Characterizing Demand Graphs for (Fixed-Parameter) Shallow-Light Steiner Network

We consider the Shallow-Light Steiner Network problem from a fixed-parameter perspective. Given a graph G, a distance bound L, and p pairs of vertices (s1, t1), . . . , (sp, tp), the objective is to find a minimum-cost subgraph G′ such that si and ti have distance at most L in G′ (for every i ∈ [p]). Our main result is on the fixed-parameter tractability of this problem with parameter p. We exactly characterize the demand structures that make the problem “easy”, and give FPT algorithms for those cases. In all other cases, we show that the problem is W[1]-hard. We also extend our results to handle general edge lengths and costs, precisely characterizing which demands allow for good FPT approximation algorithms and which demands remain W[1]-hard even to approximate.