An Exact Parallel Algorithm for the Maximum Clique Problem

1998

Abstract We present an approach to clique computations in very large multi-digraphs. We discuss graph decomposition schemes used to break up the problem into several pieces of manageable dimensions. A two-stage (out-of-memory and in-memory) greedy randomized adaptive search procedure (GRASP) for nding approximate solutions to the maximum clique problem in very large sparse graphs is presented. We experiment with this heuristic on real data sets collected in the telecomunications industry.

Computers & Operations Research, 2011

This paper presents a new exact maximum clique algorithm which improves the bounds obtained in state of the art approximate coloring by reordering the vertices at each step. Moreover the algorithm can make full use of bit strings to sort vertices in constant time as well as to compute graph transitions and bounds efficiently, exploiting 10 the ability of CPUs to process bitwise operations in blocks of size the ALU register word. As a result it significantly outperforms a current leading algorithm.

… 2009. AICCSA 2009 …, 2009

A collection of practical methods is presented for solving the maximum clique problem exactly on graphs that are too large to fit within core memory. These methods use a combination of in-core and out-of-core techniques, recursively dissecting large graphs into manageable components. A global solution to the maximum clique problem is derived from local solutions generated for each of the individual components. Parallelizing the search within these components is instrumental in improving the running times of the algorithms.

Informatica

The problem of selecting a vertex to start an enumerative algorithm for the maximum clique problem is addressed in this paper. In this paper, we propose a new vertex selection strategy that is useful in exact algorithms for the maximum clique problem. We compare our approach with the myopic greedy selection strategy. Computational results for randomly generated graphs show that our strategy performs better than the myopic greedy selection strategy.

The purpose of this paper is to investigate a number of recently reported exact algorithms for the maximum clique problem. The actual program code used is presented and critiqued. The computational study aims to show how implementation details, problem features and hardware platforms influence algorithmic behaviour in those algorithms.

Journal of Heuristics, 2000

In this work, the NP-hard maximum clique problem on graphs is considered. Starting from basic greedy heuristics, modifications and improvements are proposed and combined in a two-phase heuristic procedure. In the first phase an improved greedy procedure is applied starting from each node of the graph; on the basis of the results of this phase a reduced subset of nodes is selected and an adaptive greedy algorithm is repeatedly started to build cliques around such nodes. In each restart the selection of nodes is biased by the maximal clique generated in the previous execution. Computational results are reported on the DIMACS benchmarks suite. Remarkably, the two-phase procedure successfully solves the difficult Brockington-Culberson instances, and is generally competitive with state-of-the-art much more complex heuristics.

Journal of Heuristics, 2008

Starting from an algorithm recently proposed by Pullan and Hoos, we formulate and analyze iterated local search algorithms for the maximum clique problem. The basic components of such algorithms are a fast neighbourhood search (not based on node evaluation but on completely random selection) and simple, yet very effective, diversification techniques and restart rules. A detailed computational study is performed in order to identify strengths and weaknesses of the proposed algorithms and the role of the different components on several classes of instances. The tested algorithms are very fast and reliable: most of the DIMACS benchmark instances are solved within very short CPU times. For one of the hardest tests, a new putative optimum was discovered by one of our algorithms. Very good performances were also shown on recently proposed and more difficult instances. It is important to remark that the heuristics tested in this paper are basically parameter free (the appropriate value for the unique parameter is easily identified and was, in fact, the same value for all problem instances used in this paper).

International Journal of Metaheuristics, 2019

A hybrid algorithm for the maximum clique problem is presented. A heuristic is used to generate cliques and these are improved by some simple optimizations and tabu search. All components of the algorithm make use of a pseudoexact algorithm, which is an exact algorithm with some specialized pruning. Preprocessing is useful for some instances. The algorithm is shown to be successful using standard and new benchmarks.

Annals of Operations Research, 1993

We describe two variants of a tabu search heuristic, a deterministic one and a probabilistic one, for the maximum clique problem. This heuristic may be viewed as a natural alternative implementation of tabu search for this problem when compared to existing ones. We also present a new random graph generator, the/S-generator, which produces graphs with larger clique sizes than comparable ones obtained by classical random graph generating techniques. Computational results on a large set of test problems randomly generated with this new generator are reported and compared with those of other approximate methods.

A B S T R A C T The maximum vertex weight clique problem (MVWCP) is a challenging NP-Hard combinatorial optimization problem that searches for a clique with maximum total sum of vertices' weights. In this study, we propose a robust and cooperative parallel tabu search algorithm (PTC) for the MVWCP. Our proposed algorithm uses a dedicated tabu search algorithm with a multistart strategy for the diversification of search space on a parallel computation environment. An effective seeding mechanism is developed with respect to the rank of the processors to choose diversified starting points for a better exploration of the search space. Classical add, swap and drop operators of tabu search are improved with parallel computation and a combined neighborhood approach. The PTC algorithm is evaluated on a set of 120 problem instances from DIMACS-W and BHOSLIB-W benchmarks. Computational results show that the PTC algorithm competes with state-of-the-art heuristic algorithms by reporting average best (optimal) result hit ratios up to 99.0%.

This paper presents algorithmic and implementation enhancements,of Reactive Local Search algorithm for the Maximum,Clique problem [3]. In addition, we build an empirical complexity model for the CPU time required for a single iteration, and we show that with a careful imple- mentation of the data-structures one can achieve a speedup of at least an order of magnitude dierence,for large size graphs.

2007

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.

Journal of Chemical Information and Modeling, 2013

A new exact parallel maximum clique algorithm MaxCliquePara, which finds the maximum clique (the fully connected subgraph) in undirected general and protein graphs, is presented. First, a new branch and bound algorithm for finding a maximum clique on a single computer core, which builds on ideas presented in two published state of the art sequential algorithms is implemented. The new sequential MaxCliqueSeq algorithm is faster than the reference algorithms on both DIMACS benchmark graphs as well as on protein-derived product graphs used for protein structural comparisons. Next, the MaxCliqueSeq algorithm is parallelized by splitting the branch-and-bound search tree to multiple cores, resulting in MaxCliquePara algorithm. Ability to exploit all cores efficiently makes the new parallel MaxCliquePara algorithm markedly superior to other tested algorithms. On a 12-core computer the parallelization provides up to two orders of magnitude faster execution on the large DIMACS benchmark graphs and up to an order of magnitude faster execution on protein product graphs. The algorithms are freely accessible on http://commsys.ijs.si/~matjaz/maxclique.

Algorithms, 2020

Let G=(V,E) be an undirected graph with vertex set V and edge set E. A clique C of G is a subset of the vertices of V with every pair of vertices of C adjacent. A maximum clique is a clique with the maximum number of vertices. A tabu search algorithm for the maximum clique problem that uses an exact algorithm on subproblems is presented. The exact algorithm uses a graph coloring upper bound for pruning, and the best such algorithm to use in this context is considered. The final tabu search algorithm successfully finds the optimal or best known solution for all standard benchmarks considered. It is compared with a state-of-the-art algorithm that does not use exact search. It is slower to find the known optimal solution for most instances but is faster for five instances and finds a larger clique for two instances.