Covering the edges of bipartite graphs using K2,2 graphs

  • Authors:

  • Dorit S. Hochbaum;Asaf Levin

  • Affiliations:

  • Department of Industrial Engineering and Operations Research, University of California, Berkeley, United States and Walter A. Haas School of Business, University of California, Berkeley, United St ...;Faculty of Industrial Engineering and Management, The Technion, 32000 Haifa, Israel

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2010

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Abstract

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 K"2","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 four-node 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 G"1,G"2,...,G"p, of minimum size, where each G"i 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 H"4-98195~1.58077 (where H"p~lnp denotes the p-th harmonic number), we show that, for every @e0, there is a polynomial time (1310+@e)-approximation algorithm for our problem. Our analysis of the greedy algorithm shows that, when applied to covering a bipartite graph using copies of K"q","q bicliques, it returns a feasible solution whose cost is at most (H"q"^"2-H"q+1)OPT+1 where OPT denotes the optimal cost, thus improving the approximation bound for unweighted q^2-set cover by a factor of almost 2.