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

The Minimum Weight t-partite Clique Problem MWtCP is the problem of finding a t-clique with minimum weight in a complete edgeweighted 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 MWtCP is NP-hard, APX-hard in the general case. We also present a 2-approximation algorithm that runs in O(n) for the metric case and has 1+ 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 MWtCP to gene cluster discovery are presented. Submitted: August 2018 Reviewed: January 2019 Revised: March 2019 Reviewed: July 2019 Revised: September 2019 Accepted: March 2020 Final: March 2020 Published: April 2020 Article type: Regular Paper Communicated by: F. Vandin E-mail addresses: gasolano@up.edu.ph (Geoffrey Solano) guillaume.blin@labri.fr (Guillaume Blin) mathieu.raffinot@labri.fr (Mathieu Raffinot) jhoiclemente@gmail.com (Jhoirene Clemente) jdlcaro@up.edu.ph (Jaime Caro) 172 Solano, Blin, Raffinot, Clemente & Caro Approximability of MWtCP