Minimum augmentation of edge-connectivity with monotone requirements in undirected graphs

Abstract

For a finite ground set V , we call a set-function r : 2 V → Z + monotone, if r ( X ′ ) ≥ r ( X ) holds for each X ′ ⊆ X ⊆ V , where Z + is the set of nonnegative integers. Given an undirected multigraph G = ( V , E ) and a monotone requirement function r : 2 V → Z + , we consider the problem of augmenting G by a smallest number of new edges, so that the resulting graph G ′ satisfies d G ′ ( X ) ≥ r ( X ) for each 0̸ ≠ X ⊂ V , where d G ( X ) denotes the degree of a vertex set X in G . This problem includes the edge-connectivity augmentation problem, and in general, it is NP-hard, even if a polynomial time oracle for r is available. In this paper, we show that the problem can be solved in O ( n 4 ( m + n log n + q ) ) time, under the assumption that each 0̸ ≠ X ⊂ V satisfies r ( X ) ≥ 2 whenever r ( X ) > 0 , where n = | V | , m = | { { u , v } ∣ ( u , v ) ∈ E } | , and q is the time required to compute r ( X ) for each X ⊆ V .