Packing cycles in planar and bounded-genus graphs

TL;DR

This paper introduces constant-factor approximation algorithms for finding many disjoint cycles in planar and bounded-genus graphs, generalizing previous problems and improving approximation constants, including for vertex-disjoint paths.

Contribution

It develops a unified framework for approximating disjoint cycles in various graph families, extending known results and providing the first algorithms for vertex-disjoint paths in such graphs.

Findings

Achieved constant-factor approximation algorithms for disjoint cycles.

Improved approximation constants for planar graphs.

First algorithms for vertex-disjoint paths in these settings.

Abstract

We devise constant-factor approximation algorithms for finding as many disjoint cycles as possible from a certain family of cycles in a given planar or bounded-genus graph. Here disjoint can mean vertex-disjoint or edge-disjoint, and the graph can be undirected or directed. The family of cycles under consideration must satisfy two properties: it must be uncrossable and allow for an oracle access that finds a weight-minimal cycle in that family for given nonnegative edge weights or (in planar graphs) the union of all remaining cycles in that family after deleting a given subset of edges. Our setting generalizes many problems that were studied separately in the past. For example, three families that satisfy the above properties are (i) all cycles in a directed or undirected graph, (ii) all odd cycles in an undirected graph, and (iii) all cycles in an undirected graph that contain precisely one demand edge, where the demand edges form a subset of the edge set. The latter family (iii) corresponds to the classical disjoint paths problem in fully planar and bounded-genus instances. While constant-factor approximation algorithms were known for edge-disjoint paths in such instances, we improve the constant in the planar case and obtain the first such algorithms for vertex-disjoint paths. We also obtain approximate min-max theorems of the Erd\H{o}s--P\'osa type. For example, the minimum feedback vertex set in a planar digraph is at most 12 times the maximum number of vertex-disjoint cycles.