Hitting Weighted Even Cycles in Planar Graphs

TL;DR

This paper presents a primal-dual approximation algorithm with a ratio of approximately 6.71 for the minimum-weight hitting set of even cycles in planar graphs, addressing a problem with no known PTAS and establishing an integrality gap.

Contribution

It introduces the first constant-factor approximation algorithm for the even-cycle transversal problem in planar graphs and proves the tightness of the LP relaxation's integrality gap.

Findings

Achieves a 6.71-approximation ratio for ECT in planar graphs.

Establishes the integrality gap of the LP relaxation as 6.71.

Provides the first constant-factor approximation for this problem.

Abstract

A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph $G$ which intersects all copies of subgraphs~$F$ from a fixed family $\mathcal F$. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTAS) for planar input graphs $G$, using a variety of techniques like the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al., SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These techniques do not seem to apply to graph transversals with parity constraints, which have recently received significant attention, but for which no PTASs are known. In the even-cycle transversal (\ECT) problem, the goal is to find a minimum-weight hitting set for the set of even cycles in an undirected graph. For ECT, Fiorini et al. (IPCO 2010) showed that the integrality gap of the standard covering LP relaxation is $\Theta(\log n)$, and that adding sparsity inequalities reduces the integrality gap to~10. Our main result is a primal-dual algorithm that yields a $47/7\approx6.71$-approximation for ECT on node-weighted planar graphs, and an integrality gap of the same value for the standard LP relaxation on node-weighted planar graphs.