arXiv subjects
Publications and source records attributed to Václav Rozhoň.
Suppose that each of $n$ players rolls a different die and the players are ordered by their rolls. Can the dice use disjoint labels such that ties are impossible, while all $n!$ orders remain equally likely? We give two general constructions of such dice and solve some related problems.
Large language models are increasingly contributing to mathematical research, where progress often depends on efficient proof search, incremental improvements and careful verification. We describe Bolzano, a multi-agent open-source system that uses parallel prover agents with a verifier agent and maintains a human-readable research state. Initial manual use on expert-selected problems yielded 8 results whose proofs were checked by domain experts. Motivated by these case studies, we ran Bolzano without problem-specific human guidance on about 3,800 open problems extracted from four sets of papers, solving about 200 open problems. One experiment used papers accepted to STOC 2026, a top conference in theoretical computer science. There, we answered four questions raised in the papers, as confirmed by their authors.
Can structural knowledge about a hash function help accelerate the (black box) detection of collisions in it? This question is fundamental to cryptography theory given the importance of collision-resistant hash functions, and in this paper we tackle it from the angle of instance optimality, an ultimate notion of beyond worst case algorithm analysis that has gained significant traction in recent years. Instance optimality asks for a single algorithm that, on every input, performs nearly as well as the best correct algorithm that ``knows the structure'' of that specific input. Here we measure algorithms by the number of queries they make to the hash function $f\colon [n]\to [n]$, and we say that an algorithm ``knows the structure'' of the input if, in addition to query access to $f$, it has free access to an unlabeled copy $π^{-1}\circ f\circπ$ of $f$, for an unknown permutation $π$ on $[n]$. We prove the existence of an (almost) instance-optimal algorithm for collision detection in the regime most interesting from a cryptographic perspective: among functions where finding a collision takes significantly less than $\sqrt{n}$ queries. Specifically, we prove the existence of a single algorithm $A$ that, for any input $f$ in which a structure-aware algorithm can find a collision using $q\leq O(\sqrt{n/\log n})$ queries in expectation, $A$ can find a collision in at most $O(q\log n)$ queries. The $O(\log n)$ multiplicative overhead is tight, matching a lower bound of Ben-Eliezer, Grossman, and Naor [ICALP'25], and partially resolving their main open question. Our result implies, in particular, that it is impossible for a cryptographic designer to plant purely structural backdoors for collision finding (for this unlabeled notion of structure): whatever collisions the designer's secret knowledge finds, the public can find with a multiplicative overhead of $O(\log n)$.
A machine faces many jobs whose lengths are hidden. Spending one unit of time to inspect a job may reveal a short job that should be finished now, or it may reveal nothing useful while every other job waits. When should the machine keep looking, and when should it start working? We study natural variants of this question and provide optimal algorithms in both the worst-case and instance-optimal frameworks. The resulting algorithms are often quite simple, which may make them useful in practice.
We show that every increasing union of Borel graphs of pointwise volume growth at most $\exp(O(r^γ))$ is hyperfinite, where $γ\approx 0.1523$ is the unique real root of $(1 - γ)^3 - 4γ= 0$. This implies a positive answer to the special case of Weiss's question on the hyperfiniteness of Borel actions of countable amenable groups, for countable amenable groups locally of volume growth $\exp(O(r^γ))$ for $γ$ as above. We also show that every bounded degree Borel graph of subexponential volume growth has Borel Følner tilings, improving a result of Downarowicz and Zhang for graphs generated by free Borel actions of groups of subexponential volume growth. Our main tools are the development of a Borel analogue of the two-parameter dimension growth function of a metric space as introduced by Dranishnikov and Sapir, and ball carving algorithms from theoretical computer science. We end with a pair of conjectures relating Borel amenability, Borel subexponential dimension growth, and hyperfiniteness, which would imply a positive answer to Weiss's question.
We construct a heap with strong beyond-worst-case performance guarantees and explore the analysis of such heaps. First, we unify existing notions of the working-set bound for heaps by proving that essentially all of them are equivalent - with the notable exception of the so-called stack-like bound, which is strictly stronger. This equivalence simplifies the theoretical landscape and extends the range of applications of heaps with working-set bounds. Second, we present the first heap implementation that has the amortized stack-like bound and supports $\mathcal O(1)$-time decrease-key and $o(\log^*n)$-time insert.
Sleator and Tarjan [JACM, 1985] conjectured that splay trees are dynamically optimal -- that on every access sequence, they perform within a constant factor of the optimal offline dynamic binary search tree. Despite four decades of work, no $o(\log n)$ competitive ratio was known. We prove that splay trees are $O(\log\log n \cdot \log^2\log\log n)=\tilde{O}(\log\log n)$-competitive.
We report new results on eight problems in mathematics and theoretical computer science, produced with the assistance of Bolzano, an open-source multi-agent LLM system. Bolzano orchestrates rounds of interaction between parallel prover agents and a verifier agent while maintaining a persistent knowledge base that is carried across rounds. Classified using the significance-autonomy taxonomy of Feng et al., six of the eight results reach the level of publishable research, and five of the eight were produced essentially autonomously by Bolzano. Our results provide evidence that LLMs can contribute meaningfully to mathematical research, complementing recent reports by Bubeck et al., Woodruff et al., and others.
