The time complexity of maximum matching by simulated annealing

Reviewer: Patrick J. Ryan

Simulated annealing has recently been introduced as a heuristic method for solving optimization problems. The authors remark that no analysis of the asymptotic complexity of this method has been done. The purpose of this paper is to study the application of a particular class of algorithms to the maximum matching problem in graph theory. The maximum matching problem is to find a matching (i.e., a set of edges such that no two share a vertex) of maximum cardinality for a given (undirected) graph. Deterministic algorithms exist that solve this problem in polynomial time. The authors consider an algorithm that generates a sequence of matchings, beginning with the empty set, by successive moves in which either one edge is added (probability p), one edge is removed (probability q), or the configuration remains unchanged (probability 1 ? p ? q). The probabilities are allowed to be functions of all previous states of the system, including the current state. For each type of move, all legal moves of that type are equally likely to be chosen. The performance of such an algorithm on a specific family of graphs G n is analyzed. The authors show that positive constants a and b exist such that, no matter how cleverly the functions p and q are chosen, the expected number of moves required to find a maximum matching is greater than a exp( bn). The graph G n, defined explicitly in the paper, has 2( n + 1) 2 vertices (most of which have degree n + 2) and a total number of edges ( n + 1) 3. A result is also obtained in the opposite direction for a slightly different algorithm and an arbitrary graph. The algorithm again starts from the empty set, but proceeds by choosing an edge of the original graph at random (all edges are equally likely). If the edge is not already in the current matching and can be added legally, this is done. If, on the other hand, the edge is already in the matching, either it is removed (probability p) or the matching remains unchanged (probability 1 ? p). In this algorithm, p depends on the original graph but is otherwise constant; in particular, p does not depend on the current or previous states. The authors also obtain an upper bound for such an algorithm on the expected number of moves required to find a matching whose cardinality comes within a prescribed tolerance factor of the maximum cardinality. They use a value of p that depends on the number of vertices, the maximum degree, and the tolerance factor itself. If the tolerance factor is fixed, the upper bound obtained is polynomial in the number of vertices. Further, if the maximum degree is bounded, the upper bound is O( v 5), where v is the number of vertices. The paper requires about 13 pages of calculations to obtain these two results. This seems to indicate that asymptotic analysis for even the most simple Monte Carlo algorithms is much more difficult than for many deterministic algorithms, and that at least for finding exact solutions the Monte Carlo algorithms tend to be less efficient. If approximate solutions are adequate, however, the Monte Carlo approach holds some promise.