Isomorphism of tournaments with bounded VC dimension
2026-08-14 • Data Structures and Algorithms
Data Structures and AlgorithmsDiscrete Mathematics
AI summaryⓘ
The authors address the problem of determining if two tournaments (special types of directed graphs) are the same, a problem important for understanding graph isomorphism. They focus on tournaments with bounded VC dimension, a measure related to complexity, and develop a new method to break these tournaments into simpler parts. Their algorithm decides isomorphism for these tournaments efficiently and can also find their symmetries in polynomial time. Additionally, they prove that tournaments with a limited number of colors (bounded chromatic number) also allow for polynomial-time isomorphism testing.
tournament isomorphismgraph isomorphismVC dimensionautomorphism grouppolynomial timechromatic numberhereditary class2-colorable tournamentdecompositioncombinatorial tameness
Authors
Simon Raßmann, Pascal Schweitzer
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.