A constant factor approximation algorithm for boxicity of circular arc graphs

  • Authors:

  • Abhijin Adiga;Jasine Babu;L. Sunil Chandran

  • Affiliations:

  • Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India;Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India;Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India

  • Venue:
  • WADS'11 Proceedings of the 12th international conference on Algorithms and data structures
  • Year:
  • 2011

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Abstract

Boxicity of a graph G(V, E) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional axis parallel boxes in Rk. Equivalently, it is the minimum number of interval graphs on the vertex set V such that the intersection of their edge sets is E. It is known that boxicity cannot be approximated even for graph classes like bipartite, co-bipartite and split graphs below O(n0.5-ε)-factor, for any ε 0 in polynomial time unless NP = ZPP. Till date, there is no well known graph class of unbounded boxicity for which even an nε-factor approximation algorithm for computing boxicity is known, for any ε k)-factor polynomial time approximation algorithm for computing the boxicity of any circular arc graph along with a corresponding box representation, where k ≥ 1 is its boxicity. For Normal Circular Arc(NCA) graphs, with an NCA model given, this can be improved to an additive 2-factor approximation algorithm. The time complexity of the algorithms to approximately compute the boxicity is O(mn+n2) in both these cases and in O(mn+kn2) which is at most O(n3) time we also get their corresponding box representations, where n is the number of vertices of the graph and m is its number of edges. The additive 2-factor algorithm directly works for any Proper Circular Arc graph, since computing an NCA model for it can be done in polynomial time.