# Inapproximability of Maximum Weighted Edge Biclique and Its Applications

## TL;DR

This paper proves strong inapproximability results for the maximum weighted edge biclique problem in bipartite graphs with both positive and negative weights, showing that efficient approximation algorithms are unlikely to exist.

## Contribution

It establishes new hardness bounds for the MWEB problem for a broad class of weight sets, and applies these results to bioinformatics and data compression problems.

## Key findings

- No polynomial-time approximation within a factor of n^{} exists unless RP=NP.
- The hardness results justify the use of heuristics in practical applications.
- Specific inapproximability results for biclustering in microarray analysis and data compression.

## Abstract

Given a 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 bipartite 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 $| \frac{\min\mathcal{S}} {\max \mathcal{S}} | \in \Omega(\eta^{\delta-1/2}) \cap O(\eta^{1/2-\delta})$ (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 $\mathsf{RP = NP}$. This hardness result gives justification of the heuristic approaches adopted for various applied problems in the aforementioned areas, and indicates that good approximation algorithms are unlikely to exist. Specifically, we give two applications by showing that: 1) finding statistically significant biclusters in the SAMBA model, proposed in \cite{Tan02} for the analysis of microarray data, is $n^{\epsilon}$-inapproximable; and 2) no polynomial time algorithm exists for the Minimum Description Length with Holes problem \cite{Bu05} unless $\mathsf{RP=NP}$.