# Approximation algorithms for the robust facility leasing problem

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## Abstract

In this paper, we consider the robust facility leasing problem (RFLE), which is a variant of the well-known facility leasing problem. In this problem, we are given a facility location set, a client location set of cardinality *n*, time periods \(\{1, 2, \ldots , T\}\) and a nonnegative integer \(q < n\). At each time period *t*, a subset of clients \(D_{t}\) arrives. There are *K* lease types for all facilities. Leasing a facility *i* of a type *k* at any time period *s* incurs a leasing cost \(f_i^{k}\) such that facility *i* is opened at time period *s* with a lease length \(l_k\). Each client in \(D_t\) can only be assigned to a facility whose open interval contains *t*. Assigning a client *j* to a facility *i* incurs a serving cost \(c_{ij}\). We want to lease some facilities to serve at least \(n-q\) clients such that the total cost including leasing and serving cost is minimized. Using the standard primal–dual technique, we present a 6-approximation algorithm for the RFLE. We further offer a refined 3-approximation algorithm by modifying the phase of constructing an integer primal feasible solution with a careful recognition on the leasing facilities.

## Keywords

Facility leasing problem Robust Approximation algorithm Primal–dual## Notes

### Acknowledgements

The second author is supported by Natural Science Foundation of China (No. 11531014). The third author is supported by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and by Higher Educational Science and Technology Program of Shandong Province (No. J17KA171). The fourth author is supported by Higher Educational Science and Technology Program of Shandong Province (No. J15LN22) and the Science and Technology Development Plan Project of Jinan City (No. 201401211).

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