# Improved Approximation Algorithm for Fault-Tolerant Facility Placement

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8952)

## Abstract

We consider the Fault-Tolerant Facility Placement problem ($$FTFP$$), which is a generalization of the classical Uncapacitated Facility Location problem ($$UFL$$). In the $$FTFP$$ problem we have a set of clients $$C$$ and a set of facilities $$F$$. Each facility $$i \in F$$ can be opened many times. For each opening of facility $$i$$ we pay $$f_i \ge 0$$. Our goal is to connect each client $$j \in C$$ with $$r_j \ge 1$$ open facilities in a way that minimizes the total cost of open facilities and established connections.

In a series of recent papers $$FTFP$$ was essentially reduced to Fault-Tolerant Facility Location problem ($$FTFL$$) and then to $$UFL$$ showing it could be approximated with ratio $$1.575$$. In this paper we show that $$FTFP$$ can actually be approximated even better. We consider approximation ratio as a function of $$r = min_{j \in C}~r_j$$ (minimum requirement of a client). With increasing $$r$$ the approximation ratio of our algorithm $$\lambda _r$$ converges to one. Furthermore, for $$r > 1$$ the value of $$\lambda _r$$ is less than 1.463 (hardness of approximation of $$UFL$$). We also show a lower bound of 1.278 for the approximability of the $$FTFL$$ for arbitrary $$r$$. Already for $$r > 3$$ we obtain that $$FTFP$$ can be approximated with ratio 1.275, showing that under standard complexity theoretic assumptions $$FTFP$$ is strictly better approximable than $$FTFL$$.

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