An improved integrality gap for disjoint cycles in planar graphs

TL;DR

This paper introduces a new greedy rounding algorithm for the Cycle Packing Problem in planar graphs, significantly improving the integrality gap bounds and the Erd"H{o}s-P\'osa ratio for uncrossable cycle families.

Contribution

It presents the first constant-factor approximation for the integrality gap of the packing LP in planar graphs, reducing it to slightly less than 3.5, and improves the Erd"H{o}s-P\'osa ratio.

Findings

Integrality gap bound improved to slightly less than 3.5

Erd"H{o}s-P\'osa ratio less than 8.38 for planar graphs

Applicable to various cycle families including all cycles and odd cycles

Abstract

We present a new greedy rounding algorithm for the Cycle Packing Problem for uncrossable cycle families in planar graphs. This improves the best-known upper bound for the integrality gap of the natural packing LP to a constant slightly less than 3.5. Furthermore, the analysis works for both edge- and vertex-disjoint packing. The previously best-known constants were 4 for edge-disjoint and 5 for vertex-disjoint cycle packing. This result also immediately yields an improved Erd\H{o}s-P\'osa ratio: for any uncrossable cycle family in a planar graph, the minimum number of vertices (edges) needed to hit all cycles in the family is less than 8.38 times the maximum number of vertex-disjoint (edge-disjoint, respectively) cycles in the family. Some uncrossable cycle families of interest to which the result can be applied are the family of all cycles in a directed or undirected graph, in undirected graphs also the family of all odd cycles and the family of all cycles containing exactly one edge from a specified set of demand edges. The last example is an equivalent formulation of the fully planar Disjoint Paths Problem. Here the Erd\H{o}s-P\'osa ratio translates to a ratio between integral multi-commodity flows and minimum cuts.