip distances. More precisely, this asks for a triangulation that minimizes the sum of flip distances to all elements of the input

Authors: C. S. Elder, Guillaume Marçais, Carl Kingsford

We study Turnpike with uncertain measurements: reconstructing a one-dimensional point set from an unlabeled multiset of pairwise distances under bounded noise and rounding. We give a combinatorial characterization of realizability via a multi-matching that labels interval indices by distinct distance values while satisfying all triangle equalities. This yields an ILP based on the triangle equality whose constraint structure depends only on the two-partition set $\mathcal{P}_y=\{(r,s,t): y_r+y_s=y_t\}$ and a natural LP relaxation with $\{0,1\}$-coefficient constraints. Integral solutions certify realizability and output an explicit assignment matrix, enabling an assignment-first, regression-second pipeline for downstream coordinate estimation. Under bounded noise followed by rounding, we prove a deterministic separation condition under which $\mathcal{P}_y$ is recovered exactly, so the ILP/LP receives the same combinatorial input as in the noiseless case. Experiments illustrate integrality behavior and degradation outside the provable regime.

Authors: Rui Chai

We study online resource allocation among N interacting modules over T rounds. Unlike standard online optimization, costs are endogenous: they depend on the full allocation vector through an interaction matrix W encoding pairwise cooperation and competition. We analyze three paradigms: (I) uniform allocation (cost-ignorant), (II) gated allocation (cost-estimating), and (III) competitive allocation via multiplicative weights update with interaction feedback (cost-revealing). Our main results establish a strict separation under adversarial sequences with bounded variation: uniform incurs Omega(T) regret, gated achieves O(T^{2/3}), and competitive achieves O(sqrt(T log N)). The performance gap stems from competitive allocation's ability to exploit endogenous cost information revealed through interactions. We further show that W's topology governs a computation-regret tradeoff. Full interaction (|E|=O(N^2)) yields the tightest bound but highest per-step cost, while sparse topologies (|E|=O(N)) increase regret by at most O(sqrt(log N)) while reducing per-step cost from O(N^2) to O(N). Ring-structured topologies with both cooperative and competitive links - of which the five-element Wuxing topology is canonical - minimize the computation x regret product. These results provide the first formal regret-theoretic justification for decentralized competitive allocation in modular architectures and establish cost endogeneity as a fundamental challenge distinct from partial observability. Keywords: online learning, regret bounds, resource allocation, endogenous costs, interaction topology, multiplicative weights, modular systems, Wuxing topology

Authors: Danny Segev

The continuous-time joint replenishment problem has long served as a foundational inventory management model. Even though its unconstrained setting has seen recent algorithmic advances, the incorporation of resource constraints into this domain precludes the application of newly discovered synchronization techniques. Such constraints arise in a broad spectrum of practical environments where resource consumption is bounded as an aggregate rate over time. However, for nearly four decades, the prevailing approximation guarantee for resource-constrained joint replenishment has remained $\frac{ 1 }{ \ln 2 } \approx 1.4427$, achieved via classical power-of-$2$ policies. In this paper, we circumvent these structural policy restrictions by devising generalized rounding frameworks, demonstrating that a well-known convex relaxation is much tighter than previously established. In particular, we expand our analytical scope to encompass fractional base expansion factors, randomized shifting, and staggered interleaved grids. Through this multifaceted methodology, we present a sequence of gradually improving performance guarantees. First, by proposing a best-of-two framework that exploits structural asymmetries between deterministic power-of-$m^{1/k}$ policies, we surpass the classical barrier to obtain a $1.3776$-approximation. Second, by injecting a random shift into the logarithmic grid domain and formulating a factor-revealing linear program to optimize a dual-policy approach, we attain a $1.2512$-approximation. Finally, by superimposing a secondary offset grid to subdivide rounding intervals and suppress holding cost inflation, we utilize interleaved policies to arrive at our ultimate approximation ratio of $\frac{5}{6\ln 2} \approx 1.2023$, which is proven to be best-possible for the class of interleaved power-of-$m^{1/k}$ policies.

