Backtracking based iterated tabu search for equitable coloring

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.

On feasible and infeasible search for equitable graph coloring

“…e equitable coloring problem (ECP) involves finding the smallest number of colors k such that an equitable legal k-coloring exists for a given graph G. Like for the conventional GCP [11], the ECP can be approximated by finding a series of equitable legal k-colorings for decreasing k values. To seek an equitable legal k-coloring for a given k, one typically explores the space of equity-feasible colorings while minimizing a fitness function f which counts the number of conflicting edges [19,21]. e ECP problem with a given k is called the k-ECP problem.…”

“…If a legal k-coloring is found, the k-ECP problem is solved with the current k value and we continue with the new k-ECP problem by se ing k = k − 1. To be effective, the first phase is based on the basic tabu search procedure of the BITS algorithm [19]. If the first phase fails to find a legal k-coloring with the equityfeasible space, the second phase is invoked to enlarge the search to include equity-infeasible colorings.…”

“…e infeasible search phase terminates if an equitable coloring is found or if the best solution found so far cannot be improved during 10000 consecutive iterations. e pseudo-code of the FISA algorithm is given in Algorithm 1. e algorithm starts with an initial equity-feasible solution which is generated with a simple greedy heuristic presented in [19]. In the next sections, we explain the search strategies of both phases of the FISA algorithm.…”

“…In the next sections, we explain the search strategies of both phases of the FISA algorithm. s 1 ← per tur bat ion (s 2 ) 14: end while 3.1 Searching equity-feasible solutions [19] e first phase of the proposed FISA algorithm searches the space of candidate solutions which verifies the equity constraint and tries to find a legal k-coloring. is is achieved by minimization of the number of conflicting edges of candidate equitable k-colorings, an edge is conflicting if its endpoints belong to the same color class.…”

“…e first heuristics are based on greedy constructive principles [8]. More recently, two powerful heuristics were proposed, which are based on the tabu search method: TabuEqCol [21] and BITS [19]. TabuEqCol is a straightforward adaptation of the well-known Tabu-Col algorithm designed for the classical graph coloring problem [10,14].…”