Constant-Factor Approximations for Cycle Cover Problems

Abstract

A cycle cover of a graph is a set of cycles such that every vertex lies in exactly one cycle. We consider the following two cycle cover problems. Problem A: given a complete undirected graph \(G=(V,E)\), edge costs \(c: E \rightarrow R_+\) and positive integers \(s_1, \ldots , s_m\), find a cycle cover \(C_1, C_2, \ldots , C_m\) of minimum total cost subject to \(C_i\) has length \(s_i\) for each \(i=1,\ldots m\). Problem B: given a complete undirected graph \(G=(V,E)\), edge costs \(c: E \rightarrow R_+\), special vertices (depots) \(w_1 \ldots , w_m \in V\) and positive integers \(s_1, \ldots , s_m\), find a cycle cover \(C_1, C_2, \ldots , C_m\) of minimum total cost subject to \(C_i\) has length \(s_i\) and contains vertex \(w_i\) for each \(i=1,\ldots m\). Problem B is a version of vehicle routing problem with m vehicles and routings of given lengths. Both problems include the TSP as a special case and so do not admit constant-factor approximations unless P\(=\)NP. We consider the metric case. Goemans and Williamson established that a case of Problem A when all cycles have length k is approximable within a factor of 4. In this paper we present the following results. Problem B can be solved by a 4-approximation algorithm in \(O(n^2\log n)\) time for \(m=2\) (\(n=|V|\)). Problem A can be solved by a 4-approximation algorithm in \(O(n^3\log n)\) time for \(m=2\). Problem A can be solved by a 8-approximation algorithm in \(O(n^{m+1}\log n)\) time for any \(m\ge 3\).