On Maximum Weight Clique Algorithms, and How They Are Evaluated

Abstract

Abstract. Maximum weight clique and maximum weight independent set solvers are often benchmarked using maximum clique problem instances, with weights allocated to vertices by taking the vertex number mod 200 plus 1. For constraint programming approaches, this rule has clear implications, favouring weight-based rather than degree-based heuristics. We show that similar implications hold for dedicated algorithms, and that additionally, weight distributions affect whether certain inference rules are cost-effective. We look at other families of benchmark instances for the maximum weight clique problem, coming from winner determination problems, graph colouring, and error-correcting codes, and introduce two new families of instances, based upon kidney exchange and the Research Excellence Framework. In each case the weights carry much more interesting structure, and do not in any way resemble the 200 rule. We make these instances available in the hopes of improving the quality of future experiments.

Reduction and Local Search for Weighted Graph Coloring Problem

“…These graphs are originally unweighted, and we use two weighting functions. (1) We employ the same method as in (Cai and Lin 2016;Wang, Cai, and Yin 2017), i.e., for the ith vertex v i , the weighting function w 1 (v i )=(i mod 200)+1; (2) Ob-2 https://mat.gsia.cmu.edu/COLOR02/ served from the real weighting functions of error-correcting codes and winner determination problem (McCreesh et al 2017), vertices with low degree have high weight values, while vertices with high degree have light weights. According to our experiments, the following weighting function can well simulate the weight distributions from those real world instances, and thus is adopted to generate weights.…”

Conflict Directed Clause Learning for Maximum Weighted Clique Problem

“…We implemented our approach in C++, denoted cdcl, on top of the MINICSP solver 2 . We compared it with the solvers mwclq [Fang et al, 2016], wlmc [Jiang et al, 2017], cliquer [Östergård, 2001], OTClique [Shimizu et al, 2017] and an implementation of Tavares' method by the authors of [McCreesh et al, 2017]. We used the benchmarks introduced in [McCreesh et al, 2017] divided in four classes:…”

“…REF Research Excellence Network, where the clique stands for the optimal set of publications that a university department can provide to the authority assessing it (generated by [McCreesh et al, 2017]); EC-CODE Error-correcting Codes, where the clique stands for a set of words maximally pair-wise distant (instances due to [Östergård, 1999], reconstructed by [McCreesh et al, 2017]);…”

“…KIDNEY Kidney Exchange, where the clique stands for a maximally desirable set of donor/patient exchanges. The instances were generated by [McCreesh et al, 2017] using data from [Dickerson et al, 2012] and a weighting scheme from [Manlove and O'malley, 2015]. Moreover, we also provide results on the classic DIMACS and BHOSLIB benchmarks.…”

Common Object Discovery as Local Search for Maximum Weight Cliques in a Global Object Similarity Graph

“…where w V : V → R and w E : E → R are the weight functions for the vertices and edges respectively. Successfully solving the MWC problem has various applications not only in computer vision [17,12] but in many different domains from wireless to social networks [21,19]. The main contributions of our paper include:…”