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Serving Rides of Equal Importance for Time-Limited Dial-a-Ride

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12755)

Abstract

We consider a variant of the offline Dial-a-Ride problem with a single server where each request has a source, destination, and a prize earned for serving it. The goal for the server is to serve requests within a given time limit so as to maximize the total prize money. We consider the variant where prize amounts are uniform which is equivalent to maximizing the number of requests served. This setting is applicable when all rides may have equal importance such as paratransit services. We first prove that no polynomial-time algorithm can be guaranteed to serve the optimal number of requests, even when the time limit for the algorithm is augmented by any constant factor \(c \ge 1\). We also show that if \(\lambda = t_{max}/t_{min}\), where \(t_{max}\) and \(t_{min}\) denote the largest and smallest edge weights in the graph, the approximation ratio for a reasonable class of algorithms for this problem is unbounded, unless \(\lambda \) is bounded. We then show that the segmented best path (\(\textsc {sbp}\)) algorithm from [8] is a 4-approximation. We then present our main result, an algorithm, k-Sequence, that repeatedly serves the fastest set of k remaining requests, and provide upper and lower bounds on its performance. We show k-Sequence has approximation ratio at most \(2+\lceil \lambda \rceil /k\) and at least \(1 + \lambda /k\) and that \(1 + \lambda /k\) is tight when \(1 + \lambda /k \ge k\). Thus, for the case of \(k=1\), i.e., when the algorithm repeatedly serves the quickest request, it has approximation ratio \(1+\lambda \), which is tight for all \(\lambda \). We also show that even as k grows beyond the size of \(\lambda \), the ratio never improves below 9/7.

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Fig. 1.
Fig. 2.
Fig. 3.

Notes

  1. 1.

    We note that any simple, undirected, connected, weighted graph is allowed as input, with the simple pre-processing step of adding an edge wherever one is not present whose distance is the length of the shortest path between its two endpoints.

  2. 2.

    Note that the algorithm need not take a full time segment to move from one set of requests to another, but it is specified this way for convenience of analysis. Excluding this buffer time in the algorithm specification does not improve its approximation ratio since one can construct an instance where each move requires the full time segment.

  3. 3.

    Note that when the graph is complete, \(t_{max}\) (\(t_{min}\)) is the maximum (minimum) distance over all pairs of nodes. Otherwise, using the pre-processing described in the footnote 1 in Sect. 2, we have that the distance between any two non-adjacent nodes is the shortest distance between those nodes, and \(t_{max}\) (\(t_{min}\)) is the maximum (minimum) distance over all of these distances.

  4. 4.

    Note that if \(1+\lambda /k \ge k\), then \(\lambda (k-1)/k + 1 \ge k\).

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Anthony, B.M., Christman, A.D., Chung, C., Yuen, D. (2021). Serving Rides of Equal Importance for Time-Limited Dial-a-Ride. In: Pardalos, P., Khachay, M., Kazakov, A. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2021. Lecture Notes in Computer Science(), vol 12755. Springer, Cham. https://doi.org/10.1007/978-3-030-77876-7_3

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  • DOI: https://doi.org/10.1007/978-3-030-77876-7_3

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