Reactive and dynamic local search for Max-Clique , an experimental comparison

This paper presents results of an ongoing investigation about how different algorithmic building blocks contribute to solving the Maximum Clique problem. We consider greedy constructions, plateau searches, and more complex schemes based on dynamic penalties and/or prohibitions, in particular the recently proposed technique of Dynamic Local Search and the previously proposed Reactive Local Search. In addition we consider in detail the effect of the low-level implementation choices on the CPU time per iteration. We present experimental results on randomly generated graphs with different statistical properties, showing the crucial effects of the implementation, the robustness of different techniques, and their empirical scalability. 1 Reactive search for maximum clique Reactive Search, see [4, 5] for seminal papers, advocates the use of machine learning to automate the parameter tuning process and make it an integral and fully documented part of the algorithm. Learning is performed on-line, and therefore task-dependent and local properties of the configuration space can be used. In this way a single algorithmic framework maintains the flexibility to deal with related problems through an internal feedback loop that considers the previous history of the search, see also [7] for a proposal for learning domain-specific backtracking-based algorithms. The Maximum Clique problem in graphs (MC for short) is a paradigmatic combinatorial optimization problem with relevant applications [13], including information retrieval, computer vision, and social network analysis. Recent interest includes computational biochemistry, bio-informatics and genomics, see for example [12, 9]. The problem is NPhard and strong negative results have been shown about its approximability [10], making it an ideal testbed for search heuristics. Let G = (V,E) be an undirected graph, V = {1, 2, . . . , n} its vertex set, E ⊆ V × V its edge set, and G(S) = (S,E ∩ S × S) the subgraph induced by S, where S is a subset of V . A graph G = (V,E) is complete if all its vertices are pairwise adjacent, i.e., ∀i, j ∈ V, (i, j) ∈ E. A clique K is a subset of V such that G(K) is complete. The Maximum Clique problem asks for a clique of maximum cardinality. A Reactive Local Search (RLS) algorithm for the solution of the Maximum-Clique problem is proposed in [3, 6]. RLS is based on local search complemented by a feedback (history-sensitive) scheme to determine the amount of diversification. The reaction acts on the single parameter that decides the temporary prohibition of selected moves in the neighborhood. The performance obtained in computational tests appears to be significantly better with respect to all algorithms tested at the the second DIMACS implementation challenge (1992/93). Recently, a stochastic local search algorithm (DLS-MC) is developed in [15]. It is based on a clique expansion phase followed by a plateau search after a maximal clique is encountered. Diversification uses vertex penalties which are dynamically adjusted during the search, a ”forgetting” mechanism decreasing the penalties is added, and vertex degrees are not considered in the selection. The authors report a very good performance on the DIMACS instances after preliminary extensive optimization phase to determine the optimal penalty delay parameter for each instance. While the number of iterations (additions or deletions of nodes to the current clique) is in some cases larger than that of competing techniques, the small complexity of each iteration when the algorithm is realized through efficient supporting data structures leads to smaller overall CPU times. The initial motivation of this work is threefold. First, we want to investigate how the different algorithmic building blocks contribute to effectively solving max-clique instances corresponding to random graphs with different statistical properties. In particular, the investigation considers the effects of using the vertex degree information during the search, starting from simple to more complex techniques. Second, we want to assess how different implementations of the supporting data structures affect CPU times. For example, it may be the case that larger CPU times are caused by using a high-level language implementation w.r.t. low-level ”pointer arithmetic”. Having available the original software simplified the starting point for this analysis. Third, the DIMACS benchmark set (developed in 1992) has been around for more than a decade and there is a growing risk that the desire to get better and better results on http://dimacs.rutgers.edu/Challenges/