Variations on Memetic Algorithms for Graph Coloring Problems

11 pages, 8 figures, 3 tables, 2 algorithmsInternational audienceGraph vertex coloring with a given number of colors is a well-known and much-studied NP-complete problem.The most effective methods to solve this problem are proved to be hybrid algorithms such as memetic algorithms or quantum annealing. Those hybrid algorithms use a powerful local search inside a population-based algorithm.This paper presents a new memetic algorithm based on one of the most effective algorithms: the Hybrid Evolutionary Algorithm HEA from Galinier and Hao (1999).The proposed algorithm, denoted HEAD - for HEA in Duet - works with a population of only two individuals.Moreover, a new way of managing diversity is brought by HEAD.These two main differences greatly improve the results, both in terms of solution quality and computational time.HEAD has produced several good results for the popular DIMACS benchmark graphs, such as 222-colorings for , 81-colorings for and even 47-colorings for and 82-colorings for

Network Community Detection On Small Quantum Computers

In recent years a number of quantum computing devices with small numbers of qubits became available. We present a hybrid quantum local search (QLS) approach that combines a classical machine and a small quantum device to solve problems of practical size. The proposed approach is applied to the network community detection problem. QLS is hardware-agnostic and easily extendable to new quantum computing devices as they become available. We demonstrate it to solve the 2-community detection problem on graphs of size up to 410 vertices using the 16-qubit IBM quantum computer and D-Wave 2000Q, and compare their performance with the optimal solutions. Our results demonstrate that QLS perform similarly in terms of quality of the solution and the number of iterations to convergence on both types of quantum computers and it is capable of achieving results comparable to state-of-the-art solvers in terms of quality of the solution including reaching the optimal solutions

Evolutionary Approaches to Optimization Problems in Chimera Topologies

Chimera graphs define the topology of one of the first commercially available quantum computers. A variety of optimization problems have been mapped to this topology to evaluate the behavior of quantum enhanced optimization heuristics in relation to other optimizers, being able to efficiently solve problems classically to use them as benchmarks for quantum machines. In this paper we investigate for the first time the use of Evolutionary Algorithms (EAs) on Ising spin glass instances defined on the Chimera topology. Three genetic algorithms (GAs) and three estimation of distribution algorithms (EDAs) are evaluated over $1000$ hard instances of the Ising spin glass constructed from Sidon sets. We focus on determining whether the information about the topology of the graph can be used to improve the results of EAs and on identifying the characteristics of the Ising instances that influence the success rate of GAs and EDAs.Comment: 8 pages, 5 figures, 3 table

A Distribution Evolutionary Algorithm for Graph Coloring

Graph Coloring Problem (GCP) is a classic combinatorial optimization problem that has a wide application in theoretical research and engineering. To address complicated GCPs efficiently, a distribution evolutionary algorithm based on population of probability models (DEA-PPM) is proposed. Based on a novel representation of probability model, DEA-PPM employs a Gaussian orthogonal search strategy to explore the probability space, by which global exploration can be realized using a small population. With assistance of local exploitation on a small solution population, DEA-PPM strikes a good balance between exploration and exploitation. Numerical results demonstrate that DEA-PPM performs well on selected complicated GCPs, which contributes to its competitiveness to the state-of-the-art metaheuristics

Optimization by quantum annealing for the graph colouring problem

Quantum annealing is the quantum equivalent of the well known classical simulated annealing algorithm for combinatorial optimization problems. Despite the appeal of the approach, quantum annealing algorithms competitive with the state of the art for specific problems hardly exist in the literature. Graph colouring is a difficult problem of practical significance that can be formulated as combinatorial optimization. By introducing a symmetry-breaking problem representation, and finding fast incremental techniques to calculate energy changes, a competitive graph colouring algorithm based on quantum annealing is derived. This algorithm is further enhanced by tuning simplification techniques; replica spacing techniques to increase robustness; and a messaging protocol, which enables quantum annealing to efficiently take advantage of multiprocessor environments. Additionally, observations of some patterns in the tuning for random graphs led to a more effective algorithm able to find new upper bounds for several widely-used benchmark graphs, some of which had resisted improvement in the last two decades