Local Search Algorithms for k-Median and k-Facility Location Problems with Linear Penalties

We present two local search algorithms for the k-median and k-facility location problems with linear penalties k-MLP and k-FLPLP, two extensions of the classical k-median and k-facility location problems respectively. We show that the approximation ratios of these two algorithms are $$3+2/p+\epsilon $$3+2/p+∈ for the k-MLP, and $$2 + 1/p + \sqrt{3+ 2/p+ 1/p^2} + \epsilon $$2+1/p+3+2/p+1/p2+∈ for the k-FLPLP, respectively, where $$p \in {\mathbb {Z}}_+$$p∈Z+ is a parameter of the algorithms and $$\epsilon >0$$∈>0 is a positive number. In particular, the $$3+2/p+\epsilon $$3+2/p+∈-approximation improves the best known 4-approximation for the k-MLP for any $$p>2$$p>2.