We connect three distinct lines of research that have recently explored extensions of the classical LOCAL model of distributed computing: A. distributed quantum computing and non-signaling distributions [e.g. STOC 2024], B. finitely-dependent processes [e.g. Forum Math. Pi 2016], and C. locality in online graph algorithms and dynamic graph algorithms [e.g. ICALP 2023]. We prove new results on the capabilities and limitations of all of these models of computing, for locally checkable labeling problems (LCLs). We show that all these settings can be sandwiched between the classical LOCAL model and what we call the randomized online-LOCAL model. Our work implies limitations on the quantum advantage in the distributed setting, and we also exhibit a new barrier for proving tighter bounds. Our main technical results are these: 1. All LCL problems solvable with locality $O(\log^\star n)$ in the classical deterministic LOCAL model admit a finitely-dependent distribution with locality $O(1)$. This answers an open question by Holroyd [2024], and also presents a new barrier for proving bounds on distributed quantum advantage using causality-based arguments. 2. In rooted trees, if we can solve an LCL problem with locality $o(\log \log \log n)$ in the randomized online-LOCAL model (or any of the weaker models, such as quantum-LOCAL), we can solve it with locality $O(\log^\star n)$ in the classical deterministic LOCAL model. One of many implications is that in rooted trees, $O(\log^\star n)$ locality in quantum-LOCAL is not stronger than $O(\log^\star n)$ locality in classical LOCAL.
We investigate the connections between the fields of distributed computing and measurable combinatorics by considering complexity classes of locally checkable labeling problems on regular forests. We show that the most important deterministic complexity classes from the LOCAL model of distributed computing exactly coincide with well-studied classes in measurable combinatorics. Namely, first we show that a locally checkable labeling problem admits a continuous solution if and only if it can be solved by a deterministic local algorithm with complexity $O(\log^* n)$. Second, our main result states that, surprisingly, a locally checkable labeling problem admits a Baire measurable solution if and only if it can be solved by a local algorithm with complexity $O(\log n)$. These theorems suggest the existence of deeper connections between the two frameworks. Furthermore, the latter result relies on a complete combinatorial characterization of the classes in question, and as a by-product, it shows that membership in these classes is decidable.
In this paper we prove that Dijkstra's shortest-path algorithm, if implemented with a sufficiently efficient heap, is universally optimal in its running time, and with suitable small additions is also universally optimal in its number of comparisons. Universal optimality is a powerful beyond-worst-case performance guarantee for graph algorithms that informally states that a single algorithm on a problem involving graphs with arc and/or vertex weights performs as well as possible on every graph, assuming a worst-case choice of weights. We give the first application of this notion to any sequential algorithm. We design a new heap data structure with a working-set bound, which guarantees that the heap takes advantage of a certain kind of locality in the heap operations. Our heap has the optimal (amortized) bounds of Fibonacci heaps but also has the beyond-worst-case guarantee that the cost of deleting the minimum item is logarithmic in the number of items inserted after it but before it is deleted, instead of logarithmic in the size of the heap when the item is deleted. That is, deletion of recently inserted items is especially efficient. We prove that our working-set bound guarantees universal optimality for the problem of ordering vertices by their distance from the source vertex, which we call the distance order problem. Our result relies on the observation that the sequence of heap operations generated by any run of Dijkstra's algorithm on a fixed graph possesses enough locality that one can couple the number of comparisons performed by any heap with our working-set bound to the minimum number of comparisons required to solve the distance order problem on this graph for a worst-case choice of arc lengths.
This text provides an introduction to distributed local algorithms -- an area at the intersection of theoretical computer science and discrete mathematics. We collect recent results in the area and demonstrate how they lead to a clean theory. We also discuss many connections of local algorithms to fields such as parallel, distributed, and sublinear algorithms, or descriptive combinatorics.
Suppose we have a memory storing $0$s and $1$s and we want to estimate the frequency of $1$s by sampling. We want to do this I/O-efficiently, exploiting that each read gives a block of $B$ bits at unit cost; not just one bit. If the input consists of uniform blocks: either all 1s or all 0s, then sampling a whole block at a time does not reduce the number of samples needed for estimation. On the other hand, if bits are randomly permuted, then getting a block of $B$ bits is as good as getting $B$ independent bit samples. However, we do not want to make any such assumptions on the input. Instead, our goal is to have an algorithm with instance-dependent performance guarantees which stops sampling blocks as soon as we know that we have a probabilistically reliable estimate. We prove our algorithms to be instance-optimal among algorithms oblivious to the order of the blocks, which we argue is the strongest form of instance optimality we can hope for. We also present similar results for I/O-efficiently estimating mean with both additive and multiplicative error, estimating histograms, quantiles, as well as the empirical cumulative distribution function. We obtain our above results on I/O-efficient sampling by reducing to corresponding problems in the so-called sequential estimation. In this setting, one samples from an unknown distribution until one can provide an estimate with some desired error probability. We then provide non-parametric instance-optimal results for several fundamental problems: mean and quantile estimation, as well as learning mixture distributions with respect to $\ell_\infty$ and the so-called Kolmogorov-Smirnov distance.
