Results

**1 - 2**of**2**### APPROXIMATION ALGORITHMS FOR SOME MIN-MAX VEHICLE ROUTING PROBLEMS

, 2013

"... Vehicle routing problems are optimization problems that deal with the location and routing of vehi-cles. A set of clients based in different locations need to be served by a fleet of vehicles. The clients and vehicle depots are modeled as being placed on the vertices of a graph and the distances b ..."

Abstract
- Add to MetaCart

Vehicle routing problems are optimization problems that deal with the location and routing of vehi-cles. A set of clients based in different locations need to be served by a fleet of vehicles. The clients and vehicle depots are modeled as being placed on the vertices of a graph and the distances between them as a metric. Thus, a solution to a vehicle routing problem corresponds to covering the graph using a number of subgraphs, each denoting the route of a vehicle. In this thesis, we consider min-max vehicle routing problems, in which the maximum cost incurred by the subgraph corresponding to each vehicle is to be minimized. We study two types of covering problems and present new or improved approximation algorithms for them.

### Approximation Algorithms for Minimum-Load k-Facility Location

"... We consider a facility-location problem that abstracts settings where the cost of serving the clients as-signed 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 o ..."

Abstract
- Add to MetaCart

We consider a facility-location problem that abstracts settings where the cost of serving the clients as-signed 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 F ⊆ F of k facilities, and assign each client j to a facility f(j) ∈ F so as to minimize maxf∈F j∈C:f(j)=f d(f, j); we call j∈C:f(j)=f d(f, j) the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria ap-proximation algorithms for MLkFL for when F = C. MLkFL 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 polynomial time 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