A Primal-Dual Approximation Algorithm for the Survivable Network Design Problem in Hypergraph

Abstract

Given a hypergraph with nonnegative costs on hyperedge and a requirement function r : 2^V → Z ⁺, where V is the vertex set, we consider the problem of finding a minimum cost hyperedge set F such that for all S ⊑ V, F contains at least r(S) hyperedges incident to S. In the case that r is weakly supermodular (i.e., r(V) = 0 and r(A) + r(B) ≤ max{r(A ∩ B) + r(A ∪ B), r(A - B) + r(B - A)} for any A,B ⊑ V), and the so-called minimum violated sets can be computed in polynomial time, we present a primal-dual approximation algorithm with performance guarantee d _maxH(r _max), where d _max is the maximum degree of the hyperedges with positive cost, r _max is the maximum requirement, and H(i) = ∑ⁱ _j=11/j is the harmonic function. In particular, our algorithm can be applied to the survivable network design problem in which the requirement is that there should be at least r _st hyperedge-disjoint paths between each pair of distinct vertices s and t, for which r _st is prescribed.