Odd Cycle Transversal on $H$-free graphs

2026-09-25

AI Summary

This study investigates the odd cycle transversal problem on $H$-free graphs, aiming to establish a complete computational complexity classification. Methodologically, it introduces the $H$-amiable structural analysis tool, which optimizes the reduction overhead from independent set to $n^{O(\log n)}$, and devises a quasi-polynomial time algorithm for $kP_4$-free graphs. The core contribution lies in establishing a full complexity dichotomy for this problem on $H$-free graphs, thereby resolving an open question posed at SODA 2024. Furthermore, this work provides a systematic theoretical framework for characterizing the complexity of related graph-theoretic optimization problems.

Abstract

\textsc{Odd Cycle Transversal} is a classic $\mathsf{NP}$-hard graph optimization problem asking for a minimum-weight set of vertices whose deletion makes the input graph bipartite, or equivalently, a maximum-weight induced bipartite subgraph. We show that \textsc{Odd Cycle Transversal} is quasi-polynomial-time solvable on $kP_4$-free graphs, for every fixed $k \in \mathbb{N}$. In fact, we provide an $n^{O_k(\log n)}$-time algorithm for the more general \textsc{Max-Weight List $2$-Colorable Induced Subgraph}, where the notation $O_{k}(\cdot)$ hides factors depending on $k$. Paired with known results from the literature, this allows us to obtain a complete complexity dichotomy for these two problems on $H$-free graphs into cases solvable in quasi-polynomial time and cases which are $\mathsf{NP}$-hard, in particular resolving an open problem of Agrawal, Lima, Lokshtanov, Saurabh, and Sharma [SODA 2024]. Our algorithms are based on a new structural tool that may be of independent interest. We introduce the notion of $H$-amiable family and show that, for every fixed graph $H$ without isolated vertices and every fixed $k\ge2$, every $kH$-free graph admits an $H$-amiable family of quasi-polynomial size that can be constructed in quasi-polynomial time. Besides yielding the aforementioned algorithms, this result gives, for every fixed connected graph $H$ and every fixed $k\ge2$, a reduction from \textsc{Max-Weight Independent Set} on $kH$-free graphs to the same problem on $H$-free graphs with $n^{O_{H,k}(\log n)}$ overhead. In this setting, it improves the $n^{O_{H,k}(\log^3 n)}$ overhead obtained by specializing the general reduction of Gartland and Lokshtanov [FOCS 2020].

Problem

• Odd Cycle Transversal
• H-free graphs
• quasi-polynomial time
• complexity dichotomy
• Max-Weight Independent Set

Innovation

• Odd Cycle Transversal
• H-free graphs
• quasi-polynomial time
• H-amiable family
• Max-Weight Independent Set