Locating Depots for Capacitated Vehicle Routing

Inge Li Gørtz, Viswanath Nagarajan

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    Abstract

    We study a location-routing problem in the context of capacitated vehicle routing. The input to k-LocVRP is a set of demand locations in a metric space and a fleet of k vehicles each of capacity Q. The objective is to locate k depots, one for each vehicle, and compute routes for the vehicles so that all demands are satisfied and the total cost is minimized. Our main result is a constant-factor approximation algorithm for k-LocVRP. To achieve this result, we reduce k-LocVRP to the following generalization of k median, which might be of independent interest. Given a metric (V, d), bound k and parameter ρ ∈ R+, the goal in the k median forest problem is to find S ⊆ V with |S| = k minimizing: E u∈V d(u, S) + ρ · d(MST(V/S) ), where d(u, S) = minw∈S d(u,w) and MST(V/S) is a minimum spanning tree in the graph obtained by contracting S to a single vertex. We give a (3+E)-approximation algorithm for k median forest, which leads to a (12+E)-approximation algorithm for k-LocVRP, for any constant E > 0. The algorithm for k median forest is t-swap local search, and we prove that it has locality gap 3 + 2 t ; this generalizes the corresponding result for k median [3]. Finally we consider the k median forest problem when there is a different (unrelated) cost function c for the MST part, i.e. the objective is Eu∈V d(u, S) + c(MST(V/S) ). We show that the locality gap for this problem is unbounded even under multi-swaps, which contrasts with the c = d case. Nevertheless, we obtain a constant-factor approximation algorithm, using an LP based approach along the lines of [12].
    Original languageEnglish
    Title of host publicationApproximation, Randomization, and Combinatorial Optimization : 14th InternationalWorkshop, APPROX 2011 and 15th InternationalWorkshop, RANDOM 2011 Princeton, NJ, USA, August 17-19, 2011 Proceedings
    Volume6845
    PublisherSpringer
    Publication date2011
    Pages230-241
    ISBN (Print)978-3-642-22934-3
    ISBN (Electronic)978-3-642-22935-0
    DOIs
    Publication statusPublished - 2011
    EventInternational Workshop on Approximation Algorithms for Combinatorial Optimization Problems, and the International Workshop on Randomization and Computation - Princeton, New Jersey, USA
    Duration: 1 Jan 2011 → …
    Conference number: 14 & 15

    Conference

    ConferenceInternational Workshop on Approximation Algorithms for Combinatorial Optimization Problems, and the International Workshop on Randomization and Computation
    Number14 & 15
    CityPrinceton, New Jersey, USA
    Period01/01/2011 → …
    SeriesLecture Notes in Computer Science
    ISSN0302-9743

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