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Approximation Algorithms for Minimum-Load k-Facility Location

Published: 16 April 2018 Publication History

Abstract

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (MLkFL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k≥ 0. Assigning client j to a facility f incurs a connection cost d(f,j). The goal is to open a set FF of k facilities and assign each client j to a facility f(j)∈ F so as to minimize maxf FjC:f(j)=fd(f,j); we call ∑ jC:f(j)=fd(f,j) the load of facility f. This problem was studied under the name of min-max star cover in References [3, 7], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F=C. MLk FL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics.
Our main result is the first polytime approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics and resilient to attack by a variety of techniques that have been successfully applied to facility-location problems. For instance, we show that (a) even a configuration-style LP-relaxation has a bad integrality gap and (b) a multi-swap k-median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL.
Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding of the problem and argue that a MILP-solution minimizing a certain potential function possesses the desired structure and then use a rounding algorithm for the generalized-assignment problem to “transfer” this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming.

References

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cover image ACM Transactions on Algorithms
ACM Transactions on Algorithms  Volume 14, Issue 2
April 2018
339 pages
ISSN:1549-6325
EISSN:1549-6333
DOI:10.1145/3196491
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 16 April 2018
Accepted: 01 December 2017
Revised: 01 August 2017
Received: 01 October 2016
Published in TALG Volume 14, Issue 2

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Author Tags

  1. Approximation algorithms
  2. lower bound
  3. min-max star cover
  4. minimum load k-facility location
  5. polynomial time approximation scheme

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  • Research-article
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  • Refereed

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  • Canada Research Chairs award
  • Ontario Early Researcher Award
  • NSERC
  • NSERC Discovery Accelerator Supplement Award

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