Authors: Jean-Guillaume Dumas, Clément Pernet, Alexandre Sedoglavic

We propose a more accurate variant of an algorithm for multiplying 4x4 matrices using 48 multiplications over any ring containing an inverse of 2. This algorithm has an error bound exponent of only log 4 $γ$$\infty$,2 $\approx$ 2.386. It also reaches a better accuracy w.r.t. max-norm in practice, when compared to previously known such fast algorithms. Furthermore, we propose a straight line program of this algorithm, giving a leading constant in its complexity bound of 387 32 n 2+log 4 3 + o n 2+log 4 3 operations over any ring containing an inverse of 2. Introduction: An algorithm to multiply two 4x4 complex-valued matrices requiring only 48 non-commutative multiplications was introduced in [16] 1 using a pipeline of large language models orchestrated by an evolutionary coding agent. A matrix multiplication algorithm with that many non-commutative multiplications is denoted by ___4x4x4:48___ in the sequel. An equivalent variant of the associated tensor decomposition defining this algorithm, but over the rationals (more precisely over any ring containing an inverse of 2), was then given in [8]. Most error analysis of sub-cubic time matrix multiplication algorithms [3, 4, 2, 1, 17] are given in the max-norm setting: bounding the largest output error as a function of the max-norm product of the vectors of input matrix coefficients. In this setting, Strassen's algorithm has shown the best accuracy bound, (proven minimal under some assumptions in [2]). In [6, 8], the authors relaxed this setting by shifting the focus to the 2-norm for input and/or output; that allowed them to propose a ___2x2x2:7___ variant with an improved accuracy bound. Experiments show that this variant performs best even when measuring the max-norm of the error bound. We present in this note a variant of the recent ___4x4x4:48___ algorithm over the rationals (again in the same orbit under De Groot isotropies [10]) that is more numerically accurate w.r.t. max-norm in practice. In particular, our new variant improves on the error bound exponent, from log 2 $γ$ $\infty$,2 $\approx$ 2.577 Consider the product of an M x K matrix A by a K x N matrix B. It is computed by a ___m, k, n___ algorithm represented by the matrices L, R, P applied recursively on ${\ell}$ recursive levels and the resulting m 0 x k 0 by k 0 x n 0 products are performed using an algorithm $β$. Here M = m 0 m ${\ell}$ , K = k 0 k ${\ell}$ and n = n 0 n ${\ell}$ . The accuracy bound below uses any (possibly different) p-norms and q-norms for its left-handside, ___$\bullet$___ p and right-hand side, ___$\bullet$___ q . The associated dual norms, are denoted by ___$\bullet$___ p $\star$ and ___$\bullet$___ q $\star$ respectively. Note that, these are vector norms, hence ___A___ p for matrix A in R mxn denotes ___Vect(A)___ p and is the p-norm of the mn dimensional vector of its coefficients, and not a matrix norm.

Authors: Eduar Castrillo Velilla

In this paper we study the single-deletion variant $Δ$-DRESS, part of the broader DRESS framework. We demonstrate empirically that $Δ$-DRESS, a single level of vertex deletion applied to the DRESS graph fingerprint, achieves unique fingerprints within each tested SRG parameter family across all 51,718 non-isomorphic strongly regular graphs (SRGs) considered, spanning 16 parameter families: the complete Spence collection (12 families, 43,703 graphs on up to 64 vertices) plus four additional SRG families with up to 4,466 graphs per family. Combined with 18 additional hard graph families (102 graphs including Miyazaki, Chang, Paley, Latin square, and Steiner constructions), $Δ$-DRESS achieves 100% within-family separation across 34 benchmark families covering 51,816 distinct graph instances, implicitly resolving over 576 million within-family non-isomorphic pairs. Moreover, the classical Rook $L_2(4)$ vs. Shrikhande pair, SRG(16,6,2,2), is known to be indistinguishable by the original 3-WL algorithm, yet $Δ$-DRESS separates it, proving that $Δ$-DRESS escapes the theoretical boundaries of 3-WL. The method runs in polynomial time $\mathcal{O}(n \cdot I \cdot m \cdot d_{\max})$ per graph; a streamed implementation of the combined fingerprint uses $\mathcal{O}(m + B + n)$ memory, where $B$ is the number of histogram bins, while the experiments reported here additionally retain the full deleted-subgraph multiset matrix for post-hoc analysis.

Authors: Kangyi Tian, Mingyu Xiao

The Kidney Exchange Problem is a prominent challenge in healthcare and economics, arising in the context of organ transplantation. It has been extensively studied in artificial intelligence and optimization. In a kidney exchange, a set of donor-recipient pairs and altruistic donors are considered, with the goal of identifying a sequence of exchange -- comprising cycles or chains starting from altruistic donors -- such that each donor provides a kidney to the compatible recipient in the next donor-recipient pair. Due to constraints in medical resources, some limits are often imposed on the lengths of these cycles and chains. These exchanges create a network of transplants aimed at maximizing the total number, $t$, of successful transplants. Recently, this problem was deterministically solved in $O^*(14.34^t)$ time (IJCAI 2024). In this paper, we introduce the representative set technique for the Kidney Exchange Problem, showing that the problem can be deterministically solved in $O^*(6.855^t)$ time.

Authors: Mingda Qiao

The distance from calibration, introduced by Błasiok, Gopalan, Hu, and Nakkiran (STOC 2023), has recently emerged as a central measure of miscalibration for probabilistic predictors. We study the fundamental problems of computing and estimating this quantity, given either an exact description of the data distribution or only sample access to it. We give an efficient algorithm that exactly computes the calibration distance when the distribution has a uniform marginal and noiseless labels, which improves the $O(1/\sqrt{|\mathcal{X}|})$ additive approximation of Qiao and Zheng (COLT 2024) for this special case. Perhaps surprisingly, the problem becomes $\mathsf{NP}$-hard when either of the two assumptions is removed. We extend our algorithm to a polynomial-time approximation scheme for the general case. For the estimation problem, we show that $Θ(1/ε^3)$ samples are sufficient and necessary for the empirical calibration distance to be upper bounded by the true distance plus $ε$. In contrast, a polynomial dependence on the domain size -- incurred by the learning-based baseline -- is unavoidable for two-sided estimation. Our positive results are based on simple sparsifications of both the distribution and the target predictor, which significantly reduce the search space for computation and lead to stronger concentration for the estimation problem. To prove the hardness results, we introduce new techniques for certifying lower bounds on the calibration distance -- a problem that is hard in general due to its $\textsf{co-NP}$-completeness.

Authors: Jeanne Clelland, Kristopher Tapp

We develop effective methods for constructing an ensemble of district plans via independent sampling from a reasonable probability distribution on the space of graph partitions. We compare the performance of our algorithms to that of standard Markov Chain based algorithms in the context of grid graphs and state congressional and legislative maps. For the case of perfect population balance between districts, we provide an explicit description of the distribution from which our method samples.

Authors: Martin Nägele, Christian Nöbel, Rico Zenklusen

The odd-red bipartite perfect matching problem asks to find a perfect matching containing an odd number of red edges in a given red-blue edge-colored bipartite graph. While this problem lies in $\mathsf{P}$, its polyhedral structure remains elusive, despite renewed attention to achieving better polyhedral understanding, nurtured by recent advances from two complementary angles. Apart from being a special case of bimodular integer programs, whose polyhedral structure is also badly understood, it is related to one of the most notorious open derandomization questions in theoretical computer science: whether there is a deterministic efficient algorithm for the exact bipartite perfect matching problem, which asks to find a perfect matching with exactly $k$ red edges. Recent progress towards deterministic algorithms for this problem crucially relies on a good polyhedral understanding. Motivated by this, Jia, Svensson, and Yuan show that the extension complexity of the exact bipartite perfect matching polytope is exponential in general. Interestingly, their result is true even for the easier odd-red bipartite perfect matching problem. For this problem, they introduce an exponential-size relaxation and leave open whether it is an exact description. Apart from showing that this description is not exact and even hard to separate over, we show, more importantly, that the red-odd bipartite perfect matching polytope exhibits complex facet structure: any exact description needs constraints with large and diverse coefficients. This rules out classical relaxations based on constraints with all coefficients in $\{0,\pm1\}$, such as the above-mentioned one, and suggests that significant deviations from prior approaches may be needed to obtain an exact description. More generally, we obtain that also polytopes corresponding to bimodular integer programs have complex facet structure.