We introduce a new type of examples of bounded degree acyclic Borel graphs and study their combinatorial properties in the context of descriptive combinatorics, using a generalization of the determinacy method of Marks. The motivation for the construction comes from the adaptation of this method to the LOCAL model of distributed computing. Our approach unifies the previous results in the area, as well as produces new ones. In particular, we show that for $Δ>2$ it is impossible to give a simple characterization of acyclic $Δ$-regular Borel graphs with Borel chromatic number at most $Δ$: such graphs form a $\mathbfΣ^1_2$-complete set. This implies a strong failure of Brooks'-like theorems in the Borel context.
We present a novel technique for work-efficient parallel derandomization, for algorithms that rely on the concentration of measure bounds such as Chernoff, Hoeffding, and Bernstein inequalities. Our method increases the algorithm's computational work and depth by only polylogarithmic factors. Before our work, the only known method to obtain parallel derandomization with such strong concentrations was by the results of [Motwani, Naor, and Naor FOCS'89; Berger and Rompel FOCS'89], which perform a binary search in a $k$-wise independent space for $k=poly(\log n)$. However, that method blows up the computational work by a high $poly(n)$ factor and does not yield work-efficient parallel algorithms. Their method was an extension of the approach of [Luby FOCS'88], which gave a work-efficient derandomization but was limited to algorithms analyzed with only pairwise independence. Pushing the method from pairwise to the higher $k$-wise analysis resulted in the $poly(n)$ factor computational work blow-up. Our work can be viewed as an alternative extension from the pairwise case, which yields the desired strong concentrations while retaining work efficiency up to logarithmic factors. Our approach works by casting the problem of determining the random variables as an iterative process with $poly(\log n)$ iterations, where different iterations have independent randomness. This is done so that for the desired concentrations, we need only pairwise independence inside each iteration. In particular, we model each binary random variable as a result of a gradual random walk, and our method shows that the desired Chernoff-like concentrations about the endpoints of these walks can be boiled down to some pairwise analysis on the steps of these random walks in each iteration (while having independence across iterations).
We present an intimate connection among the following fields: (a) distributed local algorithms: coming from the area of computer science, (b) finitary factors of iid processes: coming from the area of analysis of randomized processes, (c) descriptive combinatorics: coming from the area of combinatorics and measure theory. In particular, we study locally checkable labellings in grid graphs from all three perspectives. Most of our results are for the perspective (b) where we prove time hierarchy theorems akin to those known in the field (a) [Chang, Pettie FOCS 2017]. This approach that borrows techniques from the fields (a) and (c) implies a number of results about possible complexities of finitary factor solutions. Among others, it answers three open questions of [Holroyd et al. Annals of Prob. 2017] or the more general question of [Brandt et al. PODC 2017] who asked for a formal connection between the fields (a) and (b). In general, we hope that our treatment will help to view all three perspectives as a part of a common theory of locality, in which we follow the insightful paper of [Bernshteyn 2020+] .
The $k$-means++ algorithm by Arthur and Vassilvitskii [SODA 2007] is a classical and time-tested algorithm for the $k$-means problem. While being very practical, the algorithm also has good theoretical guarantees: its solution is $O(\log k)$-approximate, in expectation. In a recent work, Bhattacharya, Eube, Roglin, and Schmidt [ESA 2020] considered the following question: does the algorithm retain its guarantees if we allow for a slight adversarial noise in the sampling probability distributions used by the algorithm? This is motivated e.g. by the fact that computations with real numbers in $k$-means++ implementations are inexact. Surprisingly, the analysis under this scenario gets substantially more difficult and the authors were able to prove only a weaker approximation guarantee of $O(\log^2 k)$. In this paper, we close the gap by providing a tight, $O(\log k)$-approximate guarantee for the $k$-means++ algorithm with noise.
We study the consistent k-center clustering problem. In this problem, the goal is to maintain a constant factor approximate $k$-center solution during a sequence of $n$ point insertions and deletions while minimizing the recourse, i.e., the number of changes made to the set of centers after each point insertion or deletion. Previous works by Lattanzi and Vassilvitskii [ICML '12] and Fichtenberger, Lattanzi, Norouzi-Fard, and Svensson [SODA '21] showed that in the incremental setting, where deletions are not allowed, one can obtain $k \cdot \textrm{polylog}(n) / n$ amortized recourse for both $k$-center and $k$-median, and demonstrated a matching lower bound. However, no algorithm for the fully dynamic setting achieves less than the trivial $O(k)$ changes per update, which can be obtained by simply reclustering the full dataset after every update. In this work, we give the first algorithm for consistent $k$-center clustering for the fully dynamic setting, i.e., when both point insertions and deletions are allowed, and improves upon a trivial $O(k)$ recourse bound. Specifically, our algorithm maintains a constant factor approximate solution while ensuring worst-case constant recourse per update, which is optimal in the fully dynamic setting. Moreover, our algorithm is deterministic and is therefore correct even if an adaptive adversary chooses the insertions and deletions.