## Abstract

In the Movement Repairmen (MR) problem we are given a metric space (V, d) along with a set R of k repairmen r_{1}, r_{2},...,r_{k} with their start depots s_{1}, s_{2},...,s_{k} ∈ V and speeds v_{1}, v_{2},...,v_{k} ≥ 0 respectively and a set C of m clients c_{1}, c_{2},...,c_{m} having start locations s_{1}′, s_{2}′,...,s _{m}′ ∈ V and speeds v_{1}′, v _{2}′,...,v_{m}′ ≥ 0 respectively. If t is the earliest time a client c_{j} is collocated with any repairman (say, r_{i}) at a node u, we say that the client is served by r_{i} at u and that its latency is t. The objective in the (Sum-MR) problem is to plan the movements for all repairmen and clients to minimize the sum (average) of the clients latencies. The motivation for this problem comes, for example, from Amazon Locker Delivery [Ama10] and USPS gopost [Ser10]. We give the first O(log n)-approximation algorithm for the Sum-MR problem. In order to solve Sum-MR we formulate an LP for the problem and bound its integrality gap. Our LP has exponentially many variables, therefore we need a separation oracle for the dual LP. This separation oracle is an instance of Neighborhood Prize Collecting Steiner Tree (NPCST) problem in which we want to find a tree with weight at most L collecting the maximum profit from the clients by visiting at least one node from their neighborhoods. The NPCST problem, even with the possibility to violate both the tree weight and neighborhood radii, is still very hard to approximate. We deal with this difficulty by using LP with geometrically increasing segments of the time line, and by giving a tricriteria approximation for the problem. The rounding needs a relatively involved analysis. We give a constant approximation algorithm for Sum-MR in Euclidean Space where the speed of the clients differ by a constant factor. We also give a constant approximation for the makespan variant.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 16th International Workshop, APPROX 2013 and 17th International Workshop, RANDOM 2013, Proceedings |

Pages | 218-232 |

Number of pages | 15 |

DOIs | |

State | Published - 2013 |

Externally published | Yes |

Event | 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013 and the 17th International Workshop on Randomization and Computation, RANDOM 2013 - Berkeley, CA, United States Duration: 21 Aug 2013 → 23 Aug 2013 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 8096 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2013 and the 17th International Workshop on Randomization and Computation, RANDOM 2013 |
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Country/Territory | United States |

City | Berkeley, CA |

Period | 21/08/13 → 23/08/13 |