Maximum Cardinality Matching by Evolutionary Algorithms

The analysis of time complexity of evolutionary algorithms has always focused on some artificial binary problems. This p aper considers the average time complexity of an evolutionary al gorithm for maximum cardinality matching in a graph. It is sh own that the evolutionary algorithm can produce matchings with nearly maximum cardinality in average polynomial time. I. I NTRODUCTION In spite of many experimental studies on combinatorial opti misation by EAs [13], [8], [14], the time complexity of evolutionary algorithms (EAs in short) for combinatorial o ptimisation problems have not been well understood [10], [3 ]. Up to now, most of the results on this topic are limited to some simple EAs on Boolean optimisation problems [1], [4], [5], [2], [6]. This paper considers a well-known combinatorial optimisat ion problem, i.e., the maximum cardinality matching proble m [9], [7]. It is an easy problem in the sense that there are know n deterministic algorithms that solve it in a polynomial tim e of |E|, where|E| indicate the number of edges in a graph [9], [7]. However, the problem is by no means trivial and it has been shown that neither simulated annealing nor a Boltzmann machine can find the maximum cardinality matching in average polynomial time for a certain family of bipartite graphs, al though they can produce matchings with near maximum cardina lity for any graphs in average polynomial time [11], [12]. It is in teresting to investigate what EAs can achieve in solving thi s problem. Section II of this paper introduces the EA used to solve the ma xi um matching problem. Section III describes drift analys is [5] that are necessary to establish our main result. Section IV gives the main result of this paper and its proof. It is show n that our EA can find a matching with a near maximum cardinality in average polynomial time. Section V concludes the paper with a few remarks. II. EVOLUTIONARY ALGORITHM FOR MAXIMUM MATCHING Given an undirected graph G = (V, E), whereV denotes the set of nodes of G and E the set of (undirected) edges, a matchingM in G is a subset ofE such that no two edges in M share a node. The maximum matching problem for G is to find a matching inG with the maximum cardinality [9], [7]. Denote m the maximum cardinality among all matchings in G. In this paper, each individual of our EA is a matching. A popul ation consists of 2N individuals, whereN > 0 is an integer. For an individualM , define its fitness to be the cardinality |M | of the matching. For a population ξ, define its fitness as |ξ| = max{|M |; M ∈ ξ}. The evolutionary algorithm used to solve the maximum matchi ng problem can be described as follows. Crossover: Att-th generation (or timet), let M1 andM2 be two individuals in the population ξt. DenoteQM1(M2) (andQM2(M1)) to be the set of edges in M1 (andM2) that are matchable to M2 (andM1). Then the crossover can be described as: M ′ 1 = M1 + QM2(M1), M ′ 2 = M2 + QM1(M2). Let the intermediate population generated by crossover be ξ t . Mutation: For each individual M in populationξ t , choose an edge e uniformly at random fromE and generate a mutant as follows:

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