Constant factor approximation of vertex-cuts in planar graphs

  • Authors:

  • Eyal Amir;Robert Krauthgamer;Satish Rao

  • Affiliations:

  • University of California, Berkeley, CA;University of California, Berkeley, CA;University of California, Berkeley, CA

  • Venue:
  • Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
  • Year:
  • 2003

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Abstract

We devise the first constant factor approximation algorithm for minimum quotient vertex-cuts in planar graphs. Our algorithm achieves approximation ratio 1+4/3(1+ε) with running time O(W• n3+2/ε), where W is the total weight of the vertices. The approximation ratio improves to 4/3(1+ε+o(1)) if there is an optimal quotient vertex-cut (A*,B*,C*) where the weight of C* is of low order compared to those of A* and B*; this holds, for example, when the input graph has uniform weights and costs. The ratio further improves to 1+ε+o(1) if, in addition, min[w(A*),w(B*)] ≤ 1/3 W.We use our algorithm for quotient vertex-cuts to achieve the first constant-factor pseudo-approximation for vertex separators in planar graphs.Our technical contribution is two-fold. First, we prove a structural theorem for planar graphs, showing the existence of a near-optimal quotient vertex-cut whose high-level structure is that of a bounded-depth tree. Second, we develop an algorithm that optimizes over such complex structures in running time that depends (exponentially) not on the size of the structure, but rather only on its depth. These techniques may be applicable in other problems.