Approximation Algorithms for the k-Clique Covering Problem

The problem of covering edges and vertices in a graph (or in a hypergraph) was motivated by a problem arising in the context of the component assembly problem. The problem is as follows: given a graph and a clique size $k$, find the minimum number of $k$-cliques such that all edges and vertices of the graph are covered by (included in) the cliques. This paper provides a collection of approximation algorithms for various clique sizes with proven worst-case bounds. The problem has a natural extension to hypergraphs, for which we consider one particular class. The $k$-clique covering problem can be formulated as a set covering problem. It is shown that the algorithms we design, which exploit the structure of this special set covering problem, have better performance than those derived from direct applications of general purpose algorithms for the set covering. In particular, these special classes of set covering problems can be solved with better worst-case bounds and/or complexity than if treated as general set covering problems.