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arXiv 2007.07554cs.DS
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最小权值成对距离保持器

Minimum Weight Pairwise Distance Preservers

Mojtaba Abdolmaleki, Yafeng Yin, Neda Masoud

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摘要：

本文研究了最小权值两两距离保持算法（MWPDP）问题。考虑正加权无向/有向连通图$G = (V, E, c)$和顶点对（也称为需求对）的子集$P$。一个子图$G'$是关于$P$的一个距离转换器当且仅当每对$(u, w) \in P$都满足$dist_{G'} (u, w) = dist_{G}(u, w)$。在MWPDP问题中，我们的目标是找到最小权重子图$G^*$，它是关于$P$的距离的一个子集。在$P$中的每一对之间采取最短路径给我们一个权重至多为$U=\sum_{(u,v) \in P} dist_{G} (u, w)$的平凡解。随后，我们问我们可以在$U$上做多大的改进。换句话说，我们选择找到最大化$U-c(G^*)$的距离转换器$G^*$。将此问题称为成本分担成对距离保持器（CSPDP），它在运输系统的规划和运营中有几个应用。唯一已知的可以为CSPDP提供非平凡解决方案的工作是Chlamtáet al.（SODA，2017）。该算法适用于无权图，只有当最优解相对于平凡解非常稀疏时才能保证非零目标。我们解决这个问题，提出了一个$O(|E|^{1/2+ε})$近似算法CSPDP在加权图中运行在$O((|P||E|)^{2.38} (1/ε))$的时间。此外，我们还证明了CSPDP至少和$\text{LABEL-COVER}_{\max}$一样难.这意味着CSPDP不能在多项式时间内在$O(|E|^{1/6-ε})$因子内近似，除非在众所周知的困难的$\text{LABEL-COVER}_{\max}$中有改进。

英文摘要：

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 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 $\text{LABEL-COVER}_{\max}$. 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 $\text{LABEL-COVER}_{\max}$.

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