最大权边二局部的不可逼近性及其应用
Inapproximability of Maximum Weighted Edge Biclique and Its Applications
摘要：
给定一个二部图$G = (V_1,V_2,E)$，其中边从集合$\mathcal{S}$中取{ \it两者}正和负权重，{ \it最大加权边biclique}问题，或简称$\mathcal{S}$ -MWEB，要求找到一个边权重和最大化的二部子图。这个问题在生物信息学、机器学习和数据库中有各种应用，它的（不）近似性仍然是开放的。在本文中，我们表明，对于广泛的$\mathcal{S}$的选择，特别是当$| \frac{\min\mathcal{S}} {\max \mathcal{S}} | \in Ω(η^{δ-1/2}) \cap O(η^{1/2-δ})$（其中$η= \max\{|V_1|, |V_2|\}$，和$δ\in (0,1/2]$），没有多项式时间算法可以近似$\mathcal{S}$-MWEB在$n^ε$的一个因素，一些$ε> 0$，除非$\mathsf{RP = NP}$。这个硬度的结果给出了理由的启发式方法，在上述领域的各种应用问题，并表明，良好的近似算法是不可能存在的。具体来说，我们给出了两个应用程序，表明：1）发现统计上显着的双聚类的SAMBA模型，提出\cite{Tan02}的微阵列数据的分析，是$n^ε$ -不可逼近的;和2）没有多项式时间算法存在的最小描述长度与孔问题\cite{Bu05}，除非$\mathsf{RP=NP}$。
英文摘要：
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 $| \frac{\min\mathcal{S}} {\max \mathcal{S}} | \in Ω(η^{δ-1/2}) \cap O(η^{1/2-δ})$ (where $η= \max\{|V_1|, |V_2|\}$, and $δ\in (0,1/2]$), no polynomial time algorithm can approximate $\mathcal{S}$-MWEB within a factor of $n^ε$ for some $ε> 0$ 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 $n^ε$-inapproximable; and 2) no polynomial time algorithm exists for the Minimum Description Length with Holes problem \cite{Bu05} unless $\mathsf{RP=NP}$.
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