Linear Programming Heuristics for the Graph Isomorphism Problem

An isomorphism between two graphs is a bijection between their vertices that preserves the edges. We consider the problem of determining whether two finite undirected weighted graphs are isomorphic, and finding an isomorphism relating them if the answer is positive. In this paper we introduce effective probabilistic linear programming (LP) heuristics to solve the graph isomorphism problem. We motivate our heuristics by showing guarantees under some conditions, and present numerical experiments that show effectiveness of these heuristics in the general case. 1 Graph isomorphism problem 1.1 Problem statement Consider two weighted undirected graphs, each with n vertices labeled 1, . . . , n, described by their adjacency matrices A, à ∈ Rn×n, where Aij is the weight on the edge in the first graph between vertices i and j, and zero if there is no edge between vertices i and j (and similarly for Ã). Since the graphs are undirected, the adjacency matrices are symmetric. The two graphs are isomorphic if there is a permutation of the vertices of the first graph that makes the first graph the same as the second. This occurs if and only if there is a permutation matrix P ∈ Rn×n (i.e., a matrix with exactly one entry in each row and column that is one, with the other zero) that satisfies PAP T = Ã. We will say that the permutation matrix P transforms A to à if PAP T = Ã, or equivalently, PA = ÃP . The graph isomorphism problem (GIP) is to determine whether such a permutation matrix exists, and to find one if so. GIP can be formulated as the (feasibility) optimization problem find P subject to PA = ÃP P1 = 1, P 1 = 1 Pij ∈ {0, 1}, i, j = 1, . . . , n, (1) with variable P ∈ Rn×n, where 1 is the vector with all entries one. The data for this problem are the adjacency matrices of the two graphs, A and Ã. The constraints on the last two lines