Approximation Algorithms and Hardness Results for Labeled Connectivity Problems

Let G = (V, E) be a connected multigraph, whose edges are associated with labels specified by an integer-valued function L : E → N. In addition, each label ` ∈ N has a non-negative cost c(`). The minimum label spanning tree problem (MinLST) asks to find a spanning tree in G that minimizes the overall cost of the labels used by its edges. Equivalently, we aim at finding a minimum cost subset of labels I ⊆ N such that the edge set {e ∈ E : L(e) ∈ I} forms a connected subgraph spanning all vertices. Similarly, in the minimum label s-t path problem (MinLP) the goal is to identify an s-t path minimizing the combined cost of its labels. The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP.