Dynamic Local Search for the Maximum Clique Problem

Abstract

In this paper, we introduce DLS-MC, a new stochastic local search algorithm for the maximum clique problem. DLS-MC alternates between phases of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, during which vertices of the current clique are swapped with vertices not contained in the current clique. The selection of vertices is solely based on vertex penalties that are dynamically adjusted during the search, and a perturbation mechanism is used to overcome search stagnation. The behaviour of DLS-MC is controlled by a single parameter, penalty delay, which controls the frequency at which vertex penalties are reduced. We show empirically that DLS-MC achieves substantial performance improvements over state-of-the-art algorithms for the maximum clique problem over a large range of the commonly used DIMACS benchmark instances

An new immune genetic algorithm based on uniform design sampling

“…Algorithm parameters are as follows: The population size is 100; the crossover and mutation probability: P c = 0.9, P m = 0.05; the maximum iteration is 100. The last column is known best results of DLS-MC and QUALEX in [10] So we can say UIGA better than the other two algorithms in searching capability, convergence speed, avoiding prematurity, etc., and UIGA could strengthen the local search, and also considered the global search simultaneously.…”

“…The experiments are carried on the simulate examples in DIMACS [12], and the result comparing with reference [10] is showed in Table 1.…”

“…1, 2, 3, we can get the results as follows: The solutions gained by UIGA are not inferior to reference [10], and UIGA can search for completely the best solutions to complex Hamming10-2 and Hamming10-4. To "Keller 4", UIGA gained the solutions superior to [10]; SGA and GGA can only get the solutions partly. Moreover, UIGA is superior to SGA and GGA on Ave, and so to search capability, convergence rate, etc.…”

“…A maximum clique is a clique of the largest possible size in a given graph. Maximum clique problem (MCP) [9][10][11] is to find the greatest clique for a given graph; it has important use in market analysis, fault diagnosis and bioinformation. MCP is also a typical NP-hard problem, so traditional deterministic algorithm cannot solve it effectively.…”

“…This paper combines genetic with immunity mechanism and presents an immune genetic algorithm based on uniform design sampling (UIGA). To prove the validity of UIGA, we solve the instance of the maximum clique problem (MCP) in DIMACS database [9][10][11][12]. The simulation results show that the UIGA is superior to simple GA and GGA, and also to DLS-MC, QUALEX.…”