Approximating min-max k-clustering

Abstract

We consider the
problems
  of set partitioning into $k$ clusters with  minimum of the maximum cost of a cluster. The  cost function is given by an oracle, and we assume that it satisfies some natural  structural constraints.  That is, we assume that the cost function is monotone, the cost  of a singleton is zero, and we assume that for all $S cap S' 
eq emptyset$ the following holds
  $c(S) + c(S') geq c(S cup S')$.   For this problem we present
a $(2k-1)$-approximation algorithm for $kgeq 3$, a
2-approximation algorithm for $k=2$, and we also show a lower
bound of $k$ on the performance guarantee of any
  polynomial-time algorithm.

We then consider special cases of this problem arising in vehicle routing problems, and present improved results.