On Independent Sets and Bicliques in Graphs

Serge Gaspers, Dieter Kratsch, Mathieu Liedloff
2010 Algorithmica  
Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this paper we improve Prisner's upper bound on the number of maximal bicliques [Combinatorica, 2000] and show that the maximum number of maximal bicliques in a graph on n vertices is Θ(3 n/3 ). Our major contribution is an exact exponential-time algorithm. This branching algorithm computes the number of distinct maximal independent sets in a graph in time O(1.3642 n ), where n is the
more » ... mber of vertices of the input graph. We use this counting algorithm and previously known algorithms for various other problems related to independent sets to derive algorithms for problems related to bicliques via polynomial-time reductions. application in data mining is based on the formal concept analysis [16] where each concept is a maximal biclique of a bipartite graph. Previous work. The complexity of algorithmic problems on bicliques has been studied extensively. First results were mentioned by Garey and Johnson [15], among them the NP-completeness of the balanced complete bipartite subgraph problem. The maximum biclique problem is polynomial for bipartite graphs [7] , and NP-hard for general graphs [32] . The maximum edge biclique problem was shown to be NP-hard by Peeters [26] . Approximation algorithms for node and edge deletion biclique problems are given by Hochbaum [18] . Enumerating maximal bicliques has attracted a lot of attention in the last decade. The algorithms in [23, 24] enumerate all maximal bicliques of a bipartite graph as concepts during the construction of the concept lattice. Nowadays there are polynomial delay enumeration algorithms for maximal (proper) bicliques in bipartite graphs [10, 21] and general graphs [9] . There are also polynomial delay algorithms to enumerate all maximal non-induced bicliques of a graph [2, 10] .
doi:10.1007/s00453-010-9474-1 fatcat:mfks63jsivgf3j5waurcs7klt4