Isomorphism of tournaments with bounded VC dimension

Simon Raßmann; Pascal Schweitzer

Home/Archive/arXiv

arXiv · 2608.14486

Isomorphism of tournaments with bounded VC dimension

Simon Raßmann·Pascal Schweitzer

View original source ↗Read source document ↗Export metadata ↓

Abstract

The tournament isomorphism problem is one of the two fundamental bottlenecks to designing better algorithms for the graph isomorphism problem. Though the problem has been investigated for more than five decades, compared to graphs, there are only very few results on the isomorphism problem of tournaments. For most classes of tournaments neither hardness nor polynomial-time solvability is known. Tournaments of bounded VC dimension are such a class for which no results are available, even though the VC dimension is arguably one of the most robust and central notions of combinatorial tameness. Resolving an open problem of Neuen and Grohe, we show that the isomorphism problem for tournaments of VC dimension $d$ can be decided in time $n^{O(d\log d)}$. Consequently, automorphism groups of tournaments of bounded VC dimension can be computed in polynomial time. To this end, we develop a new method to isomorphism-invariantly decompose tournaments. To facilitate recursion, we introduce the notion of a patched tournament and analyze bounded VC dimension in patched tournaments. We design a recursive algorithm that balances the size of the decomposed pieces against their number and makes use of the structure of near twins. In an orthogonal direction, it is known that a hereditary class of tournaments has unbounded VC dimension if and only if it contains all 2-colorable tournaments. As a second result, we show that also this class does not form an obstruction towards polynomial-time isomorphism testing and indeed show that isomorphism of tournaments of bounded chromatic number is polynomial-time decidable.

Explore related subjects

cs.DS cs.DM math.CO

Keep this discovery

BibTeX RIS

Simon Raßmann, Pascal Schweitzer. 2026-08-14. Isomorphism of tournaments with bounded VC dimension. https://arxiv.org/abs/2608.14486

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quasi-Monte Carlo Beyond Hardy-Krause II: $(1 + \varepsilon)n$ Samples Suffice

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma--Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy--Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy--Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

cs.DS↗

arXiv2026

A deterministic $(1+\varepsilon)^n$ approximation for the permanent of a nonnegative matrix

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

cs.DS↗

arXiv2026

Single-Exponential Algorithms and a Polynomial Kernel for Strong Connectivity Augmentation

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

cs.DS↗