The Capacitated Vehicle Routing problem is to find a minimum-cost set of tours that collectively cover clients in a graph, such that each tour starts and ends at a specified depot and is subject to a capacity bound on the number of clients it can serve. In this paper, we present a polynomial-time approximation scheme (PTAS) for instances in which the input graph is planar and the capacity is bounded. Previously, only a quasipolynomial-time approximation scheme was known for these instances. To obtain this result, we show how to embed planar graphs into bounded-treewidth graphs while preserving, in expectation, the client-to-client distances up to a small additive error proportional to client distances to the depot.
Approximation Algorithms for Replenishment Problems with Fixed Turnover Times
We introduce and study a class of optimization problems we call replenishment problems with fixed turnover times: a very natural model that has received little attention in the literature. Clients with capacity for storing a certain commodity are located at various places; at each client the commodity depletes within a certain time, the turnover time, which is constant but can vary between locations. Clients should never run empty. The natural feature that makes this problem interesting is that we may schedule a replenishment (well) before a client becomes empty, but then the next replenishment will be due earlier also. This added workload needs to be balanced against the cost of routing vehicles to do the replenishments. In this paper, we focus on the aspect of minimizing routing costs. However, the framework of recurring tasks, in which the next job of a task must be done within a fixed amount of time after the previous one is much more general and gives an adequate model for many practical situations. Note that our problem has an infinite time horizon. However, it can be fully characterized by a compact input, containing only the location of each client and a turnover time. This more »
- Publication Date:
- NSF-PAR ID:
- 10369259
- Journal Name:
- Algorithmica
- Volume:
- 84
- Issue:
- 9
- Page Range or eLocation-ID:
- p. 2597-2621
- ISSN:
- 0178-4617
- Publisher:
- Springer Science + Business Media
- Sponsoring Org:
- National Science Foundation
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