Backtracking based iterated tabu search for equitable coloring
Introduction
Given an undirected graph with vertex set and edge set , a k-coloring of G is a partition of its vertices into k disjoint independent sets (also called color classes), i.e., any two vertices of any are not linked by an edge. For a given k, G is k-colorable if a k-coloring exists for G. The problem of finding a k-coloring for a given number of k colors is called the graph k-coloring problem. The classical graph coloring problem (GCP) is to determine the smallest k (the chromatic number of G) such that G is k-colorable.
An equitable k-coloring of G (or k-eqcol) is a k-coloring such that the numbers of vertices in any two color classes differ by at most one (i.e., , ∀ . This is called the equity constraint Méndez-Díaz et al., 2015). In other words, an equitable k-coloring is a conflict-free k-coloring satisfying the equity constraint. The equitable coloring problem (ECP) is to determine the smallest number k of colors such that an equitable k-coloring of G exists (Meyer, 1973). This minimum k is called the equitable chromatic number of G and denoted by .
Clearly the ECP is tightly related to the classical GCP. Like the GCP, the ECP is NP-hard (Furmańczyk, 2005) and thus computationally challenging. In addition to its theoretical significance, the ECP has a variety of applications, like garbage collection (Tucker, 1973), partitioning and load balancing (Blazewicz et al., 2001), scheduling (Ding et al., 2015, Irani and Leung, 1996, Meyer, 1973), etc. For a review of possible applications of the ECP, the reader is referred to recent papers like Bahiense et al. (2014), Méndez-Díaz et al. (2014b).
Due to its relevance, much effort has been devoted to the studies of the ECP from a theoretical point of view. For example, Meyer suggested a conjecture that for any connected graph except the complete graphs and the odd circuits, where is the maximum vertex degree of G (Meyer, 1973). Lih and Wu showed that holds for any connected bipartite graph , where is the chromatic number of G (Lih and Wu, 1996). Lam et al. obtained an explicit formula to calculate the equitable chromatic number of a complete n-partite graph (Lam et al., 2001). Bodlaender and Fomin proved that the ECP can be solved in polynomial time for graphs with bounded treewidth (Bodlaender and Fomin, 2005). Kostochka and Nakprasit proved that a graph with maximum degree is equitably k-colorable for every if the average degree of vertices are at most (Kostochka and Nakprasit, 2005). Furmańczyk discussed the computational complexity of the ECP for some special graphs (Furmańczyk, 2005). Wu and Wang investigated the planar graphs with large girth (Wu and Wang, 2008). Chen and Lih investigated equitable colorings for trees (Chen and Lih, 1994) and Chang studied equitable colorings for forests (Chang, 2009). Nakprasit and Nakprasit obtained some results on the ECP for planar graphs with some special properties (Nakprasit and Nakprasit, 2012). Yan and Wang investigated the ECP for Kronecker products of the complete multipartite graphs and complete graphs (Yan and Wang, 2014).
From the computational point of view, several exact and heuristic algorithms have been proposed in the literature for solving the ECP for arbitrary graphs. Specifically, Furmanczyk and Kubale proposed two constructive heuristics (Naive and Subgraph) (Furmanczyk and Kubale, 2004). Bahiense et al. presented two effective branch-and-cut algorithms (Bahiense et al., 2009, Bahiense et al., 2014). Méndez-Díaz et al. investigated a polyhedral approach (Méndez-Díaz et al., 2014a), a DSatur-based exact algorithm (Méndez-Díaz et al., 2015), and a tabu search heuristic (Méndez-Díaz et al., 2014b). Kierstead et al. proposed a fast algorithm to find an equitable k-coloring where (Kierstead et al., 2010).
Given the NP-hard nature of the ECP, it is unlikely that an exact algorithm will be found that is able to determine the equitable chromatic number of arbitrary graphs in polynomial time. Consequently, as for any NP-hard problem, heuristics constitute a very appealing and indispensable alternative which can be used to approximate the equitable chromatic number of a graph. On the other hand, the literature review shows that there are only limited studies on heuristic algorithms for the ECP.
In this paper, we provide a method to solve the ECP approximately by means of a backtracking based iterated tabu search (BITS) algorithm. BITS solves a series of k-ECP, i.e., equitable k-coloring instances with different fixed k values. For a given k-ECP (with a particular k), the purpose of the iterated tabu search (ITS) is to seek a conflict-free equitable k-coloring. The backtracking scheme is used to adjust k to an appropriate value. A technique based on binary search is used to determine a good initial k value.
We perform a computational study testing our proposed BITS algorithm on a large number of benchmark instances widely used in the literature. The computational results show that our new algorithm is very competitive in terms of both solution quality and computation efficiency compared to the state-of-the-art heuristics. Specifically, we are able to find new upper bounds of the equitable chromatic number for 21 benchmark instances, matching the previous best upper bound for the remaining instances.
The rest of the paper is organized as follows. In Section 2, we describe the components of the BITS algorithm, including the search space and evaluation function, the initial solution generation procedure, the iterated tabu search procedure, and the binary search procedure for determining an initial k. In Section 3, we present the computational benchmark assessments and compare our results with those of the state-of-the-art heuristic algorithms in the literature. Section 4 analyzes and discusses some important components of the proposed algorithm. Finally, we provide concluding comments in Section 5.
Section snippets
Backtracking based iterated tabu search for the ECP
In this section, we present the general solution approach and the supporting procedures that compose it.
Computational results and comparisons
In this section, we present computational results and comparisons to assess the performance of the proposed BITS algorithm.
Analysis and discussions
In this section, we study several key ingredients of the BITS algorithm to get some insight into its behavior.
Conclusions
Our backtracking based iterated tabu search (BITS) approach to solve the equitable coloring problem (ECP) achieves a high level of performance by integrating several components: a backtracking scheme to define different k-ECP instances, a tabu search procedure with a hybrid tabu list management strategy to solve each associated k-ECP instance, two perturbation operators to jump out of local optima, and a binary search method to determine the initial value of k.
The effectiveness of the BITS
Acknowledgments
We are grateful to the reviewers for their useful comments which help us to improve the paper. The work is partially supported by the LigeRo project (2009–2013, Region of Pays de la Loire, France), the PGMO project (2013–2015, Jacques Hadamard Mathematical Foundation) and a post-doc Grant (for X.J. Lai) from the Region of Pays de la Loire (France).
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