Covering a graph with cycles

Abstract

In several areas of engineering and telecommunications, it is necessary to determine a least cost cover of a graph by means of cycles. We propose a highly efficient yet simple heuristic for this difficult problem. On test problems it consistently produces optimal or near optimal solutions.

This article describes a lower bounding procedure and heuristics for the Cycle Cover Problem which consists of determining a least cost cover of an undirected graph with simple cycles. Applications of this problem arise in network design and in telecommunications. Computational results demonstrate the quality of the proposed heuristics. On 100 vertex graphs, the best of these consistently produces optimal or quasi-optimal solutions.

Introduction

The purpose of this paper is to present a lower bounding procedure and heuristics for the cycle cover problem (CCP) defined as follows. Let G=(V, E) be a connected undirected graph where V={v₁, …, v_n} is the vertex set and E={(v_i, v_j):v_i, v_j∈V, i<j} is the edge set. With each edge (v_i, v_j) is associated a non-negative cost or length c_ij. If all costs are equal to 1, then G is said to be unweighted. The CCP consists of determining a least cost cover of G with simple cycles, each containing at least three different edges. Applications of the CCP arise in the design of irrigation systems[1] and in the analysis of electrical circuits[2], where it is necessary to cover a graph with circular piping or wiring while minimizing the total cover length. Generalizations of the CCP in which side constraints are imposed on the cycles are frequently encountered in telecommunications. In these applications designing two-connected networks and networks that can be covered by cycles is dictated by reliability considerations3, 4, 5, 6.

A well-known relaxation of the CCP is the undirected Chinese postman problem (CPP) which consists of determining a least cost traversal of G[7]. Low order polynomial algorithms exist for the CPP[8]. Any Chinese postman solution can be decomposed into a number of cycles, but some of these cycles are illegal for the CCP since they are of the form (v_i, v_j, v_i). Itai and Rodeh[2] have shown that in some graphs, the cost of a shortest cycle cover exceeds that of an optimal Chinese postman solution. For example, in the unweighted Petersen graph (Fig. 1), the CPP optimal solution (1, 3, 9, 3, 5, 6, 5, 2, 8, 2, 4, 10, 4, 1, 7, 8, 9, 10, 6, 7, 1) has a value of 20, whereas the CCP optimal solution has a value of 21. It consists of the four cycles (6, 7, 8, 9, 10, 6), (1, 7, 8, 2, 5, 3, 1), (2, 5, 6, 10, 4, 2), and (1, 3, 9, 10, 4, 1).

Itai et al.[9] have shown that a 2-connected (or bridgeless) unweighted graph has an optimal cover of size at most equal to min {3|E|−6, |E|+6|V|−7}, and this cover can be determined in O(|V|²) operations. Improvements to this bound were proposed by Bermond et al.[10], Alon and Tarsi[11], Fraisse[12], Jackson[13], and more recently by Fan[14]. This last author shows how to determine in polynomial time a cover of size at most |E|+6(|V|−1)/5. The complexity of the CCP is unknown, but Kesel'man[15] has shown that when G is planar, the CCP and the CPP are equivalent. This result also holds if G has a nowhere zero 4-flow, but determining whether G possesses this property is itself NP-complete[13].

To our knowledge, no heuristic for the cycle cover problem has ever been coded and tested. Our aim is to devise such heuristics. We are interested in empirical performance rather than worst-case results, since this is what counts in practice. These heuristics exploit the similarity between the CCP and the CPP. They first construct a Chinese postman solution and determine whether it constitutes a feasible cycle cover. If so, a proven optimal cycle cover has been found. Otherwise, a good feasible cover can be constructed from the Chinese postman solution. Thus, the Chinese postman solution provides a lower bound on the value of the optimal cycle cover solution which can be used to assess the empirical quality of any CCP heuristic.

The remainder of this paper is organized as follows. The CPP algorithm is summarized in Section 2, followed by a description of a CCP heuristic in Section 3, computational results are presented in Section 4, and the conclusion is in Section 5.