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 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 for any two vertices at distance $d$, the corresponding super-vertices remain at distance at least $\varphi(d)$ in the contracted graph, where $\varphi$ is a tolerance function bounding the permitted distance distortion. We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions $\varphi(x)=x/\alpha-\beta$, where $\alpha\geq 1$ and $\beta\geq 0$ 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. Further we analyze the asymptotic behavior of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results.