A two-stage discrete optimization method for largest common subgraph problems

A novel combinatorial optimization algorithm called 2-stage discrete optimization method (2DOM) is proposed for the largest common subgraph problem (LCSP) in this paper. Given two graphs G = (V-1, E-1) and H = (V-2, E-2), the goal of LCSP is to find a subgraph G' = (V-1', E-1') of G and a subgraph H' = (V-2', E-2') of H such that G' and H' are not only isomorphic to each other but also their number of edges is maximized. The two graphs G' and H' are isomorphic when \V-1'\ = \V-2'\ and \E-1'\ = \E-2'\, and there exists one-to-one vertex correspondence f : V-1' --> V-2' such that {u, v} is an element of E-1' if and only if {f(u), f(v)} is an element of E-2'. LCSP is known to be NP-complete in general. The 2DOM consists of a construction stage and a refinement stage to achieve the high solution quality and the short computation time for large size difficult combinatorial optimization problems. The construction stage creates a feasible initial solution with considerable quality, based on a greedy heuristic method. The refinement stage improves it keeping the feasibility, based on a random discrete descent method. The performance is evaluated by solving two types of randomly generated 1200 LCSP instances with a maximum of 500 vertices for G and 1000 vertices for H. The simulation result shows the superiority of 2DOM to the simulated annealing in terms of the solution quality and the computation time.