# Minimum Weight Pairwise Distance Preservers

In this paper, we study the Minimum Weight Pairwise Distance Preservers (MWPDP) problem. Consider a positively weighted undirected/directed connected graph G = (V, E, c) and a subset P of pairs of vertices, also called demand pairs. A subgraph G' is a distance preserver with respect to P if and only if every pair (u, w) ∈ P satisfies dist_G' (u, w) = dist_G(u, w). In MWPDP problem, we aim to find the minimum-weight subgraph G^* that is a distance preserver with respect to P. Taking a shortest path between each pair in P gives us a trivial solution with the weight of at most U=∑_(u,v) ∈ P dist_G (u, w). Subsequently, we ask how much improvement we can make upon U. In other words, we opt to find a distance preserver G^* that maximizes U-c(G^*). Denote this problem as Cost Sharing Pairwise Distance Preservers (CSPDP), which has several applications in the planning and operations of transportation systems. The only known work that can provide a nontrivial solution for CSPDP is that of Chlamtáč et al. (SODA, 2017). This algorithm works for unweighted graphs and guarantees a non-zero objective only if the optimal solution is extremely sparse with respect to the trivial solution. We address this issue by proposing an O(|E|^1/2+ϵ)-approximation algorithm for CSPDP in weighted graphs that runs in O((|P||E|)^2.38 (1/ϵ)) time. Moreover, we prove CSPDP is at least as hard as LABEL-COVER_max. This implies that CSPDP cannot be approximated within O(|E|^1/6-ϵ) factor in polynomial time, unless there is an improvement in the notoriously difficult LABEL-COVER_max.

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