Backtracking based iterated tabu search for equitable coloring

Highlights

• The equitable coloring problem (ECP) is a NP-hard combinatorial problem.

• We introduce a backtracking based iterated tabu search (BITS) algorithm for the ECP.

• We assess the algorithm׳s performance on a large set of ECP benchmark instances.

• We report new upper bounds for 21 benchmark instances.

Abstract

An equitable k-coloring of an undirected graph $G=(V,E)$ is a partition of its vertices into k disjoint independent sets, such that the cardinalities of any two independent sets differ by at most one. As a variant of the graph coloring problem (GCP), the equitable coloring problem (ECP) concerns finding a minimum k for which an equitable k-coloring exists. In this work, we propose a backtracking based iterated tabu search (BITS) algorithm for solving the ECP approximately. BITS uses a backtracking scheme to define different k-ECP instances, an iterated tabu search approach to solve each particular k-ECP instance for a fixed k, and a binary search approach to find a suitable initial value of k. We assess the algorithm׳s performance on a set of commonly used benchmarks. Computational results show that BITS is very competitive in terms of solution quality and computing efficiency compared to the state-of-the-art algorithm in the literature. Specifically, BITS obtains new upper bounds for 21 benchmark instances, while matching the previous best upper bound for the remaining instances. Finally, to better understand the proposed algorithm, we study how its key ingredients impact its performance.

Introduction

Given an undirected graph $G=(V,E)$ with vertex set $V=\{v_1,v_2,\dots ,v_n\}$ and edge set $E\subset V\times V$, a k-coloring of G is a partition of its vertices into k disjoint independent sets $\{V_1,V_2,\dots ,V_k\}$ (also called color classes), i.e., any two vertices of any $V_i(i=1,\dots ,k)$ 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 $\{V_1,V_2,\dots ,V_k\}$ such that the numbers of vertices in any two color classes differ by at most one (i.e., $\parallel V_i|-|V_j\parallel \le 1$, ∀ $i\ne j$. 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 ${\chi }_e\left(G\right)$.

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 ${\chi }_e\left(G\right)<\Delta \left(G\right)$ for any connected graph except the complete graphs and the odd circuits, where $\Delta \left(G\right)$ is the maximum vertex degree of G (Meyer, 1973). Lih and Wu showed that ${\chi }_e\left(G\right)=\chi \left(G\right)$ holds for any connected bipartite graph $G(X,Y)$, where $\chi \left(G\right)$ 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 $\Delta \left(G\right)$ is equitably k-colorable for every $k\ge \Delta +1$ if the average degree of vertices are at most $\Delta /5$ (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 $\left(O\right(n^2\left)\right)$ to find an equitable k-coloring where $k=\Delta \left(G\right)+1$ (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.