### Abstract

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization : 14th InternationalWorkshop, APPROX 2011 and 15th InternationalWorkshop, RANDOM 2011 Princeton, NJ, USA, August 17-19, 2011 Proceedings |

Volume | 6845 |

Publisher | Springer |

Publication date | 2011 |

Pages | 230-241 |

ISBN (Print) | 978-3-642-22934-3 |

ISBN (Electronic) | 978-3-642-22935-0 |

DOIs | |

Publication status | Published - 2011 |

Event | International 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

Conference | International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, and the International Workshop on Randomization and Computation |
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Number | 14 & 15 |

City | Princeton, New Jersey, USA |

Period | 01/01/2011 → … |

Series | Lecture Notes in Computer Science |
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ISSN | 0302-9743 |

### Cite this

*Approximation, Randomization, and Combinatorial Optimization: 14th InternationalWorkshop, APPROX 2011 and 15th InternationalWorkshop, RANDOM 2011 Princeton, NJ, USA, August 17-19, 2011 Proceedings*(Vol. 6845, pp. 230-241). Springer. Lecture Notes in Computer Science https://doi.org/10.1007/978-3-642-22935-0_20

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*Approximation, Randomization, and Combinatorial Optimization: 14th InternationalWorkshop, APPROX 2011 and 15th InternationalWorkshop, RANDOM 2011 Princeton, NJ, USA, August 17-19, 2011 Proceedings.*vol. 6845, Springer, Lecture Notes in Computer Science, pp. 230-241, International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, and the International Workshop on Randomization and Computation, Princeton, New Jersey, USA, 01/01/2011. https://doi.org/10.1007/978-3-642-22935-0_20

**Locating Depots for Capacitated Vehicle Routing.** / Gørtz, Inge Li; Nagarajan, Viswanath.

Research output: Chapter in Book/Report/Conference proceeding › Article in proceedings › Research › peer-review

TY - GEN

T1 - Locating Depots for Capacitated Vehicle Routing

AU - Gørtz, Inge Li

AU - Nagarajan, Viswanath

PY - 2011

Y1 - 2011

N2 - 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].

AB - 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].

U2 - 10.1007/978-3-642-22935-0_20

DO - 10.1007/978-3-642-22935-0_20

M3 - Article in proceedings

SN - 978-3-642-22934-3

VL - 6845

SP - 230

EP - 241

BT - Approximation, Randomization, and Combinatorial Optimization

PB - Springer

ER -