Maximum Clique Search in Circulant k-Hypergraphs

Abstract

The search for max-cliques in graphs is a well established NP-complete problem in graph theory and algorithm design, with many algorithms designed to make use of internal structures of specific types of graphs. We study the extension of the problem of searching for max-cliques in graphs to hypergraphs with constant edge size k, and adapt existing algorithms for graphs to work in k-hypergraphs. In particular, we are interested in the generalization of circulant graphs to circulant k-hypergraphs, and provide a definition of this type of hypergraph. We design and implement a new algorithm to perform max-clique searches on circulant k-hypergraphs. This algorithm combines ideas from a Russian doll algorithm for max-cliques in graphs (Ostergard 2002) with an algorithm based on necklaces for a class of circulant k-hypergraphs (Tzanakis, Moura, Stevens and Panario 2016). We examine the performance of our new algorithm against a set of adapted algorithms (backtracking and Russian doll search for general k-hypergraphs, and necklace-based search for circulant k-hypergraphs) in a set of benchmarking experiments across various densities and edge sizes. This study reveals that the new algorithm outperforms the others when edge density of the hypergraph is high, and that the pure necklace-based algorithm is best in the case of low densities. Finally, we use our new algorithm to perform an exhaustive search on circulant 4-hypergraphs constructed from linear feedback shift register sequences on finite fields of order q that yields covering arrays. The search is completed for 2 <= q <= 5 which solves the open case of q=5 left by Tzanakis et al.