Distance-Preserving Graph Contractions

Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework for contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible. Formally, given an edge-weighted graph, the contraction should guarantee that any two vertices at distance d remain at distance at least φ(d) in the resulting graph, where φ is a non-decreasing tolerance function that bounds the permitted distance distortion. In this paper, we present a comprehensive picture of the algorithmic complexity of the corresponding maximization problem for affine tolerance functions φ(x) = x/α−β, where α ≥ 1 and β ≥ 0 are arbitrary real-valued parameters. Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases.