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.

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 $\mathsf{RP=NP}$.

Equations

$$\begin{array}[]{lllr}Pr\left\{\mathbb{E}_{1}\right\}&\leq Pr\left\{\mbox{at least one element in $\mathcal{H}$ is a $\lambda$-approximation of $G^{N}$}\right\}\\ &\leq 4^{N\eta}\cdot Pr\left\{H^{\prime}\mbox{ is a $\lambda$-approximation of $G^{N}$}\right\}\\ &=4^{N\eta}\cdot Pr\{\mathbb{E}_{2}\}\end{array}$$

$$\begin{array}[]{cc}w_{-\gamma}(H^{\prime})+x_{1}\gamma\leq N^{2}(\frac{K}{\lambda}-1)\\ w_{-\gamma}(H^{\prime})+w_{\gamma}(H^{\prime})\geq\frac{1}{\lambda}(w_{-\gamma}(U_{1}^{N}\times U_{2}^{N})+w_{\gamma}(U_{1}^{N}\times U_{2}^{N}))\\ \end{array}$$

$$(w_{\gamma }(H^{\prime })-x_1\gamma )-\frac{1}{\lambda }(w_{\gamma }(U_1^N\times U_2^N)-x_2\gamma )\ge N^2$$

$$\begin{array}[]{lr}Pr\{\mathbb{E}_{2}\}&\leq Pr\left\{(w_{\gamma}(H^{\prime})-x_{1}\gamma)-\frac{1}{\lambda}(w_{\gamma}(U_{1}^{N}\times U_{2}^{N})-x_{2}\gamma)\geq N^{2}\right\}\end{array}$$

$$\begin{array}[]{llr}&Pr\left\{(w_{\gamma}(H^{\prime})-x_{1}\gamma)-\frac{1}{\lambda}(w_{\gamma}(U_{1}^{N}\times U_{2}^{N})-x_{2}\gamma)\geq N^{2}\right\}\\ =&Pr\left\{\sum_{i=1}^{z_{1}}{Y_{i}}-\frac{1}{\lambda}\sum_{j=1}^{z_{2}}{Y_{j}}+\frac{\lambda-1}{\lambda}\sum_{k=1}^{z_{3}}{Y_{k}}\geq N^{2}\right\}\\ =&Pr\left\{\sum_{i=1}^{z_{1}}{Y_{i}}+\frac{1}{\lambda}\sum_{j=1}^{z_{2}}{(-Y_{j})}+\frac{\lambda-1}{\lambda}\sum_{k=1}^{z_{3}}{Y_{k}}\geq N^{2}\right\}\\ \leq&Pr\left\{\sum_{i=1}^{z_{1}}{Y_{i}}\geq\frac{N^{2}}{3}\right\}+Pr\left\{\frac{1}{\lambda}\sum_{j=1}^{z_{2}}{(-Y_{j})}\geq\frac{N^{2}}{3}\right\}+Pr\left\{\frac{\lambda-1}{\lambda}\sum_{k=1}^{z_{3}}{Y_{k}}\geq\frac{N^{2}}{3}\right\}\\ \leq&Pr\left\{\sum_{i=1}^{z_{1}}{Y_{i}}\geq\frac{N^{2}}{3}\right\}+Pr\left\{\sum_{j=1}^{z_{2}}{(-Y_{j})}\geq\frac{N^{2}}{3}\right\}+Pr\left\{\sum_{k=1}^{z_{3}}{Y_{k}}\geq\frac{N^{2}}{3}\right\}\\ \leq&\sum_{i\in\{1,2,3\}}{\left(\exp\left(-2z_{i}\left(\frac{N^{2}}{3z_{i}(c_{1}f(N\eta))}\right)^{2}\right)\right)}&\mbox{ (Hoeffding bound)}\\ \leq&3\cdot\exp\left(-c_{2}\cdot\frac{N^{1+2\delta}}{\eta^{3-2\delta}}\right)&\mbox{ ($z_{i}\leq\eta^{2}N^{2}$)}\\ \end{array}$$

$$Pr\left\{{\mathbb{E}}_1\right\}\le 4^{N\eta }\cdot Pr\left\{{\mathbb{E}}_2\right\}\le 3\cdot \exp \left(\ln 4\cdot {\eta }^{\frac{4}{(1+2\delta )}+\theta }-c_2\cdot {\eta }^{(1+2\delta )\theta }\right)$$

$$\begin{array}[]{lllr}E^{\prime}&=&E_{1}\cup E_{2}\cup E_{3}\\ E_{1}&=&\{(u,v)~{}|~{}u\in V_{1},v\in V_{2}\mbox{ and }(\phi_{1}(u),\phi_{2}(v))\in E\}\\ E_{2}&=&\{(u,v)~{}|~{}u\in V_{1},v\in V_{2},\phi_{1}(u)\neq\phi_{2}(v)\mbox{ and }(\phi_{1}(u),\phi_{2}(v))\notin E\}\\ E_{3}&=&\{(u,v)~{}|~{}u\in V_{1},v\in V_{2},\mbox{ and }\phi_{1}(u)=\phi_{2}(v)\}\end{array}$$

