Minimum Back-Walk-Free Latency Problem

Abstract

Consider a graph with n nodes V = {1, 2,..., n} and let d(i,j) denote the distance between nodes i,j. Given a permutation π on {1, 2,..., n} such that π(1) = 1, the back-walk-free latency from node 1 to node j is defined by l _π (j) = l_πj - 1) + min{d(π(k),π(j))| 1 ≤ k ≤j - 1}. Note that lπ(1) = d(1, 1) = 0. Each vertex i is associated with a nonnegative weight w(i). The (weighted) minimum back-walk-free latency problem (MBLP) is to find a permutation π such that the total back-walk-free latency Σ ⁿ=_i w(i)π(i) is minimized.

In this paper, we show an O(n log n) time algorithm when the given graph is a tree. For a k-path trees, we derive an O(n log k) time algorithm; the algorithm is shown to be optimal in term of time complexity on any comparison based computational model. Further, we show that the optimal tour on weighted paths can be found in O(n) time.

No previous hardness results were known for MBLP on general graphs. Here we settle the problem by showing that MBLP is NP-complete even when the given graph is a direct acyclic graph whose vertex weights are either 0 or 1.

The work is supported in part by the National Science Council, Taiwan, R.O.C, grant NSC-89-2218-E-126-004.