Covering the edges of bipartite graphs using K2,2gaphs

  • Authors:

  • Dorit S. Hochbaum;Asaf Levin

  • Affiliations:

  • Department of Industrial Engineering and Operations Research and Walter A. Haas School of Business, University of California, Berkeley;Department of Statistics, The Hebrew University, Jerusalem, Israel

  • Venue:
  • WAOA'07 Proceedings of the 5th international conference on Approximation and online algorithms
  • Year:
  • 2007

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Abstract

We consider an optimization problem arising in the design of optical networks. We are given a bipartite graph G = (L,R,E) over the node set L ∪ R where the edge set is E ⊆ {[u, v] : u ∈ L, v ∈ R}, and implicitly a collection of all four-nodes cycles in the complete graph over V. The goal is to find a minimum size sub-collection of graphs G1,G2, . . . , Gp where for each i Gi is isomorphic to a cycle over four nodes, and such that the edge set E is contained in the union (over all i) of the edge sets of Gi. Noting that every four edge cycle can be a part of the solution, 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 - 196/390 ≅ 1.58077 (where Hp denotes the p-th harmonic number), we show that for every Ɛ 0 there is a polynomial time (13/10 + Ɛ)-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 by a factor of almost 2.