$$p^{\ast }(H)=2^{|V_1^{\prime }||V_2^{\prime }|}{p}^{|E^{\prime }|}(1-p{)}^{|V_1^{\prime }||V_2^{\prime }|-|E^{\prime }|}$$

$$-\log p^{\ast }(H)=|E^{\prime }|(-1-\log p)+(|V_1^{\prime }||V_2^{\prime }|-|E^{\prime }|)(-1-\log (1-p))$$

$$\log L(H)=\sum _{(u,v)\in E^{\prime }}\log \frac{p_c}{p_{u,v}}+\sum _{(u,v)\in \overline{E^{\prime }}}\log \frac{1-p_c}{1-p_{u,v}}$$

$$(n_1-|R|)+(n_2-|C|)+(|Z|-z)+w=(n_1+n_2+|Z|)-(|R|+|C|+z-w)$$

Full text

1 Introduction

Let $G=(V_{1},V_{2},E)$ be an undirected bipartite graph. A biclique subgraph in $G$ is a complete bipartite subgraph of $G$ and maximum edge biclique (MEB) is the problem of finding a biclique subgraph with the most number of edges. MEB is a well-known problem and received much attention in recent years because of its wide range of applications in areas including machine learning [14], management science [16] and bioinformatics, where it is found particularly relevant in the formulation of numerous biclustering problems for biological data analysis [5, 2, 18, 19, 17], and we refer readers to the survey by Madeira and Oliveira [13] for a fairly extensive discussion on this. Maximum edge biclique is shown to be NP-hard by Peeters [15] via a reduction from 3SAT. Its approximability status, on the other hand, remains an open question despite considerable efforts [7, 8, 12] 111Note it might be easy to confuse the MEB problem with the Bipartite Clique problem discussed by Khot in [12]. Bipartite Clique, which also known as Balanced Complete Bipartite Subgraph [8], aims to maximize the number of vertices of a balanced subgraph whereas MEB aims to maximize the total weights on edges in a (not necessarily balanced) subgraph.. In particular, Feige and Kogan [8] conjectured that maximum edge biclique is hard to approximate within a factor of $n^{\epsilon}$ for some $\epsilon>0$. In this paper, we consider a weighted formulation of this problem defined as follows

Definition 1

$\mathcal{S}$**-Maximum Weighted Edge Biclique ($\mathcal{S}$-MWEB) **

Instance: A complete bipartite graph $G=(V_{1},V_{2},E)$ (throughout the paper, let $\eta=max\{|V_{1}|,|V_{2}|\}$ and $n=|V_{1}|+|V_{2}|$), a weight function $w_{G}:E\rightarrow\mathcal{S}$, where $\mathcal{S}$ is a set consisting of both positive and negative integers.

Question: Find a biclique subgraph of $G$ where the sum of weights on edges is maximized.

A few comments are in order. First note it is not a lose of generality but a technical convenience to require the graph be complete, one can always think of an incomplete bipartite graph as complete where non-edges are assigned weight 0. Also note we require that both positive and negative weights be in $\mathcal{S}$ at the same time because otherwise $\mathcal{S}$-MWEB becomes a trivial problem.

Our study of $\mathcal{S}$-MWEB is motivated by the problem of finding statistically significant biclusters in microarray data analysis in the SAMBA model [18] and the Minimum Description Length with Holes (MDLH) problem [3, 4, 10]; detailed discussion of the two problems can be found in Sect. 4. Our main technical contribution of this paper is to show that if $\mathcal{S}$ satisfies the condition $|\frac{\min\mathcal{S}}{\max\mathcal{S}}|\in\Omega(\eta^{\delta-1/2})\cap O(\eta^{1/2-\delta})$, where $\delta>0$ is any arbitrarily small constant, then no polynomial time algorithm can approximate $\mathcal{S}$-MWEB within a factor of $n^{\epsilon}$ for some $\epsilon>0$ unless $\mathsf{RP=NP}$. This result enables us to answer open questions regarding the hardness of the SAMBA model and the MDLH problem. Since maximum edge biclique can be characterized as a special case of $\mathcal{S}$-MWEB with $\mathcal{S}=\{-\eta,1\}$, the $n^{\epsilon}$-inapproximability result also provides interesting insights into the conjectured $n^{\epsilon}$-inapproximability [8] of maximum edge biclique.

The rest of the paper is organized in three sections. In Sect. 2, we present the main technical result by proving the aforementioned inapproximability of $\mathcal{S}$-MWEB. We give applications of this by answering hardness questions regarding two applied problems in Sect. 3. We conclude this work by raising a few open problems in the last section.

2 Approximating $\mathcal{S}$-Maximum Edge Biclique is Hard

We start this section by giving two lemmas about CLIQUE, which will be used in establishing inapproximability for the biclique problems we consider later. Lemma 1 is a recent result by Zuckerman [20], obtained by a derandomization of results of Håstad [11]; Lemma 2 follows immediately from Lemma 1.

Lemma 1

([20])* It is NP-hard to approximate CLIQUE within a factor of $n^{1-\epsilon}$, for any $\epsilon>0$.*

Lemma 2

For any constant $\epsilon>0$, no polynomial time algorithm can approximate CLIQUE within a factor of $n^{1-\epsilon}$ with probability at least $\frac{1}{poly(n)}$ unless $\mathsf{RP=NP}$.

2.1 A Technical Lemma

We first describe the construction of a structure called $\{\gamma,\{\alpha,\beta\}\}$-Product, which will be used in the proof of our main technical lemma.

Definition 2

$(\{\gamma,\{\alpha,\beta\}\}$-Product)

Input: An instance of $\mathcal{S}$-MWEB on complete bipartite graph $G=V_{1}\times V_{2}$, where $\gamma\in\mathcal{S}$ and $\alpha<\gamma<\beta$; an integer $N$.

Output: Complete bipartite graph $G^{N}=V_{1}^{N}\times V_{2}^{N}$ constructed as follows: $V_{1}^{N}$ and $V_{2}^{N}$ are $N$ duplicates of $V_{1}$ and $V_{2}$, respectively. For each edge $(i,j)\in G^{N}$, let $(\phi(i),\phi(j))$ be the corresponding edge in $G$. If $w_{G}(\phi(i),\phi(j))=\gamma$, assign weight $\alpha$ or $\beta$ to $(i,j)$ independently and identically at random with expectation being $\gamma$, denote the weight by random variable $X$. If $w_{G}(\phi(i),\phi(j))\neq\gamma$, then keep the weight unchanged. Call the weight function constructed this way $w(\cdot)$.

For any subgraph $H$ of $G^{N}$, denote by $w_{\gamma}(H)$ (resp., $w_{-\gamma}(H)$) the total weight of $H$ contributed by former-$\gamma$-edges (resp., other edges). Clearly, $w(H)=w_{\gamma}(H)+w_{-\gamma}(H)$.

With a graph product constructed in this randomized fashion, we have the following lemma.

Lemma 3

Given an $\mathcal{S}$-MWEB instance $G=(V_{1},V_{2},E)$ where $\gamma\in\mathcal{S}$, and a number $\delta\in(0,\frac{1}{2}]$; let $\eta=\max{(|V_{1}|,|V_{2}|)}$, $N=\eta^{\frac{\delta(3-2\delta)+3}{\delta(1+2\delta)}}$, $G^{N}=(V_{1}^{N},V_{2}^{N},E)$ be the $\{\gamma,\{\alpha,\beta\}\}$-product of $G$ and $\mathcal{S^{\prime}}=(\mathcal{S}\cup\{\alpha,\beta\})-\{\gamma\}$. If

1. $|\beta-\alpha|=O((N\eta)^{\frac{1}{2}-\delta})$; and

2. there is a polynomial time algorithm that approximates the $\mathcal{S^{\prime}}$-MWEB instance within a factor of $\lambda$, where $\lambda$ is some arbitrary function in the size of the $\mathcal{S^{\prime}}$-MWEB instance

then there exists a polynomial time algorithm that approximates the $\mathcal{S}$-MWEB instance within a factor of $\lambda$, with probability at least $\frac{1}{poly(n)}$.

Proof

For notational convenience, we denote $\eta^{\frac{1}{2}-\delta}$ by $f(\eta)$ throughout the proof. Define random variable $Y=X-\gamma$, clearly $E[Y]=0$. Suppose there is a polynomial time algorithm $\mathbb{A}$ that approximates $\mathcal{S^{\prime}}$-MWEB within a factor of $\lambda$, we can then run $\mathbb{A}$ on $G^{N}$, the output biclique $G_{B}^{*}$ corresponds to $N^{2}$ bicliques in $G$ (not necessarily all distinct). Let $G_{A}^{*}$ be the most weighted among these $N^{2}$ subgraphs of $G$, in the rest of the proof we show that with high probability, $G_{A}^{*}$ is a $\lambda$-approximation of $\mathcal{S}$-MWEB on $G$.

Denote by $\mathbb{E}_{1}$ the event that $G_{B}^{*}$ does not imply a $\lambda$-approximation on $G$. Let $\mathcal{H}$ be the set of subgraphs of $G^{N}$ that do not imply a $\lambda$-approximation on $G$, clearly, $|\mathcal{H}|\leq 4^{N\eta}$. Let $H^{\prime}$ be an arbitrary element in $\mathcal{H}$, we have the following inequalities

[TABLE]

where $\mathbb{E}_{2}$ is the event that $H^{\prime}$ is a $\lambda$-approximation of $G^{N}$.

Let the weight of an optimal solution $U_{1}\times U_{2}$ of $G$ be $K$, denote by $U_{1}^{N}\times U_{2}^{N}$ the corresponding $N^{2}$-duplication in $G^{N}$. Let $x_{1}$ and $x_{2}$ be the number of former-$\gamma$-edges in $H^{\prime}$ and $U_{1}^{N}\times U_{2}^{N}$, respectively. Suppose $\mathbb{E}_{2}$ happens, then we must have

[TABLE]

where the first inequality follows from the fact that we only consider integer weights. Since $w_{-\gamma}(U_{1}^{N}\times U_{2}^{N})=N^{2}K-x_{2}\gamma$, it implies

[TABLE]

so we have the following statement on probability

[TABLE]

Let $z_{1}$ (resp., $z_{2}$ and $z_{3}$) be the number of edges in $E(H^{\prime})-E(U_{1}^{N}\times U_{2}^{N})$ ( resp., $E(U_{1}^{N}\times U_{2}^{N})-E(H^{\prime})$ and $E(U_{1}^{N}\times U_{2}^{N})\cap E(H^{\prime})~{}$) transformed from former-$\gamma$-edges in $G$. We have

[TABLE]

where $c_{1},c_{2}$ are constants ($c_{2}>0$). Now if we set $N=\eta^{\frac{3-2\delta}{1+2\delta}+\theta}$ for some $\theta$, we have

[TABLE]

For this probability to be bounded by $\frac{1}{2}$ as $\eta$ is large enough, we need to have $\frac{4}{1+2\delta}+\theta<(1+2\delta)\theta$. Solving this inequality gives $\theta>\frac{2}{\delta(1+2\delta)}$. Therefore, for any $\delta\in(0,\frac{1}{2}]$, by setting $N=\eta^{\frac{\delta(3-2\delta)+3}{\delta(1+2\delta)}}$, we have $Pr\{\mathbb{E}_{1}\}$, i.e. the probability that the solution returned by $\mathbb{A}$ does not imply a $\lambda$-approximation of $G$, is bounded from above by $\frac{1}{2}$ once input size is large enough. This gives a polynomial time algorithm that approximates $\mathcal{S}$-MWEB within a factor of $\lambda$ with probability at least $\frac{1}{2}$. ∎

This lemma immediately leads to the following corollary.

Corollary 1

Following the construction in Lemma 3, if $\mathcal{S^{\prime}}$-MWEB can be approximated within a factor of $n^{\epsilon^{\prime}}$, for some $\epsilon^{\prime}>0$, then there exists a polynomial time algorithm that approximates $\mathcal{S}$-MWEB within a factor of $n^{\epsilon}$, where $\epsilon=(1+\frac{\delta(3-2\delta)+3}{\delta(1+2\delta)})\epsilon^{\prime}$, with probability at least $\frac{1}{poly(n)}$. 222Note we are slightly abusing notation here by always representing the size of a given problem under discussion by $n$. Here $n$ refers to the size of $\mathcal{S^{\prime}}$-MWEB (resp. $\mathcal{S}$-MWEB) when we are talking about approximation factor $n^{\epsilon^{\prime}}$ (resp. $n^{\epsilon}$). We adopt the same convention in the sequel.

Proof

Let $|G|$ and $|G^{N}|$ be the number of nodes in the $\mathcal{S}$-MWEB and $\mathcal{S^{\prime}}$-MWEB problem, respectively. Since $\lambda=|G^{N}|^{\epsilon^{\prime}}\leq|G|^{(1+\frac{\delta(3-2\delta)+3}{\delta(1+2\delta)})\epsilon^{\prime}}$, our claim follows from Lemma 3. ∎

2.2 $\{-1,0,1\}$-MWEB

In this section, we prove inapproximability of $\{-1,0,1\}$-MWEB by giving a reduction from CLIQUE; in subsequence sections, we prove inapproximability results for more general $\mathcal{S}$-MWEB by constructing randomized reduction from $\{-1,0,1\}$-MWEB.

Lemma 4

The decision version of the $\{-1,0,1\}$-MWEB problem is $\mathsf{NP}$-complete.

Proof

We prove this by describing a reduction from CLIQUE. Given a CLIQUE instance $G=(V,E)$, construct $G^{\prime}=(V^{\prime},E^{\prime})$ such that $V^{\prime}=V_{1}\cup V_{2}$ where $V_{1}$, $V_{2}$ are duplicates of $V$ in that there exist bijections $\phi_{1}:V_{1}\rightarrow V$ and $\phi_{2}:V_{2}\rightarrow V$. And

[TABLE]

Clearly, $G^{\prime}$ is a biclique. Now assign weight 0 to edges in $E_{1}$, $-1$ to edges in $E_{2}$ and 1 to edges in $E_{3}$. We then claim that there is a clique of size $k$ in $G$ if and only if there is a biclique of total edge weight $k$ in $G^{\prime}$.

First consider the case where there is a clique of size $k$ in $G$, let $U$ be the set of vertices of the clique, then taking the subgraph induced by $\phi_{1}^{-1}(U)\times\phi_{2}^{-1}(U)$ in $G^{\prime}$ gives us a biclique of total weight $k$.

