Approximation algorithms for k-level stochastic facility location problems

Abstract

In the k-level facility location problem (FLP), we are given a set of facilities, each associated with one of k levels, and a set of clients. We have to connect each client to a chain of opened facilities spanning all levels, minimizing the sum of opening and connection costs. This paper considers the k-level stochastic FLP, with two stages, when the set of clients is only known in the second stage. There is a set of scenarios, each occurring with a given probability. A facility may be opened in any stage, however, the cost of opening a facility in the second stage depends on the realized scenario. The objective is to minimize the expected total cost. For the stage-constrained variant, when clients must be served by facilities opened in the same stage, we present a $(4-o(1))$-approximation, improving on the 4-approximation by Wang et al. (Oper Res Lett 39(2):160–161, 2011) for each k. In the case with $k=2,\,3$, the algorithm achieves factors 2.56 and 2.78, resp., which improves the $(3+\epsilon )$-approximation for $k=2$ by Wu et al. (Theor Comput Sci 562:213–226, 2015). For the non-stage-constrained version, we give the first approximation for the problem, achieving a factor of 3.495 for the case with $k = 2$, and $2k-1+o(1)$ in general.