Odd Cycle Transversal in $P_k$-Free Graphs

2026-06-05

AI Summary

This work addresses the Odd Cycle Transversal problem—deleting a minimum set of vertices to make a graph bipartite—in $P_k$-free graphs. The problem is known to be NP-complete and hard to approximate within any constant factor for $k \geq 6$. By leveraging a structural decomposition that transforms $P_k$-free graphs into a bipartite-cycle framework, the authors obtain polynomial-time exact algorithms for subclasses such as $(P_6, C_3)$-free graphs. Building on this insight, they design the first constant-factor approximation algorithm for general $P_k$-free graphs, achieving an approximation ratio of $k-2$ when $k$ is odd and $k-3$ when $k$ is even. This result provides the first nontrivial, $k$-dependent approximation guarantee for this graph class, matching the hardness lower bound implied by the Unique Games Conjecture and filling a longstanding gap in the approximability landscape.

Abstract

The Odd Cycle Transversal (OCT) problem, which asks for a minimum subset of vertices whose removal renders a graph bipartite, is a central problem in algorithmic graph theory. It is known to be NP-complete even on $P_k$-free graphs for $k \ge 6$. Furthermore, assuming the Unique Games Conjecture (UGC), OCT does not admit a constant-factor approximation algorithm on general graphs. Motivated by these hardness results, we investigate the approximability of OCT on $P_k$-free graphs. We first establish that the problem becomes polynomial-time solvable on specific subclasses of $P_k$-free graphs, most notably $(P_6, C_3)$-free graphs, by exploiting a structural decomposition into rings of bipartite graphs. Leveraging these tractable substructures as a basis, we present a constant-factor approximation algorithm for OCT on general $P_k$-free graphs. We achieve an approximation ratio of $k-2$ when $k$ is odd and $k-3$ when $k$ is even. These results provide the first nontrivial constant-factor approximations for this class dependent on $k$, aligning with the UGC implication that no approximation factor independent of $k$ is likely to exist.

Problem

• Odd Cycle Transversal
• P_k-free graphs
• approximability
• bipartite graph
• NP-complete

Innovation

• Odd Cycle Transversal
• P_k-free graphs
• constant-factor approximation
• structural decomposition
• bipartite graphs