On the Approximability of the Minimum Weight $t$-partite Clique Problem

Abstract

The ${\rm M{\small INIMUM}~W\small{EIGHT}}$ $t$-${\rm {\small PARTITE}~C{\small LIQUE}~P{\small ROBLEM}~MW} t {\rm CP}$ is the problem of finding a $t$-clique with minimum weight in a complete edge-weighted $t$-partite graph. The motivation for studying this problem is its potential in modelling the problem of identifying sets of commonly existing putative co-regulated, co-expressed genes, called gene clusters. In this paper, we show that ${\rm MW} t {\rm CP}$ is NP-hard, APX-hard in the general case. We also present a 2-approximation algorithm that runs in $O(n^2)$ for the metric case and has 1+$ \frac{1}{t}$-approximation performance guarantee for the ultrametric subclass of instances. We further show how relaxing or tightening the application of the metricity property affects the approximation ratio. Finally insights on the application ${\rm MW} t {\rm CP}$ to gene cluster discovery are presented.