Chapter

Approximation Algorithms for Combinatorial Optimization

Volume 2462 of the series Lecture Notes in Computer Science pp 185-199

Date:

Hardness of Approximation for Vertex-Connectivity Network-Design Problems

  • Guy KortsarzAffiliated withDepartment of Computer Sciences, Rutgers University
  • , Robert KrauthgamerAffiliated withInternational Computer Science Institute (ICSI) and Computer Science Division, University of California
  • , James R. LeeAffiliated withComputer Science Division, University of California

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Abstract

In the survivable network design problem SNDP, the goal is to find a minimum-cost subgraph satisfying certain connectivity requirements. We study the vertex-connectivity variant of SNDP in which the input specifies, for each pair of vertices, a required number of vertexdisjoint paths connecting them.

We give the first lower bound on the approximability of SNDP, showing that the problem admits no efficient \( 2^{\log ^{1 - \in } n} \) ratio approximation for any fixed ∈>0 unless NP ⊆ DTIME(n polylog(n)). We also show hardness of approximation results for several important special cases of SNDP, including constant factor hardness for the k-vertex connected spanning subgraph problem (k-VCSS) and for the vertex-connectivity augmentation problem, even when the edge costs are severely restricted.