Approximation Algorithms for Capacitated k-Travelling Repairmen Problems

Authors Christopher S. Martin, Mohammad R. Salavatipour



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Christopher S. Martin
Mohammad R. Salavatipour

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Christopher S. Martin and Mohammad R. Salavatipour. Approximation Algorithms for Capacitated k-Travelling Repairmen Problems. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 56:1-56:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.ISAAC.2016.56

Abstract

We study variants of the capacitated vehicle routing problem. In the multiple depot capacitated k-travelling repairmen problem (MD-CkTRP), we have a collection of clients to be served by one vehicle in a fleet of k identical vehicles based at given depots. Each client has a given demand that must be satisfied, and each vehicle can carry a total of at most Q demand before it must resupply at its original depot. We wish to route the vehicles in a way that obeys the constraints while minimizing the average time (latency) required to serve a client. This generalizes the Multi-depot k-Travelling Repairman Problem (MD-kTRP) [Chekuri and Kumar, IEEE-FOCS, 2003; Post and Swamy, ACM-SIAM SODA, 2015] to the capacitated vehicle setting, and while it has been previously studied [Lysgaard and Wohlk, EJOR, 2014; Rivera et al, Comput Optim Appl, 2015], no approximation algorithm with a proven ratio is known. We give a 42.49-approximation to this general problem, and refine this constant to 25.49 when clients have unit demands. As far as we are aware, these are the first constant-factor approximations for capacitated vehicle routing problems with a latency objective. We achieve these results by developing a framework allowing us to solve a wider range of latency problems, and crafting various orienteering-style oracles for use in this framework. We also show a simple LP rounding algorithm has a better approximation ratio for the maximum coverage problem with groups (MCG), first studied by Chekuri and Kumar [APPROX, 2004], and use it as a subroutine in our framework. Our approximation ratio for MD-CkTRP when restricted to uncapacitated setting matches the best known bound for it [Post and Swamy, ACM-SIAM SODA, 2015]. With our framework, any improvements to our oracles or our MCG approximation will result in improved approximations to the corresponding k-TRP problem.
Keywords
  • approximation
  • capacitated
  • latency
  • group coverage

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References

  1. A. Ageev and M. Sviridenko. Pipage rounding: A new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization, 8(3):307-328, 2004. Google Scholar
  2. Kemal Altinkemer and Bezalel Gavish. Heursitics for unequal weight delivery problems with a fixed error guarantee. Operations Research Letters, 6(4):149-158, 1987. Google Scholar
  3. Aaron Archer and Anna Blasiak. Improved approximation algorithms for the minimum latency problem via prize-collecting strolls. 21st ACM SODA, pages 429-447, 2010. Google Scholar
  4. Aaron Archer, Asaf Levin, and David P Williamson. A faster, better approximation algorithm for the minimum latency problem. SIAM Journal on Computing, 37(5):1472-1498, 2008. Google Scholar
  5. A. Blum, P. Chalasani, B. Coppersmith, B. Pulleyblank, P. Raghavan, and M. Sudan. The minimum latency problem. 26th ACM STOC, pages 163-171, 1994. Google Scholar
  6. Gruia Călinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM J. Comput., 40(6):1740-1766, 2011. URL: http://dx.doi.org/10.1137/080733991.
  7. B. Carr and S. Vempala. Randomized meta-rounding. In Proceedings of STOC, 2000. Google Scholar
  8. Deeparnab Chakrabarty and Chaitanya Swamy. Facility location with client latencies: Linear-programming based techniques for minimum-latency problems. 15th IPCO, pages 92-103, 2011. Google Scholar
  9. Kamalika Chaudhuri, Godfrey Brighten, Satish Rao, and Kunal Talwar. Paths, trees, and minimum latency tours. 44th IEEE-FOCS, pages 36-45, 2003. Google Scholar
  10. Chandra Chekuri and Amit Kumar. Maximum coverage problem with group budget constraints and applications. Approximation, Randomization, and Combinatorial Optimization, Algorithms and Techniques, pages 72-83, 2004. Google Scholar
  11. Jittat Fakcharoenphol, Chris Harrelson, and Satish Rao. The k-traveling repairman problem. 14th ACM-SIAM SODA, pages 655-664, 2003. Google Scholar
  12. Zachary Friggstad and Chaitanya Swamy. Approximation algorithms for regret-bounded vehicle routing and applications to distance-constrained vehicle routing. In Proceedings of STOC, pages 744-753, 2014. Google Scholar
  13. Michel Goemans and Jon Kleinberg. An improved approximation ratio for the minimum latency problem. Mathematical Programming, 82(1-2):111-124, 1998. Google Scholar
  14. Jens Lysgaard and Sanne W ohlk. A branch-and-cut-and-price algorithm for the cumulative capacitated vehicle routing problem. European Journal of Operational Research, 236(3):800-810, 2014. Google Scholar
  15. Ian Post and Chaitanya Swamy. Linear-programming based approximation algorithms for multi-vehicle minimum latency problems. 26th ACM-SIAM SODA, pages 512-531, 2015. Google Scholar
  16. Juan Carlos Rivera, H. Murat Afsar, and Christian Prins. A multistart iterated local search for the multitrip cumulative capacitated vehicle routing problem. Computational Optimization and Applications, 61(1):159-187, 2015. Google Scholar
  17. René Sitters. The minimum latency problem is NP-hard for weighted trees. IPCO, 2337:230-239, 2002. Google Scholar
  18. René Sitters. Polynomial time approximation schemes for the travelling repairman and other minimum latency problems. 25th ACM-SIAM SODA, 2014. Google Scholar
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