Now suppose that there is a biclique $U_{1}\times U_{2}$ of total weight $k$ in $G^{\prime}$. Without loss of generality, assume $U_{1}$ and $U_{2}$ correspond to the same subset of vertices in $V$ because if $(\phi_{1}(U_{1})-\phi_{2}(U_{2}))\cup(\phi_{2}(U_{2})-\phi_{1}(U_{1}))$ is not empty, then removing $(U_{1}-U_{2})\cup(U_{2}-U_{1})$ will never decrease the total weight of the solution. Given $\phi_{1}(U_{1})=\phi_{2}(U_{2})$, we argue that there is no edge of weight $-1$ in biclique $U_{1}\times U_{2}$; suppose otherwise there exists a weight $-1$ edge $(i_{1},j_{2})$ ($i_{1}\in U_{1}$, and $j_{2}\in U_{2}$), then the corresponding edge $(j_{1},i_{2})$ ($j_{1}\in U_{1}$, and $i_{2}\in U_{2}$) must be of weight $-1$ too and removing $i_{1},i_{2}$ from the solution biclique will increase total weight by at least 1 because among all edges incident to $i_{1}$ and $i_{2}$, $(i_{1},i_{2})$ is of weight 1, $(i_{1},j_{2})$ and $(i_{2},j_{1})$ are of weight $-1$ and the rest are of weights either 0 or $-1$.

Therefore, we have shown that if there is a solution $U_{1}\times U_{2}$ of weight $k$ in $G^{\prime}$, $U_{1}$ and $U_{2}$ correspond to the same set of vertices $U\in V$ and $U$ is a clique of size $k$. It is clear that the reduction can be performed in polynomial time and the problem is $\mathsf{NP}$, and thus $\mathsf{NP}$-complete. ∎

Given Lemma 1, the following corollary follows immediately from the above reduction.

Theorem 2.1

For any constant $\epsilon>0$, no polynomial time algorithm can approximate problem $\{-1,0,1\}$-MWEB within a factor of $n^{1-\epsilon}$ unless $\mathsf{P=NP}$.

Proof

It is obvious that the reduction given in the proof of Lemma 4 preserves inapproximability exactly, and given that CLIQUE is hard to approximate within a factor of $n^{1-\epsilon}$ unless $\mathsf{P=NP}$, the theorem follows. ∎

Theorem 2.2

For any constant $\epsilon>0$, no polynomial time algorithm can approximate $\{-1,0,1\}$-MWEB within a factor of $n^{1-\epsilon}$ with probability at least $\frac{1}{poly(n)}$ unless $\mathsf{RP=NP}$.

Proof

If there exists such a randomized algorithm for $\{-1,0,1\}$-MWEB, combining it with the reduction given in Lemma 4, we obtain an $\mathsf{RP}$ algorithm for CLIQUE. This is impossible unless $\mathsf{RP=NP}$. ∎

2.3 $\{-1,1\}$-MWEB

Lemma 5

If there exists a polynomial time algorithm that approximates $\{-1,1\}$-MWEB within a factor of $n^{\epsilon}$, then there exists a polynomial time algorithm that approximates $\{-1,0,1\}$-MWEB within a factor of $n^{5\epsilon}$ with probability at least $\frac{1}{poly(n)}$.

Proof

We prove this by constructing a $\{\gamma,\{\alpha,\beta\}\}$-Product from $\{-1,0,1\}$-MWEB to $\{-1,1\}$-MWEB by setting $\gamma=0$, $\alpha=-1$ and $\beta=1$. Since $\delta=\frac{1}{2}$, according to Corollary 1, it is sufficient to set $N=\eta^{4}$ so that the probability of obtaining a $n^{5\epsilon}$-approximation for $\{-1,0,1\}$-MWEB is at least $\frac{1}{poly(n)}$. ∎

Theorem 2.3

For any constant $\epsilon>0$, no polynomial time algorithm can approximate $\{-1,1\}$-MWEB within a factor of $n^{\frac{1}{5}-\epsilon}$ with probability at least $\frac{1}{poly(n)}$ unless $\mathsf{RP=NP}$.

Proof

This follows directly from Theorem 2.2 and Lemma 5.∎

2.4 $\{-\eta^{\frac{1}{2}-\delta},1\}$-MWEB and $\{-\eta^{\delta-\frac{1}{2}},1\}$-MWEB

In this section, we consider the generalized cases of the $\mathcal{S}$-MWEB problem.

Theorem 2.4

For any $\delta\in(0,\frac{1}{2}]$, there exists some constant $\epsilon$ such that no polynomial time algorithm can approximate $\{-\eta^{\frac{1}{2}-\delta},1\}$-MWEB within a factor of $n^{\epsilon}$ with probability at least $\frac{1}{poly(n)}$ unless $\mathsf{RP=NP}$. The same statement holds for $\{-\eta^{\delta-\frac{1}{2}},1\}$-MWEB.

