Inapproximability of Maximum Weighted Edge Biclique via Randomized Reduction and Its Applications
Jinsong Tan · arXiv (Cornell University) · 2007
Given a complete bipartite graph $G = (V_1,V_2,E)$ where edges take on {\it both} positive and negative weights from set $\mathcal{S}$, the {\it maximum weighted edge biclique} problem, or $\mathcal{S}$-MWEB for short, asks to find a biclique subgraph whose sum of edge weights is maximized. This problem has various applications in bioinformatics, machine learning and databases and its (in)approximability remains open. In this paper, we show that for a wide range of choices of $\mathcal{S}$, specifically when $| \frac{\min_{w}\{w\in \mathcal{S} \}} {\max_{w}\{w\in \mathcal{S} \}} | \in \Omega(\eta^{\delta-1/2}) \cap O(\eta^{1/2-\delta})$ (where $\eta = \max\{|V_1|, |V_2|\}$, and $\delta \in (0,1/2]$), no polynomial time algorithm can approximate $\mathcal{S}$-MWEB within a factor of $n^{\epsilon}$ for some $\epsilon > 0$ unless $\mathsf{RP = NP}$. We also give applications of this result by showing that finding statistically significant biclusters in the SAMBA model \cite{Tan02} is $n^{\epsilon}$-inapproximable and no polynomial time algorithm exists for the Minimum Description Length with Holes problem \cite{Bu05} unless $\mathsf{RP=NP}$.