Speedup in the Traveling Repairman Problem with Unit Time Windows

📝 Abstract

The input to the unrooted traveling repairman problem is an undirected metric graph and a subset of nodes, each of which has a time window of unit length. Given that a repairman can start at any location, the goal is to plan a route that visits as many nodes as possible during their respective time windows. A polynomial-time bicriteria approximation algorithm is presented for this problem, gaining an increased fraction of repairman visits for increased speedup of repairman motion. For speedup $s $, we find a $6\gamma/(s + 1) $-approximation for $s$ in the range $1 \leq s \leq 2$ and a $4\gamma/s $-approximation for $s$ in the range $2 \leq s \leq 4 $, where $\gamma = 1$ on tree-shaped networks and $\gamma = 2 + \epsilon$ on general metric graphs.

💡 Analysis

The input to the unrooted traveling repairman problem is an undirected metric graph and a subset of nodes, each of which has a time window of unit length. Given that a repairman can start at any location, the goal is to plan a route that visits as many nodes as possible during their respective time windows. A polynomial-time bicriteria approximation algorithm is presented for this problem, gaining an increased fraction of repairman visits for increased speedup of repairman motion. For speedup $s $, we find a $6\gamma/(s + 1) $-approximation for $s$ in the range $1 \leq s \leq 2$ and a $4\gamma/s $-approximation for $s$ in the range $2 \leq s \leq 4 $, where $\gamma = 1$ on tree-shaped networks and $\gamma = 2 + \epsilon$ on general metric graphs.

📄 Content

arXiv:0907.5372v1 [cs.DS] 30 Jul 2009 Speedup in the Traveling Repairman Problem with Unit Time Windows Greg N. Frederickson Barry Wittman July 7, 2018 Abstract The input to the unrooted traveling repairman problem is an undirected metric graph and a subset of nodes, each of which has a time window of unit length. Given that a repairman can start at any location, the goal is to plan a route that visits as many nodes as possible during their respective time windows. A polynomial-time bicriteria approximation algorithm is presented for this problem, gaining an increased fraction of repairman visits for increased speedup of repairman motion. For speedup s, we find a 6γ/(s + 1)-approximation for s in the range 1 ≤s ≤2 and a 4γ/s-approximation for s in the range 2 ≤s ≤4, where γ = 1 on tree-shaped networks and γ = 2 + ǫ on general metric graphs. 1 Introduction When planning the route for an agent who needs to drop offsupplies, make repairs, deliver impor- tant information, or other similar tasks, distance traveled may not be the only important metric. The usefulness of visiting a particular location may be heavily dependent on time. Consider the example of a client who schedules a window of time during which he or she will be home. An appliance repairman will be able to perform repairs only during that time window. In the last decade, the algorithms community has achieved much progress with time-sensitive routing problems of this kind. These problems typically identify the locations to be visited and the cost of traveling between them as the nodes and edges, respectively, of a weighted, undirected graph. For example, the orienteering problem [1, 2, 5, 7, 9] seeks to find a path that visits as many nodes as possible before a global time deadline. The deadline traveling salesman problem [2] generalizes this problem further by allowing each location to have its own deadline. This problem can be generalized even further to the traveling repairman problem [2, 4, 8, 12, 16, 19] by allowing each location to have a time window during which it must be visited to receive credit. In [12], we introduced the first polynomial-time algorithms that give constant approximations to the traveling repairman problem with unit-time windows whenever the underlying graph is a tree or a metric graph. In this version of the problem, a time window is identified with each service request whose location is given by the node with which it is associated. Let a repairman earn a specified profit for visiting a service request during its time window. A planned sequence of visits made by traveling at a given speed is called a service run. The goal of the traveling repairman problem is to plan a service run that maximizes profit. Unlike much of the preceding literature [2, 4], we considered the unrooted version of the problem, in which the repairman may start at any time from any location and stop similarly. Although the unrooted problem is no harder than the rooted problem, which specifies a starting point, it is a fascinating and difficult problem in its own right. Both problems are NP-hard when the graph is a tree [12] and APX-hard for a general metric graph [13]. 1 As a counterpoint to the repairman problem, we also introduced the speeding deliveryman problem in [12, 13], with an alternative optimization paradigm, namely speedup. The input to the speeding deliveryman problem is the same as the input to the traveling repairman problem, but the goal is to find the minimum speed necessary to visit all service requests during their time windows and thus collect all profit. The deliveryman problem is similarly hard, and in [12, 13] we introduced polynomial-time approximation algorithms for it. In both the repairman and deliveryman problems, our algorithms from [12, 13] rely on trimming windows so that the resulting time windows are pairwise either identical or non-overlapping. To trim time windows, we first make divisions in time every .5 time units, starting at time 0. We call the time interval that starts at a particular division and continues up to the next division a period. Because we allow no window to start on a period boundary, each time window will completely overlap exactly one period and partially overlap the two neighboring periods. Trimming then removes those parts of each window that fall outside of the completely overlapped period. This process of trimming incurs a penalty in our approximations that is a reduction by a factor of 1/3 in the number of requests serviced by the repairman and an increase by a factor of 4 in the speedup needed by the deliveryman. Yet one might expect that a spectrum of performance is possible between these two extremes. If we have any particular speedup s greater than 1 but less than 4, we expect an increase in the number of serviced requests, proportionate in some sense to s. Indeed, this is so, as we show in this paper. The concept of speedup has been used in the scheduling community since its introduction in [15].

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