Reduction and Local Search for Weighted Graph Coloring Problem

Abstract

The weighted graph coloring problem (WGCP) is an important extension of the graph coloring problem (GCP) with wide applications. Compared to GCP, where numerous methods have been developed and even massive graphs with millions of vertices can be solved well, fewer works have been done for WGCP, and no solution is available for solving WGCP for massive graphs. This paper explores techniques for solving WGCP, including a lower bound and a reduction rule based on clique sampling, and a local search algorithm based on two selection rules and a new variant of configuration checking. This results in our algorithm RedLS (Reduction plus Local Search). Experiments are conducted to compare RedLS with the state-of-the-art algorithms on massive graphs as well as conventional benchmarks studied in previous works. RedLS exhibits very good performance and robustness. It significantly outperforms previous algorithms on all benchmarks.

A deep learning guided memetic framework for graph coloring problems

“…This section shows a comparative analysis on the pxx, rxx, DIMACS/COLOR small, and DIMACS/COLOR large instances with respect to the state-of-the-art methods [27,33,36]. The reference methods include the three best recent heuristics: AFISA [33], RedLS [36] and ILS-TS [27]. When they are available, we also include the optimal scores obtained with the MWSS exact algorithm [5] and reported in [27].…”

“…We carried out extensive experiments on the benchmark used in the recent papers on the WVCP [27,33,36]: the pxx, rxx, DIMACS/COLOR small, and DIMACS/COLOR large instances. The pxx and rxx instances are based on matrix-decomposition problems [31], while DIMACS/COLOR small [5,7] and DIMACS/COLOR large [33] are based on DIMACS and COLOR competitions.…”

“…As indicated in [27,36], a preprocessing procedure can be applied to reduce a graph G with the set of weight W for the WVCP. For each clique C l with l nodes, if we note w the the smallest weight of this set, all the nodes i in the graph with a degree equal to l − 1 and a weight w i < w can be removed from the graph without changing the optimal WVCP score that can be found for this instance.…”

“…As a proof of concept, we apply this approach to solve the weighted vertex graph coloring problem (WVCP) which has recently attracted a lot of interest in the literature [27,36]. In the WVCP, a strictly positive weight w i is associated to each vertex i.…”

“…The WVCP has a number of practical applications in different fields such as matrix decomposition problems [31], batch scheduling [9] and manufacturing [13]. It has been addressed by exact methods [5,7,24] and heuristics [31,34,36]. The current most performing heuristic for this problem is an iterated local search algorithm exploring two different neighborhoods and based on tabu search [27].…”