Authors: Anish Hebbar, Rong Ge, Amit Kumar, Debmalya Panigrahi

The field of learning-augmented algorithms seeks to use ML techniques on past instances of a problem to inform an algorithm designed for a future instance. In this paper, we introduce a novel model for learning-augmented algorithms inspired by online learning. In this model, we are given a sequence of instances of a problem and the goal of the learning-augmented algorithm is to use prior instances to propose a solution to a future instance of the problem. The performance of the algorithm is measured by its average performance across all the instances, where the performance on a single instance is the ratio between the cost of the algorithm's solution and that of an optimal solution for that instance. We apply this framework to the classic $k$-median clustering problem, and give an efficient learning algorithm that can approximately match the average performance of the best fixed $k$-median solution in hindsight across all the instances. We also experimentally evaluate our algorithm and show that its empirical performance is close to optimal, and also that it automatically adapts the solution to a dynamically changing sequence.

The Department of Computer Science at the University of Warwick invite applications at the ranking of Assistant/Associate Professor. Preference will be given to outstanding candidates with an independent research agenda who will complement existing strengths in the Department of Computer Science. We are especially interested in strengthening the Division of Theory and Foundations and DIMAP.

Website: https://warwick.ac.uk/fac/cross_fac/dimap/focs_post_2026/
Email: igorcarb@gmail.com

By shacharlovett

Authors: Samuel Bismuth, Erel Segal-Halevi

We present an approximation notion for NP-hard optimization problems represented by binary functions. We prove that (assuming P != NP) the new notion is strictly stronger than FPTAS, but strictly weaker than having a polynomial-time algorithm.

Authors: Sergey S. Ketkov, Oleg A. Prokopyev

A standard approach to solving optimistic bilevel linear programs (BLPs) is to replace the lower-level problem with its Karush-Kuhn-Tucker (KKT) optimality conditions and reformulate the resulting complementarity constraints using auxiliary binary variables. This yields a single-level mixed-integer linear programming (MILP) model involving big-$M$ parameters. While sufficiently large and bilevel-correct big-$M$s can be computed in polynomial time, verifying a priori that given big-$M$s do not cut off any feasible or optimal lower-level solutions is known to be computationally difficult. In this paper, we establish two complementary hardness results. First, we show that, even with a single potentially incorrect big-$M$ parameter, it is $coNP$-complete to verify a posteriori whether the optimal solution of the resulting MILP model is bilevel optimal. In particular, this negative result persists for min-max problems without coupling constraints and applies to strong-duality-based reformulations of mixed-integer BLPs. Second, we show that verifying global big-$M$ correctness remains computationally difficult a posteriori, even when an optimal solution of the MILP model is available.

Authors: Weikun K. Zhang, Rohan Pandey, Bhaumik Mehta, Kaijie Jin, Naomi Morato, Archit Ganapule, Michael Ruofan Zeng, Jarod Alper

Motivated by auto-proof generation and Valiant's VP vs. VNP conjecture, we study the problem of discovering efficient arithmetic circuits to compute polynomials, using addition and multiplication gates. We formulate this problem as a single-player game, where an RL agent attempts to build the circuit within a fixed number of operations. We implement an AlphaZero-style training loop and compare two approaches: Proximal Policy Optimization with Monte Carlo Tree Search (PPO+MCTS) and Soft Actor-Critic (SAC). SAC achieves the highest success rates on two-variable targets, while PPO+MCTS scales to three variables and demonstrates steady improvement on harder instances. These results suggest that polynomial circuit synthesis is a compact, verifiable setting for studying self-improving search policies.

Authors: A. Ramos-Cisneros, M. Skopenkov, H. Pottmann

We study analogs of planar-quadrilateral meshes in Laguerre sphere geometry and the approximation of smooth surfaces by them. These new Laguerre meshes can be viewed as watertight surfaces formed by planar quadrilaterals (corresponding to the vertices of a mesh), strips of right circular cones (representing the edges), and spherical faces. In the smooth limit, we get an analog of conjugate nets in Laguerre geometry, which we call Laguerre conjugate nets with respect to an attached sphere congruence. We introduce the notion of Laguerre conjugate directions, provide a method for computing them, and apply them to approximate surfaces by L-meshes with prescribed radii of spherical faces.

