A unified approach to source location via rooted network augmentation

In Source Location (SL) problems the goal is to select a minimum cost source set S such that the connectivity from S to any node v is at least the demand dv of v. In Network Augmentation (NA) problems we are given a graph G = (V,E) and an edge-set F on V , edge-costs on F or node-costs on V , and connectivity requirements {rsv : sv ∈ D} on a set D of “demand edges”. The goal is to compute a minimum cost edge-set I ⊆ F , such that in the graph G + I, the connectivity of each sv ∈ D is at least rsv. In Rooted NA D is a star with center s, and in a-Based NA F is a star with center a. We suggest a notion of qconnectivity, where every node u has capacity qu ≥ 1 that represents the resistance of u to failures. We observe that a large class of SL problems, including the variants that appear in the literature, is a particular case of q-connectivity s-Based Rooted NA problems. We use this to derive some approximation algorithms for SL from those for NA, as well as to derive some new results for SL problems. Some of our results are as follows. (i) We observe that directed SL with unit costs is Ω(logn)-hard to approximate even with 0, 1 demands, while some versions of SL with uniform demands can be solved in polynomial time. (ii) We show that the H(|D|)-approximation algorithm of [14] for edge costs a-Based NA has ratio H(α) for the q-connectivity version with node-costs, where pmax is the maximum number of parallel edges in F , α = min{r(D), |D| · pmax}, and H(j) = ∑j i=1 1/i is the jth harmonic number. This gives a simple unifying H(d(V ))-approximation algorithm for several SL variants considered separately by Sakashita, Makino, and Fujishige [21]. Moreover, we will show that this is so whenever the connectivity function is submodular. (iii) For undirected a-Based NA we give an approximation algorithm with ratio min{k, pmax ln k} ·O(ln k) for node-costs and O(ln k) for edgecosts, where k = maxv∈V dv is the maximum demand; the latter improves the ratio Õ(k) of Fukunaga [6]. (iv) We consider a variant of SL where we also have edge-costs and flowcost bounds {bv : v ∈ V }, and require that for every node v, the minimum cost of a flow of value dv from S to v is at most bv. We show that the problem admits approximation ratio H(d(V )) +H(c(E)).