Isomorphism for Tournaments of Small Twin Width

TL;DR

The paper addresses the isomorphism problem for tournaments under the twin width parameter. It develops a group-theoretic isomorphism tester that achieves running time $k^{O(\log k)}n^{O(1)}$ for tournaments of twin width at most $k$, implying polynomial-time solvability for classes with bounded or slowly growing twin width and, by functional relationships, for several directed-width parameters. A key technical contribution is a three-phase approach: reduction to $2$-WL-homogeneous inputs, a combinatorial partition-sequence lemma identifying a bounded-degree backbone, and a group-theoretic isomorphism routine that leverages solvable permutation groups and wreath products; the running time hinges on a lifting subroutine with $d^{O(\log d)}$ complexity. The authors further show that the combinatorial Weisfeiler-Leman algorithm cannot decide isomorphism for bounded-twin-width tournaments: for every $k \ge 2$ there exist non-isomorphic $T_k,T'_k$ with twin width at most $35$ that are indistinguishable by $k$-WL, established via a CFI-type construction. They also compare twin width to other width measures, establishing that twin width is functionally smaller than cut width, directed path width, and directed tree width on tournaments, with strict separations demonstrated by specific examples, thereby highlighting the distinct algorithmic power of twin width in this graph class.

Abstract

We prove that isomorphism of tournaments of twin width at most $k$ can be decided in time $k^{O(\log k)}n^{O(1)}$. This implies that the isomorphism problem for classes of tournaments of bounded or moderately growing twin width is in polynomial time. By comparison, there are classes of undirected graphs of bounded twin width that are isomorphism complete, that is, the isomorphism problem for the classes is as hard as the general graph isomorphism problem. Twin width is a graph parameter that has been introduced only recently (Bonnet et al., FOCS 2020), but has received a lot of attention in structural graph theory since then. On directed graphs, it is functionally smaller than clique width. We prove that on tournaments (but not on general directed graphs) it is also functionally smaller than directed tree width (and thus, the same also holds for cut width and directed path width). Hence, our result implies that tournament isomorphism testing is also fixed-parameter tractable when parameterized by any of these parameters. Our isomorphism algorithm heavily employs group-theoretic techniques. This seems to be necessary: as a second main result, we show that the combinatorial Weisfeiler-Leman algorithm does not decide isomorphism of tournaments of twin width at most 35 if its dimension is $o(n)$. (Throughout this abstract, $n$ is the order of the input graphs.)

Figures (2)

• Figure 1: The figure shows part of a tournament $T$. The colors $c_1$ and $c_2$ are shown in blue and green, respectively. Also, the parts of the partition ${\mathcal{Q}}_1$ are highlighted in gray. Note that only green edges, which are outgoing from the middle part, are shown.
• Figure 2: The figure shows the sets $W_{1,i}$ (orange), $U_{1,i}$ (blue) and $L_{1,i+1}^{u}$ computed in the proof of Lemme \ref{['lem:lift-isomorphisms']}. The color $c^*$ is shown in green and gray regions depict parts of the partition ${\mathcal{Q}}_1$.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: BonnetKTW22
• Lemma 2.2
• proof
• Theorem 2.3
• Theorem 2.4: BabaiL83
• Theorem 2.5: BabaiL83
• Theorem 2.6: Miller83
• proof