Approximation Algorithms for Restricted Cycle Covers Based on Cycle Decompositions

TL;DR

This paper investigates the computational complexity and approximation algorithms for finding maximum weight L-cycle covers in graphs, establishing hardness results and providing approximation guarantees for both directed and undirected cases.

Contribution

It proves NP-hardness and APX-hardness for most L, and presents approximation algorithms with factors of 2 and 8/3 for undirected and directed graphs respectively.

Findings

Most L-cycle cover problems are NP-hard and APX-hard.

Approximation algorithms achieve a factor of 2 for undirected graphs.

Approximation algorithms achieve a factor of 8/3 for directed graphs.

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. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges. We come close to settling the complexity and approximability of computing L-cycle covers. On the one hand, we show that for almost all L, computing L-cycle covers of maximum weight in directed and undirected graphs is APX-hard and NP-hard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing L-cycle covers of maximum weight can be approximated within a factor of 2 for undirected graphs and within a factor of 8/3 in the case of directed graphs. This holds for arbitrary sets L.