Exact algorithms for the minimum latency problem

Abstract

Let G = (V, E,w) be an undirected graph with positive weight w(e) on each edge e ∈ E. Given a starting vertex s ∈ V and a subset U ⊂ V as the demand vertex set, the minimum latency problem (MLP) asks for a tour P starting at s and visiting each demand vertex at least once such that the total latency of all demand vertices is minimized, in which the latency of a vertex is the length of the path from s to the first visit of the vertex. The MLP is an important problem in computer science and operations research, and is also known as the delivery man problem or the traveling repairman problem. Similar to the well-known traveling salesperson problem (TSP), in the MLP we are asked to find an “optimal” way for routing a server passing through the demand vertices. The difference is the objective functions. The latency of a vertex can be thought of as the delay of the service. In the MLP we care about the total delay (service quality), while the total length (service cost) is concerned in the TSP. The MLP on a metric space is NP-hard and also MAX-SNP-hard [4]. Polynomial time algorithms are only known for very special graphs, such as paths [1, 6], edge-unweighted trees [9], trees of diameter 3 [4], trees of constant number of leaves [8], or graphs with similar structure [12]. Even for caterpillars (paths with edges sticking out), no polynomial time algorithm has been reported. In a recent work, it is shown that the MLP on edge-weighted trees is NP-hard [11]. Due to the NP-hardness, many works ∗corresponding author (bangye@mail.stu.edu.tw)