Authors: Giordano Da Lozzo, Fabrizio Frati, Ignaz Rutter

In 1999, Heath, Pemmaraju, and Trenk [SIAM J. Comput. 28(4), 1999] extended the classic notion of book embeddings to digraphs, introducing the concept of upward book embeddings, in which the vertices must appear along the spine in a topological order and the edges are partitioned into pages, so that no two edges in the same page cross. For a partitioned digraph $G=(V,\bigcup^k_{i=1} E_i)$, that is, a digraph whose edge set is partitioned into $k$ subsets, an upward book embedding is required to assign edges to pages as prescribed by the given partition. In a companion paper, Heath and Pemmaraju [SIAM J. Comput 28(5), 1999] proved that the problem of testing the existence of an upward book embedding of a partitioned digraph is linear-time solvable for $k=1$ and recently Akitaya, Demaine, Hesterberg, and Liu [GD, 2017] have shown the problem NP-complete for $k\geq 3$. In this paper, we study upward book embeddings of partitioned digraphs and focus on the unsolved case $k=2$. Our first main result is a novel characterization of the upward embeddings that support an upward book embedding in two pages. We exploit this characterization in several ways, and obtain a rich picture of the complexity landscape of the problem. First, we show that the problem remains NP-complete when $k=2$, thus closing the complexity gap for the problem. Second, we show that, for an $n$-vertex partitioned digraph $G$ with a prescribed planar embedding, the existence of an upward book embedding of $G$ that respects the given planar embedding can be tested in $O(n \log^3 n)$ time. Finally, leveraging the SPQ(R)-tree decomposition of biconnected graphs into triconnected components, we present a cubic-time testing algorithm for biconnected directed partial $2$-trees.

Authors: Michał Szyfelbein

We consider the following generalization of the classic Binary Search Problem: a searcher is required to find a hidden target vertex $x$ in a graph $G$, by iteratively performing queries about vertices. A query to $v$ incurs a cost $c(v, x)$ and responds whether $v=x$ and if not, returns the connected component in $G-v$ containing $x$. The goal is to design a search strategy that minimizes the average-case search cost. Firstly, we consider the case when the cost of querying a vertex is independent of the target. We develop a $\br{4+ε}$-approximation FPTAS for trees running in $O(n^4/ε^2)$ time and an $O({\sqrt{\log n}})$-approximation for general graphs. Additionally, we give an FPTAS parametrized by the number of non-leaf vertices of the graph. On the hardness side we prove that the problem is NP-hard even when the input is a tree with bounded degree or bounded diameter. Secondly, we consider trees and assume $c(v, x)$ to be a monotone non-decreasing function with respect to $x$, i.e.\ if $u \in P_{v, x}$ then $c(u, x) \leq c(v, x)$. We give a $2$-approximation algorithm which can also be easily altered to work for the worst-case variant. This is the first constant factor approximation algorithm for both criterions. Previously known results only regard the worst-case search cost and include a parametrized PTAS as well as a $4$-approximation for paths. At last, we show that when the cost function is an arbitrary function of the queried vertex and the target, then the problem does not admit any constant factor approximation under the UGC, even when the input tree is a star.

Authors: Yiyi Cai

Experimental implementations of Hamiltonian dynamics are often affected by dissipative noise arising from interactions with the environment. This raises the question of whether one can detect the presence or absence of such dissipation using only access to the observed time evolution of the system. We consider the following decision problem: given black-box access to the time-evolution channels $e^{t\mathcal{L}}$ generated by an unknown time-independent Lindbladian $\mathcal{L}$, determine whether the dynamics are purely Hamiltonian or contain dissipation of magnitude at least $ε$ in normalized Frobenius norm. We give a randomized procedure that solves this task using total evolution time $\mathcal{O}(ε^{-1})$, which is information-theoretically optimal. This guarantee holds under the assumptions that the Lindblad generator has bounded strength and its dissipative part is of constant locality with bounded degree. Our work provides a practical method for detecting dissipative noise in experimentally implemented quantum dynamics.