Covering the Edges of Bipartite Graphs Using K 2, 2 Graphs

An optimization problem arising in the design of optical networks is shown here to be abstracted by the following model of covering the edges of a bipartite graph with a minimum number of 4-cycles, or K2,2: Given a bipartite graph G = (L,R,E) over the node set L ∪ R with E ⊆ {[u, v] : u ∈ L, v ∈ R}, and the implicit collection of all fournode cycles in the complete bipartite graph over L∪R. The goal is to cover all the edges in E with a sub-collection of graphs G1, G2, ..., Gp, of minimum size, where each Gi is a subgraph of a cycle over four nodes – a 4-cycle. Since a subgraph of a 4-cycle can cover up to 4 edges, this covering problem is a special case of the unweighted 4-set cover problem. This specialization allows us to obtain an improved approximation guarantee. Whereas the currently best known approximation algorithm for the general unweighted 4-set cover problem has an approximation ratio of H4 − 98 195 ≈ 1.58077 (where Hp ≈ ln p denotes the p-th harmonic number), we show that for every 2 > 0, there is a polynomial time ( 13 10 + 2)-approximation algorithm for our problem. Our analysis of the greedy algorithm shows that when applied to covering a bipartite graph using copies of Kq,q bicliques, it returns a feasible solution whose cost is at most (Hq2 − Hq + 1)OPT + 1 where OPT denotes the optimal cost, thus improving the approximation bound for unweighted q-set cover by a factor of almost 2.