Capacitated Facility Location with Outliers/Penalties.

In this paper, we present a framework to design approximation algorithms for capacitated facility location problems with penalties/outliers by rounding a solution to the standard LP. Primal-dual technique, which has been particularly successful in dealing with outliers and penalties, has not been very successful in dealing with capacities. For example, despite unbounded integrality gap for facility location problem with outliers (FLwO), Charikar et al. [7] were able to get around it by guessing the maximum facility opening cost in the optimal and provide a primal-dual solution for the problem. On the other hand, no primal-dual solution has been able to break the integrality gap of the capacitated facility location problem (CFL). (Standard) LP-Rounding techniques had also not been very successful in dealing with capacities until a recent work by Grover et al. [13]. Their constant factor approximation violating the capacities by a small factor (1+epsilon) is promising while dealing with capacities. Though LP-rounding has not been very promising while dealing with penalties and outliers, we successfully apply it to deal with them along with capacities. Solutions obtained by using LP-rounding techniques are interesting as they are easy to integrate with other LP-based algorithms. We apply our framework to obtain first constant factor approximations for capacitated facility location problem with outlier (CFLwO) and capacitated k-facility location problem with penalty (CkFLwP) for hard uniform capacities. Our solutions incur slight violations in capacities, (1+epsilon) for the problems without cardinality(k) constraint and (2+epsilon) for the problems with the cardinality constraint. For the outlier variant, we also incur a small loss (1+ epsilon) in outliers. Due to the unbounded integrality gaps in the underlying problems, the violations are inevitable while rounding solutions to the standard LP. Thus we achieve the best possible by rounding the solution of natural LP for these problems. To the best of our knowledge, no results are known for CFLwO and CkFLwP. For uniform facility opening cost, we get rid of violation in capacities for CFLwO. As a byproduct, we obtain first constant factor approximations for (i) Capacitated k-Median with penalties(CkMwP) (ii) the uncapacitated variants of FLwO, kMwP and kFLwP using LP-rounding.