Abstract
In this paper, we single out the following particular case of the clique search problem. The vertices of the given graph are legally colored with k colors and we are looking for a clique with k nodes in the graph. In other words, we want to decide if a given k-partite graph contains a clique with k nodes. The maximum clique problem asks for finding a maximum clique in a given finite simple graph. The problem of deciding if the given graph contains a clique with k vertices is called the k-clique problem. The first problem is NP-hard and the second one is NP-complete. The special clique search problem, we propose, is still an NP-complete problem. We will show that the k-clique problem in the special case of k-partite graphs is more tractable than in the general case. In order to illustrate the possible practical utility of this restricted type clique search problem we will show that the job shop scheduling problem can be reduced to such a clique search problem in a suitable constructed graph. We carry out numerical experiments to assess the efficiency of the approach. It is a common practice that before one embarks on a large scale clique search typically one attempts to simplify and tidy up the given graph. This procedure is commonly referred as preconditioning or kernelization of the given graph. Of course, the preconditioning or kernelization is meant with respect to the given type of clique search problem. The other main topic of the paper is to describe a number of kernelization methods tailored particularly to the proposed special k-clique problem. Some of these techniques works in connection with the generic k-clique problem. In these situations, we will see that they are more efficient in the case of k-partite graphs. Some other preconditioning methods applicable only to k-partite graphs. We illustrate how expedient these preconditioning methods can be by solving non-trivial scheduling problems to optimality employing only kernelization techniques dispensing with exhaustive clique search algorithms altogether.
Highlights
This paper is part of a larger on going project of the authors
The nodes of a finite simple graph G are legally colored with k colors and we want to decide if G admits a k-clique or not
We describe different graph kernelization methods based on legal coloring of the nodes, on dominance relations and on transformations of edges into nodes
Summary
This paper is part of a larger on going project of the authors. Namely, we are considering different combinatorial optimization problems that can be reduced to a clique search applied to a specially constructed auxiliary graph. Only finite simple graphs will appear, that is, the graphs have finitely many nodes and finitely many edges. For each finite simple graph G there is an integer k such that the nodes of G can be legally colored using k colors but the nodes of G cannot be legally colored using (k − 1) colors This well defined integer k is called the chromatic number of G and is denoted by χ( G ). A finite simple graph is called a k-partite graph if its nodes can be legally colored using k colors. For a general finite simple graph G computing χ( G ) is an NP-hard problem It is a remarkable property of the class of perfect graphs that the problem of computing their clique and chromatic numbers belong to the P complexity class
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