Proof

We prove this by first construct a $\{\gamma,\{\alpha,\beta\}\}$-Product from $\{-1,1\}$-MWEB to $\{-\eta^{\frac{1}{2}-\delta},1\}$-MWEB by setting $\gamma=-1$, $\alpha=-(N\eta)^{\frac{1}{2}-\delta}$ and $\beta=1$. By Corollary 1, we know that for any $\delta\in(0,\frac{1}{2}]$, if there exists a polynomial time algorithm that approximates $\{-\eta^{\frac{1}{2}-\delta},1\}$-MWEB within a factor of $n^{\epsilon}$, then there exists a polynomial time algorithm that approximates $\{-1,1\}$-MWEB within a factor of $n^{(1+\frac{\delta(3-2\delta)+3}{\delta(1+2\delta)})\epsilon}$ with probability at least $\frac{1}{poly(n)}$. So invoking the hardness result in Theorem 2.3 gives the desired hardness result for $\{-\eta^{\frac{1}{2}-\delta},1\}$-MWEB.

The same conclusion applies to $\{-1,\eta^{\frac{1}{2}-\delta}\}$-MWEB by setting $\gamma=1$, $\alpha=-1$ and $\beta=(N\eta)^{\frac{1}{2}-\delta}$. Since $\eta$ is a constant for any given graph, we can simply divide each weight in $\{-1,\eta^{\frac{1}{2}-\delta}\}$ by $\eta^{\frac{1}{2}-\delta}$. ∎

Theorem 2.4 leads to the following general statement.

Theorem 2.5

For any small constant $\delta\in(0,\frac{1}{2}]$, if $\left|\frac{\min\mathcal{S}}{\max\mathcal{S}}\right|\in\Omega(\eta^{\delta-1/2})\cap O(\eta^{1/2-\delta})$, then there exists some constant $\epsilon$ such that no polynomial time algorithm can approximate $\mathcal{S}$-MWEB within a factor of $n^{\epsilon}$ with probability at least $\frac{1}{poly(n)}$ unless $\mathsf{RP=NP}$.

3 Two Applications

In this section, we describe two applications of the results establish in Sect. 3 by proving hardness and inapproximability of problems found in practice.

3.1 SAMBA Model is Hard

Microarray technology has been the latest technological breakthrough in biological and biomedical research; in many applications, a key step in analyzing gene expression data obtained through microarray is the identification of a bicluster satisfying certain properties and with largest area (see the survey [13] for a fairly extensive discussion on this).

In particular, Tanay et. al. [18] considered the Statistical-Algorithmic Method for Bicluster Analysis (SAMBA) model. In their formulation, a complete bipartite graph is given where one side corresponds to genes and the other size corresponds to conditions. An edges $(u,v)$ is assigned a real weight which could be either positive or negative, depending on the expression level of gene $u$ in condition $v$, in a way such that heavy subgraphs corresponds to statistically significant biclusters. Two weight-assigning schemes are considered in their paper. In the first, or simple statistical model, a tight upper-bound on the probability of an observed biclusters in computed; in the second, or refined statistical model, the weights are assigned in a way such that a maximum weight biclique subgraph corresponds to a maximum likelihood bicluster.

The Simple SAMBA Statistical Model:

Let $H=(V_{1}^{\prime},V_{2}^{\prime},E^{\prime})$ be a subgraph of $G=(V_{1},V_{2},E)$, $\overline{E^{\prime}}=\{V_{1}^{\prime}\times V_{2}^{\prime}\}-E^{\prime}$ and $p=\frac{|E|}{|V_{1}||V_{2}|}$. The simple statistical model assumes that edges occur independently and identically at random with probability $p$. Denote by $BT(k,p,n)$ the probability of observing $k$ or more successes in $n$ binomial trials, the probability of observing a graph at least as dense as $H$ is thus $p(H)=BT(|E^{\prime}|,p,|V_{1}^{\prime}||V_{2}^{\prime}|)$. This model assumes $p<\frac{1}{2}$ and $|V_{1}^{\prime}||V_{2}^{\prime}|\ll|V_{1}||V_{2}|$, therefore $p(H)$ is upper bounded by

[TABLE]

The goal of this model is thus to find a subgraph $H$ with the smallest $p^{*}(H)$. This is equivalent to maximizing

[TABLE]

which is essentially solving a $\mathcal{S}$-MWEB problem that assigns either positive weight $(-1-\log{p})$ or negative weight $(-1-\log{(1-p)})$ to an edge $(u,v)$, depending on whether gene $u$ express or not in condition $v$, respectively. The summation of edge weights over $H$ is defined as the statistical significance of $H$.

Since $\frac{1}{\eta^{2}}\leq p<\frac{1}{2}$, asymptotically we have $\frac{-1-\log{(1-p)}}{-1-\log{p}}\in\Omega(\frac{1}{\log{\eta}})\cap O(1)$. Invoking Theorem 2.5 gives the following.

Theorem 3.1

For the Simple SAMBA Statistical model, there exists some $\epsilon>0$ such that no polynomial time algorithm, possibly randomized, can find a bicluster whose statistical significance is within a factor of $n^{\epsilon}$ of optimal unless $\mathsf{RP=NP}$.

