The Heterogeneous Capacitated $k$-Center Problem

TL;DR

This paper introduces the heterogeneous capacitated k-center problem, addressing facility capacity assignment and client assignment to minimize maximum client-facility distance, and provides approximation algorithms with capacity violations.

Contribution

It formulates the heterogeneous capacitated k-center problem, relates it to the Santa Claus problem, and develops new approximation algorithms with capacity violation guarantees.

Findings

Quasi-polynomial time $O(rac{ ext{log} n}{ ext{epsilon}})$-approximation with capacity violation by $1+ ext{epsilon}$.

Polynomial time $O(1)$-approximation with capacity violation by $O( ext{log} n)$.

Improved results for soft-capacities allowing multiple facilities at the same location.

Abstract

In this paper we initiate the study of the heterogeneous capacitated $k$-center problem: given a metric space $X = (F \cup C, d)$, and a collection of capacities. The goal is to open each capacity at a unique facility location in $F$, and also to assign clients to facilities so that the number of clients assigned to any facility is at most the capacity installed; the objective is then to minimize the maximum distance between a client and its assigned facility. If all the capacities $c_i$'s are identical, the problem becomes the well-studied uniform capacitated $k$-center problem for which constant-factor approximations are known. The additional choice of determining which capacity should be installed in which location makes our problem considerably different from this problem, as well the non-uniform generalizations studied thus far in literature. In fact, one of our contributions is in relating the heterogeneous problem to special-cases of the classical Santa Claus problem. Using this connection, and by designing new algorithms for these special cases, we get the following results: (a)A quasi-polynomial time $O(\log n/\epsilon)$-approximation where every capacity is violated by $1+\varepsilon$, (b) A polynomial time $O(1)$-approximation where every capacity is violated by an $O(\log n)$ factor. We get improved results for the {\em soft-capacities} version where we can place multiple facilities in the same location.