Approximation algorithms for movement repairmen

In the Movement Repairmen (MR) problem we are given a metric space (V, d) along with a set R of k repairmen r₁, r₂,...,r_k with their start depots s₁, s₂,...,s_k ∈ V and speeds v₁, v₂,...,v_k ≥ 0 respectively and a set C of m clients c₁, c₂,...,c_m having start locations s₁′, s₂′,...,s _m′ ∈ V and speeds v₁′, 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.