Minimum-weight Cycle Covers and Their Approximability

TL;DR

This paper studies the approximability of minimum-weight cycle covers with specific cycle length constraints in both undirected and directed graphs, providing algorithms and hardness results, and contrasting with maximum-weight cases.

Contribution

It introduces polynomial-time approximation algorithms for minimum-weight L-cycle covers and establishes hardness bounds, while also analyzing maximum-weight cases.

Findings

Constant approximation ratio for undirected graphs with all L

No approximation within 2-eps for certain L in undirected graphs

O(n) approximation ratio for directed graphs, optimal up to lower bounds

Abstract

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An L-cycle cover is a cycle cover in which the length of every cycle is in the set L. We investigate how well L-cycle covers of minimum weight can be approximated. For undirected graphs, we devise a polynomial-time approximation algorithm that achieves a constant approximation ratio for all sets L. On the other hand, we prove that the problem cannot be approximated within a factor of 2-eps for certain sets L. For directed graphs, we present a polynomial-time approximation algorithm that achieves an approximation ratio of O(n), where $n$ is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated within a factor of o(n). To contrast the results for cycle covers of minimum weight, we show that the problem of computing L-cycle covers of maximum weight can, at least in principle, be approximated arbitrarily well.