Minimum Weight Pairwise Distance Preservers

TL;DR

This paper introduces an approximation algorithm for the Minimum Weight Pairwise Distance Preservers problem in weighted graphs, improving upon previous work and establishing its computational hardness.

Contribution

It presents the first $O(|E|^{1/2+ ext{epsilon}})$-approximation algorithm for CSPDP in weighted graphs and proves the problem's hardness based on LABEL-COVER complexity.

Findings

Provides an $O(|E|^{1/2+ ext{epsilon}})$-approximation algorithm

Establishes CSPDP as hard as LABEL-COVER$_{ ext{max}}$

Shows no polynomial approximation within $O(|E|^{1/6- ext{epsilon}})$ unless LABEL-COVER$_{ ext{max}}$ improves.

Abstract

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 $(u, w) \in P$ 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 $U=\sum_{(u,v) \in P} dist_{G} (u, w)$. 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 Chlamt\'a\v{c} 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 $O(|E|^{1/2+\epsilon})$-approximation algorithm for CSPDP in weighted graphs that runs in $O((|P||E|)^{2.38} (1/\epsilon))$ time. Moreover, we prove CSPDP is at least as hard as $\text{LABEL-COVER}_{\max}$. This implies that CSPDP cannot be approximated within $O(|E|^{1/6-\epsilon})$ factor in polynomial time, unless there is an improvement in the notoriously difficult $\text{LABEL-COVER}_{\max}$.