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 = (V₁, E₁) and H = (V₂. E₂), 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} ∈ E′₁ if and only if {f(u),/(w)} ∈ 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.