Preface: graph colorings

Adaptive feasible and infeasible tabu search for weighted vertex coloring

General swap-based multiple neighborhood adaptive search for the maximum balanced biclique problem

Iterated local search with tabu search for the weighted vertex coloring problem

Spatial-domain fitness landscape analysis for combinatorial optimization

Given an undirected graph G = ( V , E ) and a positive integer k , an equitable legal k -coloring of G is a partition of the vertex set V into k disjoint independent sets such that the cardinalities of any two independent sets differ by one at most. The equitable coloring problem is to find the smallest k for which an equitable legal k -coloring exists. The problem has a number of applications. However, it is known to be NP-hard and thus computationally challenging. In this work, we present the first population-based memetic algorithm for solving the problem. The proposed algorithm combines a backbone-based crossover operator (to generate promising offspring solutions), a 2-phase tabu search procedure (to seek high-quality local optima) as well as a quality-and-distance based pool updating strategy (to maintain a healthy population). The computational results on 73 benchmark instances demonstrate that the proposed algorithm competes favorably with the state-of-the-art algorithms in the literature. Specifically, our algorithm attains the optimal results for all 41 instances with known optima and discovers improved upper bounds for 9 out of the 32 instances whose optimal solutions are still unknown. We investigate the benefits of the 2-phase tabu search procedure and the crossover operator with the memetic framework.

Memetic search for the equitable coloring problem

An equitable legal k-coloring of an undirected graph G = (V, E) is a partition of the vertex set V into k disjoint independent sets, such that the cardinalities of any two independent sets differ by at most one (this is called the equity constraint). As a variant of the popular graph coloring problem (GCP), the equitable coloring problem (ECP) involves finding a minimum k for which an equitable legal k-coloring exists. In this paper, we present a study of searching both feasible and infeasible solutions with respect to the equity constraint. The resulting algorithm relies on a mixed search strategy exploring both equitable and inequitable colorings unlike existing algorithms where the search is limited to equitable colorings only. We present experimental results on 73 DIMACS and COLOR benchmark graphs and demonstrate the competitiveness of this search strategy by showing 9 improved best-known results (new upper bounds).

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On feasible and infeasible search for equitable graph coloring

A solution-driven multilevel approach for graph coloring

Subgraph isomorphism or subgraph matching is generally considered as an NP-complete problem, made more complex in practical applications where the edge weights take real values and are subject to measurement noise and possible anomalies. To the best of our knowledge, almost all subgraph matching methods utilize node labels to perform node-node matching. In the absence of such labels (in applications such as image matching and map matching among others), these subgraph matching methods do not work. We propose a method for identifying the node correspondence between a subgraph and a full graph in the inexact case without node labels in two steps -(a) extract the minimal unique topology preserving subset from the subgraph and ﬁnd its feasible matching in the full graph, and (b) implement a consensus-based algorithm to expand the matched node set by pairing unique paths based on boundary commutativity. Going beyond the existing subgraph matching approaches, the proposed method is shown to have realistically sub-linear computational eﬃciency, robustness to random measurement noise, and good statistical properties. Our method is also readily applicable to the exact matching case without loss of generality. To demonstrate the eﬀectiveness of the proposed method, a simulation and a case study is performed on the Erdos-Renyi random graphs and the image-based aﬃne covariant features dataset respectively.

Wen Sun

Papers

Adaptive feasible and infeasible tabu search for weighted vertex coloring

Memetic search for the equitable coloring problem

On feasible and infeasible search for equitable graph coloring

A solution-driven multilevel approach for graph coloring

Efficient Subgraph Isomorphism using Graph Topology