The Refined SAMBA Statistical Model:

In the refined model, each edge $(u,v)$ is assumed to take an independent Bernoulli trial with parameter $p_{u,v}$, therefore $p(H)=(\prod_{(u,v)\in E^{\prime}}p_{u,v})(\prod_{(u,v)\in\overline{E^{\prime}}}(1-p_{u,v}))$ is the probability of observing a subgraph $H$. Since $p(H)$ generally decreases as the size of $H$ increases, Tanay et al. aims to find a bicluster with the largest (normalized) likelihood ratio $L(H)=\dfrac{(\prod_{(u,v)\in E^{\prime}}p_{c})(\prod_{(u,v)\in\overline{E^{\prime}}}(1-p_{c}))}{p(H)}$, where $p_{c}>\max_{(u,v)\in E}p_{u,v}$ is a constant probability and chosen with biologically sound assumptions. Note this is equivalent to maximizing the log-likelihood ratio

[TABLE]

With this formulation, each edge is assigned weight either $\log{\frac{p_{c}}{p_{u,v}}}>0$ or $\log{\frac{1-p_{c}}{1-p_{u,v}}}<0$ and finding the most statistically significant bicluster is equivalent to solving $\mathcal{S}$-MWEB with $\mathcal{S}=\{\log{\frac{1-p_{c}}{1-p_{u,v}}},\log{\frac{p_{c}}{p_{u,v}}}\}$. Since $p_{c}$ is a constant and $\frac{1}{\eta^{2}}\leq p_{u,v}<p_{c}$, we have $\frac{\log{(1-p_{c})}-\log{(1-p_{u,v})}}{\log{p_{c}}-\log{p_{u,v}}}\in\Omega(\frac{1}{\log{\eta}})\cap O(1)$. Invoking Theorem 2.5 gives the following.

Theorem 3.2

For the Refined SAMBA Statistical model, there exists some $\epsilon>0$ such that no polynomial time algorithm, possibly randomized, can find a bicluster whose log-likelihood is within a factor of $n^{\epsilon}$ of optimal unless $\mathsf{RP=NP}$.

3.2 Minimum Description Length with Holes (MDLH) is Hard

Bu et. al [4] considered the Minimum Description Length with Holes problem (defined in the following); the 2-dimensional case is claimed $\mathsf{NP}$-hard in this paper and the proof is referred to [3]. However, the proof given in [3] suffers from an error in its reduction333In Lemma 3.2.1 of [3], the reduction from CLIQUE to CEW is incorrect., thus whether MDLH is NP-complete remains unsettled. In this section, by employing the results established in the previous sections, we show that no polynomial time algorithm exists for MDLH, under the slightly weaker (than $\mathsf{P\neq NP}$) but widely believed assumption $\mathsf{RP\neq NP}$.

We first briefly describe the Minimum Description Length summarization with Holes problem; for a detailed discussion of the subject, we refer the readers to [3, 4].

Suppose one is given a $k$-dimensional binary matrix $M$, where each entry is of value either 1, which is of interest, or of value 0, which is not of interest. Besides, there are also $k$ hierarchies (trees) associated with each dimension, namely $T_{1},T_{2},...,T_{k}$, each of height $l_{1},l_{2},...,l_{k}$ respectively. Define $level$ $l=\text{max}_{i}(l_{i})$. For each $T_{i}$, there is a bijection between its leafs and the ’hyperplanes’ in the $i$th dimension (e.g. in a 2-dimensional matrix, these hyperplanes corresponds to rows and columns). A $region$ is a tuple $(x_{1},x_{2},...,x_{k})$, where $x_{i}$ is a leaf node or an internal node in hierarchy $T_{i}$. Region $(x_{1},x_{2},...,x_{k})$ is said to $cover$ cell $(c_{1},c_{2},...,c_{k})$ if $c_{i}$ is a descendant of $x_{i}$, for all $1\leq i\leq k$. A $k$-dimensional $l$-level MDLH summary is defined as two sets $S$ and $H$, where 1) $S$ is a set of regions covering all the 1-entries in $M$; and 2) $H$ is the set of 0-entries covered (undesirably) by $S$ and to be excluded from the summary. The length of a summary is defined as $|S|+|H|$, and the MDLH problem asks the question if there exists a MDLH summary of length at most $K$, for a given $K>0$.

In an effort to establish hardness of MDLH, we first define the following problem, which serves as an intermediate problem bridging $\{-1,1\}$-MWEB and MDLH.

Definition 3

(Problem $\mathcal{P}$)

**Instance: ** A complete bipartite graph $G=(V_{1},V_{2},E)$ where each edge takes on a value in $\{-1,1\}$, and a positive integer $k$.

**Question: ** Does there exist an induced subgraph (a biclique $U_{1}\times U_{2}$) whose total weight of edges is $\omega$, such that $|U_{1}|+|U_{2}|+\omega\geq k$.

Lemma 6

No polynomial time algorithm exists for Problem $\mathcal{P}$ unless $\mathsf{RP=NP}$.

Proof

We prove this by constructing a reduction from $\{-1,1\}$-MWEB to Problem $\mathcal{P}$ as follows: for the given input biclique $G=(V_{1},V_{2},E)$, make $N$ duplicates of $V_{1}$ and $N$ duplicates of $V_{2}$, where $N=(|V_{1}|+|V_{2}|)^{2}$. Connect each copy of $V_{1}$ to each copy of $V_{2}$ in a way that is identical to the input biclique, we then claim that there is a size $k$ solution to $\{-1,1\}$-MWEB if and only if there is a size $N^{2}k$ solution to Problem $\mathcal{P}$.

If there is a size $k$ solution to $\{-1,1\}$-MWEB, then it is straightforward that there is a solution to Problem $\mathcal{P}$ of size at least $N^{2}k$. For the reverse direction, we show that if no solution to $\{-1,1\}$-MWEB is of size at least $k$, then the maximum solution to Problem $\mathcal{P}$ is strictly less than $N^{2}k$. Note a solution $U_{1}^{N}\times U_{2}^{N}$ to Problem $\mathcal{P}$ consists of at most $N^{2}$ (not necessarily all distinct) solutions to $\{-1,1\}$-MWEB, and each of them can contribute at most $(k-1)$ in weight to $U_{1}^{N}\times U_{2}^{N}$, so the total weight gained from edges is at most $N^{2}(k-1)$. And note the total weight gained from vertices is at most $N(|V_{1}|+|V_{2}|)=N\sqrt{N}$, therefore the weight is upper bounded by $N\sqrt{N}+N^{2}(k-1)<N^{2}k$ and this completes the proof.

As a conclusion, we have a polynomial time reduction from $\{-1,1\}$-MWEB to Problem $\mathcal{P}$. Since no polynomial time algorithm exists for $\{-1,1\}$-MWEB unless $\mathsf{RP=NP}$, the same holds for Problem $\mathcal{P}$. ∎

Theorem 3.3

No polynomial time algorithm exists for MDLH summarization, even in the 2-dimension 2-level case, unless $\mathsf{RP=NP}$.

Proof

We prove this by showing that Problem $\mathcal{P}$ is a complementary problem of 2-dimensional 2-level MDLH.

Let the input 2D matrix $M$ be of size $n_{1}\times n_{2}$, with a tree of height 2 associated with each dimension. Without loss of generality, we only consider the ’sparse’ case where the number of 1-entries is less than the number of 0-entries by at least 2 so that the optimal solution will never contain the whole matrix as one of its regions. Let $S$ be the set of regions in a solution. Let $R$ and $C$ be the set of rows and columns not included in $S$. Let $Z$ be the set of all zero entries in $M$. Let $z$ be the total number of zero entries in the $R\times C$ ’leftover’ matrix and let $w$ be the total number of 1-entries in it. MDLH tries to minimize the following:

[TABLE]

Since $(n_{1}+n_{2}+|Z|)$ is a fixed quantity for any given input matrix, the 2-dimensional 2-level MDLH problem is equivalent to maximizing $(|R|+|C|+z-w)$, which is precisely the definition of Problem $\mathcal{P}$.

Therefore, 2-dimensional 2-level MDLH is a complementary problem to Problem $\mathcal{P}$ and by Lemma 6 we conclude that no polynomial time algorithm exists for 2-dimensional 2-level MDLH unless $\mathsf{RP=NP}$. ∎

4 Concluding Remarks

Maximum weighted edge biclique and its variants have received much attention in recently years because of it wide range of applications in various fields including machine learning, database, and particularly bioinformatics and computational biology, where many computational problems for the analysis of microarray data are closely related. To tackle these applied problems, various kinds of heuristics are proposed and experimented and it is not known whether these algorithms give provable approximations. In this work, we answer this question by showing that it is highly unlikely (under the assumption $\mathsf{RP\neq NP}$) that good polynomial time approximation algorithm exists for maximum weighted edge biclique for a wide range of choices of weight; and we further give specific applications of this result to two applied problems. We conclude our work by listing a few open questions.

1. We have shown that $\{\Theta(-\eta^{\delta}),1\}$-MWEB is $n^{\epsilon}$-inapproximable for $\delta\in(-\frac{1}{2},\frac{1}{2})$; also it is easy to see that (i) the problem is in $\mathsf{P}$ when $\delta\leq-1$, where the entire input graph is the optimal solution; (ii) for any $\delta\geq 1$, the problem is equivalent to MEB, which is conjectured to be $n^{\epsilon}$-inapproximable [8]. Therefore it is natural to ask what is the approximability of the $\{-n^{\delta},1\}$-MWEB problem when $\delta\in(-1,-\frac{1}{2}]$ and $\delta\in[\frac{1}{2},1]$. In particular, can this be answered by a better analysis of Lemma 3?

2. We are especially interested in $\{-1,1\}$-MWEB, which is closely related to the formulations of many natural problems [1, 3, 4, 18]. We have shown that no polynomial time algorithm exists for this problem unless $\mathsf{RP=NP}$, and we believe this problem is $\mathsf{NP}$-complete, however a proof has eluded us so far.