Approximation Algorithms and Hardness Results for Labeled Connectivity Problems

Abstract

Let G = (V,E) be a connected multigraph, whose edges are associated with labels specified by an integer-valued function ${\mathcal{L}} : E \rightarrow \mathbb{N}$. In addition, each label l∈ℕ to which at least one edge is mapped has a non-negative cost c( l). 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 $\{ e \in E : {\mathcal {L}}( e ) \in I \}$ forms a connected subgraph spanning all vertices. Similarly, in the minimum label s-tpath problem (MinLP) the goal is to identify an s-t path minimizing the combined cost of its labels, where s and t are provided as part of the input. The main contributions of this paper are improved approximation algorithms and hardness results for MinLST and MinLP. As a secondary objective, we make a concentrated effort to relate the algorithmic methods utilized in approximating these problems to a number of well-known techniques, originally studied in the context of integer covering.