Low-rank binary matrix approximation in column-sum norm

TL;DR

This paper introduces the first polynomial time approximation scheme for low-rank binary matrix approximation in column-sum norm over GF(2), providing near-optimal solutions efficiently.

Contribution

It presents a randomized PTAS for $ ext{l}_1$-Rank-$r$ Approximation over GF(2), a problem previously lacking such an efficient approximation method.

Findings

Achieves a $(1+ ext{epsilon})$-approximation with polynomial time complexity.

First PTAS for $ ext{l}_1$-Rank-$r$ Approximation over GF(2).

Provides theoretical bounds and algorithmic framework for binary matrix approximation.

Abstract

We consider $\ell_1$-Rank-$r$ Approximation over GF(2), where for a binary $m\times n$ matrix ${\bf A}$ and a positive integer $r$, one seeks a binary matrix ${\bf B}$ of rank at most $r$, minimizing the column-sum norm $||{\bf A} -{\bf B}||_1$. We show that for every $\varepsilon\in (0, 1)$, there is a randomized $(1+\varepsilon)$-approximation algorithm for $\ell_1$-Rank-$r$ Approximation over GF(2) of running time $m^{O(1)}n^{O(2^{4r}\cdot \varepsilon^{-4})}$. This is the first polynomial time approximation scheme (PTAS) for this problem.

Equations

$$\text{ minimize }\Vert \mathbf{A}-\mathbf{B}{\Vert }_{\nu }$$

$$\text{ subject to }\mathrm{rank}(\mathbf{B})\le r.$$

$$\Vert \mathbf{A}{\Vert }_1=\underset{x\ne 0}{\sup }\frac{\Vert \mathbf{Ax}{\Vert }_1}{\Vert \mathbf{x}{\Vert }_1}=\underset{1\le j\le n}{\max }\sum _{i=1}^m|a_{ij}|.$$

$$\text{ minimize }\Vert \mathbf{A}-\mathbf{B}{\Vert }_1$$

$$\text{ subject to }\mathrm{rank}(\mathbf{B})\le r.$$

$$\mathbf{c}_{1}=\left(\begin{array}[]{c}0\\ 1\\ \end{array}\right),\,\mathbf{c}_{2}=\left(\begin{array}[]{c}0\\ 0\\ \end{array}\right),\,\mathbf{c}_{3}=\left(\begin{array}[]{c}1\\ 1\\ \end{array}\right)$$

$$\mathrm{cost}(X,C)=\underset{i\in \{1,\dots ,k\},\mathbf{x}\in X_i}{\max }{d}_H(\mathbf{x},{\mathbf{c}}_i).$$

$$\mathsf{OPT}(J^{\prime })=\underset{C=({\mathbf{c}}_1,\dots ,{\mathbf{c}}_k)\text{mbox}s.t.<C,\mathcal{R}>}{\min }\left\{\underset{i\in \{1,\dots ,k\},\mathbf{x}\in X_i}{\max }{d}_H(\mathbf{x},{\mathbf{c}}_i)\right\}.$$

$$d_H(\mathbf{x},{\mathbf{c}}_i)\le \frac{1+ϵ}{1-ϵ}\ell \le (1+4ϵ)\ell ,$$

$$(1-ϵ)\alpha \cdot 2\ell \le d_H(\psi (\mathbf{x}),\psi ({\mathbf{c}}_i))\le d_H(\psi (\mathbf{x}),\psi ({\mathbf{c}}_j))\le (1+ϵ)\alpha \cdot \ell .$$

$$d_H^P(\mathbf{x},\mathbf{y})=\sum _{i\in P}|x_i-y_i|.$$

$$d_H^P(\mathbf{x},{\mathbf{y}}_{i_1})-d_H^P(\mathbf{x},{\mathbf{c}}^{\ast })\le \frac{1}{r-1}{d}^{\ast },$$

$$J_P({\ell }^{\prime })$$

$$J({\ell }^{\prime })$$

$${\rho }_s=\underset{1\le i_1,\dots ,i_s\le n}{\min }\frac{p_{i_1,\dots ,i_s}}{d^{\ast }}.$$

$$J({\ell }^{\prime })$$

$$\le$$

$$d_H^P(\mathbf{x},{\mathbf{x}}_i^{(1)})-d_H^P(\mathbf{x},{\mathbf{c}}_i^{\ast })\le \frac{1}{r-1}{d}_{opt},$$

$$Q=\left\{j\in \bigcap _{i\in [k]}{Q}_i:({\mathbf{x}}_1^{(1)}[j],{\mathbf{x}}_2^{(1)}[j],\dots ,{\mathbf{x}}_k^{(1)}[j])\in R_j\right\}.$$

$$\underset{i\in [k],\mathbf{x}\in X_i}{\max }{d}_H(\mathbf{x},{\mathbf{c}}_i)$$

$$d_H(\mathbf{x}{|}_{\overline{Q}},{\mathbf{e}}_i^{\ast })+d_{\mathbf{x}{|}_{\overline{Q}}}$$

$$\min d$$

$$\sum _{t\in R_j}{y}_{j,t}=1,\text{mbox}forallj\in [m];$$

$$\sum _{j\in [m]}\sum _{t\in R_j}{\chi }_i(\mathbf{x}[j],t)\cdot y_{j,t}\le d-d_{\mathbf{x}},\text{mbox}foralli\in [k]\text{mbox}and\mathbf{x}\in X_i$$

$$y_{j,t}\in \{0,1\},\text{mbox}forallj\in [m]\text{mbox}andt\in R_j.$$

$$E[d_H(\mathbf{x},{\mathbf{c}}_i)]$$

$$\mathrm{Pr}[d_H(\mathbf{x},{\mathbf{c}}_i)>d^{\prime }-d_{\mathbf{x}}+ϵm]\le e^{-\frac{1}{3}m{ϵ}^2}.$$

$$\mathrm{Pr}[\text{mbox}Thereexisti\in [k]\text{mbox}and\mathbf{x}\in X_i\text{mbox}suchthat{d}_H(\mathbf{x},{\mathbf{c}}_i)>d^{\prime }-d_{\mathbf{x}}+ϵm]\le n\cdot e^{-\frac{1}{3}m{ϵ}^2}$$

$$\mathrm{Pr}[\text{mbox}Thereexisti\in [k]\text{mbox}and\mathbf{x}\in X_i\text{mbox}suchthat{d}_H(\mathbf{x},{\mathbf{c}}_i)>d^{\prime }-d_{\mathbf{x}}+ϵm]\le n^{-2}.$$

$${\mathbf{c}}_i={\lambda }_i[1]{\mathbf{s}}_1\oplus \cdots \oplus {\lambda }_i[r]{\mathbf{s}}_r,$$

$$\text{ minimize }\Vert \mathbf{A}-\mathbf{B}{\Vert }_1$$

$$\text{ subject to }\mathrm{rank}(\mathbf{B})\le r.$$

Full text

1 Introduction

Low-rank matrix approximation is the method of compressing a matrix by reducing its dimension. It is the basic component of various methods in data analysis including Principal Component Analysis (PCA), one of the most popular and successful techniques used for dimension reduction in data analysis and machine learning [31, 15, 8]. In low-rank matrix approximation one seeks the best low-rank approximation of data matrix $\mathbf{A}$ with matrix $\mathbf{B}$ solving

[TABLE]

Here $\|\cdot\|_{\nu}$ is some matrix norm. The most popular matrix norms studied in the literature are the Frobenius $||\mathbf{A}||_{F}^{2}=\sum_{i,j}a_{ij}^{2}$ and the spectral $\|\mathbf{A}\|_{2}=\sup_{x\neq 0}\frac{\|\mathbf{A}\mathbf{x}\|_{2}}{\|\mathbf{x}\|_{2}}$ norms. By the Eckart-Young-Mirsky theorem [8, 27], (1) is efficiently solvable via Singular Value Decomposition (SVD) for these two norms. The spectral norm is an “extremal” norm—it measures the worst-case stretch of the matrix. On the other hand, the Frobenius norm is “averaging”. Spectral norm is usually applied in the situation when one is interested in actual columns for the subspaces they define and is of greater interest in scientific computing and numerical linear algebra. The Frobenius norm is widely used in statistics and machine learning, see the survey of Mahony [24] for further discussions.

Recently there has been considerable interest in developing algorithms for low-rank matrix approximation problems for binary (categorical) data. Such variants of dimension reduction for high-dimensional data sets with binary attributes arise naturally in applications involving binary data sets, like latent semantic analysis [4], pattern discovery for gene expression[32], or web search models [19], see [7, 17, 14, 20, 30, 37] for other applications. In many such applications it is much more desirable to approximate a binary matrix $\mathbf{A}$ with a binary matrix $\mathbf{B}$ of small (GF(2) or Boolean) rank because it could provide a deeper insight into the semantics associated with the original matrix. There is a big body of work done on binary and Boolean low-rank matrix approximation, see [2, 3, 7, 22, 25, 26, 28, 35, 34] for further discussions.

Unfortunately, SVD is not applicable for the binary case which makes such problems computationally much more challenging. For binary matrix, its Frobenius norm is equal to the number of its $1$-entries, that is $\|\mathbf{A}\|_{F}=\sum_{j=1}^{n}\sum_{i=1}^{m}|a_{ij}|$. Thus, the value $\|\mathbf{A}\ -\mathbf{B}\|_{F}$ measures the total Hamming distance from points (columns) of $\mathbf{A}$ to the subspace spanned by the columns of $\mathbf{B}$. For this variant of the low-rank binary matrix approximation, a number of approximation algorithms were developed, resulting in efficient polynomial time approximation schemes (EPTASes) obtained in [1, 9]. However, the algorithmic complexity of the problem for any vector-induced norm, including the spectral norm, remained open.

For binary matrices, the natural “extremal” norm to consider is the $\|\cdot\|_{1}$ norm, also known as column-sum norm, operator $1$-norm, or Hölder matrix 1-norm. That is, for a matrix $\mathbf{A}$,

[TABLE]

In other words, the column-sum norm is the maximum number of $1$-entries in a column in $\mathbf{A}$, whereas the Frobenius norm is the total number of $1$-entries in $\mathbf{A}$. The column-sum norm is analogous to the spectral norm, only it is induced by the $\ell_{1}$ vector norm, not the $\ell_{2}$ vector norm.

We consider the problem, where for an $m\times n$ binary data matrix $\mathbf{A}$ and a positive integer $r$, one seeks a binary matrix $\mathbf{B}$ optimizing

[TABLE]

Here, by the rank of the binary matrix $\mathbf{B}$ we mean its GF(2)-rank. We refer to the problem defined by (2) as to $\ell_{1}$-Rank-$r$ Approximation over GF(2). The value $\|\mathbf{A}\ -\mathbf{B}\|_{1}$ is the maximum Hamming distance from each of the columns of $\mathbf{A}$ to the subspace spanned by columns of $\mathbf{B}$ and thus, compared to approximation with the Frobenius norm, it could provide a more accurate dimension reduction.

It is easy to see by the reduction from the Closest String problem, that already for $r=1$, $\ell_{1}$-Rank-$r$ Approximation over GF(2) is NP-hard. The main result of this paper is that (2) admits a polynomial time approximation scheme (PTAS). More precisely, we prove the following theorem.

Theorem 1**.**

*For every $\varepsilon\in(0,1)$, there is a randomized $(1+\varepsilon)$-approximation algorithm for $\ell_{1}$-Rank-$r$ Approximation over GF(2) of running time $m^{{\cal O}(1)}n^{{\cal O}(2^{4r}\cdot\varepsilon^{-4})}$. *

In order to prove Theorem 1 we obtain a PTAS for a more general problem, namely Binary Constrained $k$-Center. This problem has a strong expressive power and can be used to obtain PTASes for a number of problems related to $\ell_{1}$-Rank-$r$ Approximation over GF(2). For example, for the variant, when the rank of the matrix $\mathbf{B}$ is not over GF(2) but is Boolean. Or a variant of clustering, where we want to partition binary vectors into groups, minimizing the maximum distance in each of the group to some subspace of small dimension. We provide discussions of other applications of our work in Section 4.

Related work.

The variant of (1) with both matrices $\mathbf{A}$ and $\mathbf{B}$ binary, and $\|\cdot\|_{\nu}$ being the Frobenius norm, is known as Low GF(2)-Rank Approximation. Due to numerous applications, various heuristic algorithms for Low GF(2)-Rank Approximation could be found in the literature [16, 17, 11, 20, 32].

When it concerns rigorous algorithmic analysis of Low GF(2)-Rank Approximation, Gillis and Vavasis  [13] and Dan et al. [7] have shown that Low GF(2)-Rank Approximation is NP-complete for every $r\geq 1$. A subset of the authors studied parameterized algorithms for Low GF(2)-Rank Approximation in [10]. The first approximation algorithm for Low GF(2)-Rank Approximation is due to Shen et al. [32], who gave a $2$-approximation algorithm for the special case of $r=1$. For rank $r>1$, Dan et al. [7] have shown that a $(r/2+1+\frac{r}{2(2^{r}-1)})$-approximate solution can be formed from $r$ columns of the input matrix $\mathbf{A}$. Recently, these algorithms were significantly improved in [1, 9], where efficient polynomial time approximation schemes (EPTASes) were obtained.

Also note that for general (non-binary) matrices a significant amount of work is devoted to $L1$-PCA, where one seeks a low-rank matrix $\mathbf{B}$ approximating given matrix $\mathbf{A}$ in entrywise $\ell_{1}$ norm, see e.g. [33].

While our main motivation stems from low-rank matrix approximation problems, $\ell_{1}$-Rank-$r$ Approximation over GF(2) extends Closest String, very well-studied problem about strings. Given a set of binary strings $S=\{s_{1},s_{2},\dots,s_{n}\}$, each of length $m$, the Closest String problem is to find the smallest $d$ and a string $s$ of length $m$ which is within Hamming distance $d$ to each $s_{i}\in S$.

A long history of algorithmic improvements for Closest String was concluded by the PTAS of running time $n^{{\cal O}(\epsilon^{-5})}$ by Li, Ma, and Wang [21], which running time was later improved to $n^{{\cal O}(\epsilon^{-2})}$ [23]. Let us note that Closest String can be seen as a special case of $\ell_{1}$-Rank-$r$ Approximation over GF(2) for $r=1$. Indeed, Closest String is exactly the variant of $\ell_{1}$-Rank-$r$ Approximation over GF(2), where columns of $\mathbf{A}$ are strings of $S$ and approximating matrix $\mathbf{B}$ is required to have all columns equal. Note that in a binary matrix $\mathbf{B}$ of rank $1$ all non-zero columns are equal. However, it is easy to construct an equivalent instance of Closest String by attaching to each string of $S$ a string $1^{m+1}$, such that the solution to $\ell_{1}$-Rank-$r$ Approximation over GF(2) for $r=1$ does not have zero columns.

Cygan et al. [6] proved that the existence of an EPTAS for Closest String, that is $(1+\varepsilon)$-approximation in time $n^{{\cal O}(1)}\cdot f(\varepsilon)$, for any computable function $f$, is unlikely, as it would imply that FPT$=$W[1], a highly unexpected collapse in the hierarchy of parameterized complexity classes. They also showed that the existence of a PTAS for Closest String with running time $f(\varepsilon)n^{o(1/\varepsilon)}$, for any computable function $f$, would contradict the Exponential Time Hypothesis. The result of Cygan et al. implies that $\ell_{1}$-Rank-$r$ Approximation over GF(2) also does not admit EPTAS (unless FPT$=$W[1]) already for $r=1$.

A generalization of Closest String, $k$-closest strings is also known to admit a PTAS [18, 12]. This problem corresponds to the variant of $\ell_{1}$-Rank-$r$ Approximation over GF(2), where approximating matrix $\mathbf{B}$ is required to have at most $k$ different columns. However, it is not clear how solution to this special case can be adopted to solve $\ell_{1}$-Rank-$r$ Approximation over GF(2).

Our approach.

The usual toolbox of techniques to handle NP-hard variants of low-rank matrix approximation problems like sketching [36], sampling, and dimension reduction [5] is based on randomized linear algebra. It is very unclear whether any of these techniques can be used to solve even the simplest case of $\ell_{1}$-Rank-$r$ Approximation over GF(2) with $r=1$. For example for sampling, the presence of just one outlier outside of a sample, makes all information we can deduce from the sample about the column sum norm of the matrix, completely useless. This is exactly the reason why approximation algorithms for Closest String do not rely on such techniques. On the other hand, randomized dimension reduction appears to be very helpful as a “preprocessing” procedure whose application allows us to solve $\ell_{1}$-Rank-$r$ Approximation over GF(2) by applying linear programming techniques similar to the ones developed for the Closest String. From a very general perspective, our algorithm consists of three steps. While each of these steps is based on the previous works, the way to combine these steps, as well as the correctness proof, is a non-trivial task. We start with a high-level description of the steps and then provide more technical explanations.

Step 1. In order to solve $\ell_{1}$-Rank-$r$ Approximation over GF(2), we encode it as the Binary Constrained $k$-Center problem. This initial step is almost identical to the encoding used in [9] for Low GF(2)-Rank Approximation. Informally, Binary Constrained $k$-Center is defined as follows. For a given set of binary vectors $X$, a positive integer $k$, and a set of constraints, we want to find $k$ binary vectors $C=(\mathbf{c}_{1},\dots,\mathbf{c}_{k})$ satisfying the constraints and minimizing $\max_{\mathbf{x}\in X}d_{H}(\mathbf{x},C)$, where $d_{H}(\mathbf{x},C)$ is the Hamming distance between $\mathbf{x}$ and the closest vector from $C$. For example, when $k=1$ and there are no constraints, then this is just the Closest String problem over binary alphabet.

In the technical description below we give a formal definition of this encoding and in Section 4 we prove that $\ell_{1}$-Rank-$r$ Approximation over GF(2) is a special case of Binary Constrained $k$-Center. Now on, we are working with Binary Constrained $k$-Center.

Step 2. We give an approximate Turing reduction which allows to find a partition of vector set $X$ into clusters $X_{1},\dots,X_{k}$ such that if we find a tuple of vectors $C=(\mathbf{c}_{1},\dots,\mathbf{c}_{k})$ satisfying the constraints and minimizing $\max_{1\leq i\leq k,\mathbf{x}\in X_{i}}d_{H}(\mathbf{x},\{\mathbf{c}_{i}\})$, then the same tuple $C$ will be a good approximation to Binary Constrained $k$-Center. In order to obtain such a partition, we use the dimension reduction technique of Ostrovsky and Rabani [29]. While this provides us with important structural information, we are not done yet. Even with a given partition, the task of finding the corresponding tuple of “closest strings” $C$ satisfying the constraints, is non-trivial.

Step 3. In order to find the centers, we implement the approach used by Li, Ma, and Wang in [21] to solve Closest String. By brute-forcing, it is possible to reduce the solution of the problem to special instances, which loosely speaking, have a large optimum. Moreover, Binary Constrained $k$-Center  has an Integer Programming (IP) formulation. Similar to [21], for the reduced instance of Binary Constrained $k$-Center (which has a “large optimum”) it is possible to prove that the randomized rounding of the corresponding Linear Program (LP) relaxation of this IP, provides a good approximation.

Now we give a more technical description of the algorithm.

Step 1. Binary Constrained $k$-Center. To define Binary Constrained $k$-Center, we need to define some notations. A $k$-ary relation $R$ is a set of binary $k$-tuples with elements from $\{0,1\}$. A $k$-tuple $t=(t_{1},\dots,t_{k})$ satisfies $R$, we write $t\in R$, if $t$ is equal to one of the $k$-tuples in $R$.

Definition 1** (Vectors satisfying $\mathcal{R}$).**

Let $\mathcal{R}=(R_{1},\dots,R_{m})$ be a tuple of $k$-ary relations. We say that a tuple $C=(\mathbf{c}_{1},\mathbf{c}_{2},\dots,\mathbf{c}_{k})$ of binary $m$-dimensional vectors satisfies $\mathcal{R}$ and write $<C,\mathcal{R}>$, if $(\mathbf{c}_{1}[i],\ldots,\mathbf{c}_{k}[i])\in R_{i}$ for all $i\in\{1,\ldots,m\}$.

For example, for $m=2$, $k=3$, $R_{1}=\{(0,0,1),(1,0,0)\}$, and $R_{2}=\{(1,1,1),(1,0,1),(0,0,1)\}$, the tuple of vectors

[TABLE]

satisfies $\mathcal{R}=(R_{1},R_{2})$ because $(\mathbf{c}_{1}[1],\mathbf{c}_{2}[1],\mathbf{c}_{3}[1])=(0,0,1)\in R_{1}$ and $({\mathbf{c}}_{1}[2],\mathbf{c}_{2}[2],\mathbf{c}_{3}[2])=(1,0,1)\in R_{2}$.

Let us recall that the Hamming distance between two vectors $\mathbf{x},\mathbf{y}\in\{0,1\}^{m}$, where $\mathbf{x}=(x_{1},\ldots,x_{m})^{\intercal}$ and $\mathbf{y}=(y_{1},\ldots,y_{m})^{\intercal}$, is $d_{H}(\mathbf{x},\mathbf{y})=\sum_{i=1}^{m}|x_{i}-y_{i}|$ or, in words, the number of positions $i\in\{1,\ldots,m\}$ where $x_{i}$ and $y_{i}$ differ. Recall that for a set of vectors $C\subseteq\{0,1\}^{m}$ and a vector $\mathbf{x}\in\{0,1\}^{m}$, $d_{H}(\mathbf{x},C)=\min_{\mathbf{c}\in C}d_{H}(\mathbf{x},\mathbf{c})$. For sets $X,C\subset\{0,1\}^{m}$, we define ${\operatorname{cost}}(X,C)=\max_{\mathbf{x}\in X}d_{H}(\mathbf{x},C).$

Now we define Binary Constrained $k$-Center formally.

Binary Constrained $k$-Center

Input: A set $X\subseteq\{0,1\}^{m}$ of $n$ vectors, a positive integer $k$, and a tuple of $k$-ary relations $\mathcal{R}=(R_{1},\dots,R_{m})$.

Task: Among all tuples $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ of vectors from $\{0,1\}^{m}$ satisfying $\mathcal{R}$, find a tuple $C$ minimizing ${\operatorname{cost}}(X,C)$.

As in the case of Low GF(2)-Rank Approximation in [9], we prove that $\ell_{1}$-Rank-$r$ Approximation over GF(2) is a special case of Binary Constrained $k$-Center, where $k=2^{r}$. For completeness, this proof and other applications of Binary Constrained $k$-Center are given in Section 4. Thus, to prove Theorem 1, it is enough to design a PTAS for Binary Constrained $k$-Center.

Theorem 2**.**

There is an algorithm for Binary Constrained $k$-Center that given an instance $J=(X,k,{\cal R})$ and $0<\varepsilon<1$, runs in time $m^{{\cal O}(1)}n^{{\cal O}((k/\varepsilon)^{4})}$, and outputs a $(1+\varepsilon)$-approximate solution with probability at least $1-2n^{-2}$.

By the argument above, Theorem 1 is an immediate corollary of Theorem 2.

Step 2: Dimension reduction. Let $J=(X,k,{\cal R}=(R_{1},\ldots,R_{m}))$ be an instance of Binary Constrained $k$-Center and $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ be a solution to $J$, that is, a tuple of vectors satisfying ${\cal R}$. Then, the cost of $C$ is ${\operatorname{cost}}(X,C)$. Given the tuple $C$, there is a natural way we can partition the set of vectors $X$ into $k$ parts $X_{1}\uplus\cdots\uplus X_{k}$ such that

[TABLE]

Thus, for each vector $\mathbf{x}$ in $X_{i}$, the closest to $\mathbf{x}$ vector from $C$ is $\mathbf{c}_{i}$. We call such a partition $X_{1}\uplus\cdots\uplus X_{k}$ the clustering of $X$ induced by $C$ and refer to the sets $X_{1},\ldots,X_{k}$ as the clusters corresponding to $C$. We use ${\sf OPT}(J)$ to denote the cost of an optimal solution to $J$. That is, ${\sf OPT}(J)=\min\{{\operatorname{cost}}(X,C)~{}|~{}<C,\mathcal{R}>\}.$ In fact, even if we know the clustering of $X$ induced by a hypothetical optimal solution, finding a good solution is not trivial as the case when $k=1$ is the same as the Closest String problem.

As mentioned before, our approach is to reduce to a version of Binary Constrained $k$-Center, where we know the partition of $X$, and solve the corresponding problem. That is, we design an approximation scheme for the following partitioned version of the problem.

Binary Constrained Partition Center

Input: A positive integer $k$, a set $X\subseteq\{0,1\}^{m}$ of $n$ vectors partitioned into $X_{1}\uplus\ldots\uplus X_{k}$, and a tuple of $k$-ary relations $\mathcal{R}=(R_{1},\dots,R_{m})$.

Task: Among all tuples $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ of vectors from $\{0,1\}^{m}$ satisfying $\mathcal{R}$, find a tuple $C$ minimizing $\max_{i\in\{1,\ldots,k\},\mathbf{x}\in X_{i}}d_{H}(\mathbf{x},\mathbf{c}_{i})$.

For an instance $J^{\prime}=(k,X=X_{1}\uplus\ldots X_{k},{\cal R})$ of Binary Constrained Partition Center, we use ${\sf OPT}(J^{\prime})$ to denote the cost of an optimal solution to $J^{\prime}$. That is,

[TABLE]

Clearly, for an instance $J=(X,k,{\cal R})$ of Binary Constrained $k$-Center and a partition of $X$ into $X_{1}\uplus\ldots\uplus X_{k}$, any solution to the instance $J^{\prime}=(k,X=X_{1}\uplus\ldots X_{k},{\cal R})$ of Binary Constrained Partition Center, of cost $d$, is also a solution to $J$ with cost at most $d$. We prove that there is a randomized polynomial time algorithm that given an instance $J=(X,k,{\cal R})$ of Binary Constrained $k$-Center and $0<\epsilon\leq\frac{1}{4}$, outputs a collection ${\cal I}$ of Binary Constrained Partition Center instances $J^{\prime}=(k,X=X_{1}\uplus\ldots\uplus X_{k},{\cal R})$ such that the cost of at least one instance in ${\cal I}$ is at most $(1+4\epsilon){\sf OPT}(J)$ with high probability.

Lemma 1**.**

There is an algorithm that given an instance $J=(X,k,{\cal R})$ of Binary Constrained $k$-Center, $0<\epsilon\leq\frac{1}{4}$, and $\gamma>0$, runs in time $m^{2}n^{{\cal O}(k/\epsilon^{4})}$, and outputs a collection ${\cal I}$ of $m\cdot n^{{\cal O}(k/\epsilon^{4})}$ instances of Binary Constrained Partition Center such that each instance in ${\cal I}$ is of the form $(k,X=X_{1}\uplus\ldots\uplus X_{k},{\cal R})$, and there exists $J^{\prime}\in{\cal I}$ such that ${\sf OPT}(J^{\prime})\leq(1+4\epsilon){\sf OPT}(J)$ with probability at least $1-n^{-\gamma}$.

To prove Lemma 1, we use the dimension reduction technique of Ostrovsky and Rabani from [29]. Loosely speaking, this technique provides a linear map $\psi$ with the following properties. For any $\mathbf{y}\in\{0,1\}^{m}$, $\psi(\mathbf{y})$ is a 0-1 vector of length ${\cal O}(\log n/\epsilon^{4})$, and for any set $Y$ of $n+k$ vectors, Hamming distances between any pair of vectors in $\psi(Y)$ are relatively preserved with high probability. So we assume that $\psi$ is “a good map” for the set of vectors $X\cup C$, where $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ is a hypothetical optimal solution to $J$. Then, we guess the potential tuples of vectors $(\phi(\mathbf{c}_{1}),\ldots,\phi(\mathbf{c}_{k}))$ for the hypothetical optimal solution $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$, and use these choices for $(\phi(\mathbf{c}_{1}),\ldots,\phi(\mathbf{c}_{k}))$ to construct partitions of $X$, and thereby construct instances in ${\cal I}$. Lemma 1 is proved in Section 2.

Step 3: LP relaxation. Because of Lemma 1, to prove Theorem 2, it is enough to design a PTAS for Binary Constrained Partition Center. So we prove the following lemma.

Lemma 2**.**

There is an algorithm for Binary Constrained Partition Center that given an instance $J=(k,X=X^{i}_{1}\uplus\ldots\uplus X^{i}_{k},{\cal R})$ and $0<\epsilon<1/2$, runs in time $m^{{\cal O}(1)}n^{{\cal O}((k/\epsilon)^{4})}$, and outputs a solution of cost at most $(1+\epsilon){\sf OPT}(J)$ with probability at least $1-n^{-2}$.

Towards the proof of Lemma 2, we encode Binary Constrained Partition Center using an Integer programming (IP) formulation (see (6) in Section 3). We show that the randomized rounding using the solution of the linear programming relaxation of this IP provides a good approximation if the optimum value is large. Here we follow the approach similar to the one used by Li, Ma, and Wang in [21] to solve Closest String. We prove that there exist $Y_{1}\subseteq X_{1},\ldots,Y_{k}\subseteq X_{k}$, each of size $r=1+\frac{4}{\epsilon}$, with the following property. Let $Q$ be the set of positions in $\{1,\ldots,m\}$ such that for each $i\in\{1,\ldots,k\}$ and $j\in Q$, all the vectors in $Y_{i}$ agree at the position $j$, and for each $j\in Q$, $(\mathbf{y}_{1}[j],\ldots,\mathbf{y}_{k}[j])\in R_{j}$, where $\mathbf{y}_{i}\in Y_{i}$ for all $i\in\{1,\ldots,k\}$. Then, for any solution of $J$ such that for each $j\in Q$ the entries at the position $j$ coincide with $(\mathbf{y}_{1}[j],\ldots,\mathbf{y}_{k}[j])$, the cost of this solution restricted to $Q$ deviates from the cost of an optimal solution restricted to $Q$ by at most $\frac{1}{r-1}{\sf OPT}(J)$. Moreover, the subproblem of $J$ restricted to $\{1,\ldots,m\}\setminus Q$ has large optimum value and we could use linear programming to solve the subproblem. Lemma 2 is proved in Section 3.

Putting together.

Next we explain how to prove Theorem 2 using Lemmata 1 and 2. Let $J=(X,k,{\cal R})$ be the input instance of Binary Constrained $k$-Center and $0<\varepsilon<1$ be the given error parameter. Let $\beta=\frac{\varepsilon}{8}$. Since $\varepsilon<1$, $\beta<\frac{1}{4}$. Now, we apply Lemma 1 on $J$, $\beta$, and $\gamma=2$. As a result, we get a collection ${\cal I}$ of instances of Binary Constrained Partition Center such that each instance in ${\cal I}$ is of the form $(k,X=X_{1}\uplus\ldots\uplus X_{k},{\cal R})$, and there exists $J^{\prime}\in{\cal I}$ such that ${\sf OPT}(J^{\prime})\leq(1+4\beta){\sf OPT}(J)$ with probability at least $1-n^{-2}$. From now on, we assume that this event happened. Next, for each instance $\widehat{J}\in{\cal I}$, we apply Lemma 2 with the error parameter $\beta$, and output the best solution among the solutions produced. Let $J^{\prime}\in{\cal I}$ be the instance such that ${\sf OPT}(J^{\prime})\leq(1+4\beta){\sf OPT}(J)\leq(1+\frac{\varepsilon}{2}){\sf OPT}(J)$. Any solution to $\widehat{J}\in{\cal I}$ of cost $d$, is also a solution to $J$ of cost at most $d$. Therefore, because of Lemmas 1 and 2, our algorithm outputs a solution of $J$ with cost at most $(1+\beta){\sf OPT}(J^{\prime})=(1+\frac{\varepsilon}{8})(1+\frac{\varepsilon}{2}){\sf OPT}(J)\leq(1+\varepsilon){\sf OPT}(J)$ with probability at least $1-2n^{-2}$, since both Lemmas 1 and 2 have the success probability of at least $1-n^{-2}$. The running time of the algorithm follows from Lemmata 1 and 2.

As Theorem 2 is already proved using Lemmas 1 and 2, the rest of the paper is devoted to the proofs of Lemmata 1 and 2, and to the examples of the expressive power of Binary Constrained $k$-Center, including $\ell_{1}$-Rank-$r$ Approximation over GF(2). In Sections 2 and 3, we prove Lemmata 1 and 2, respectively. In Section 4, we give applications of Theorem 2.

2 Proof of Lemma 1

In this section we prove Lemma 1. The main idea is to map the given instance to a low-dimensional space while approximately preserving distances, then try all possible tuples of centers in the low-dimensional space, and construct an instance of Binary Constrained Partition Center by taking the optimal partition of the images with respect to a fixed tuple of centers back to the original vectors.

To implement the mapping, we employ the notion of $(\delta,\ell,h)$-distorted maps, introduced by Ostrovsky and Rabani [29]. Intuitively, a $(\delta,\ell,h)$-distorted map approximately preserves distances between $\ell$ and $h$, does not shrink distances larger than $h$ too much, and does not expand distances smaller than $\ell$ too much. In what follows we make the definitions formal.

A metric space is a pair $(P,d)$ where $P$ is a set (whose elements are called points), and $d$ is a distance function $d:P\times P\rightarrow{\mathbb{R}}$ (called a metric), such that for every $p_{1},p_{2},p_{3}\in P$ the following conditions hold: $(i)$ $d(p_{1},p_{2})\geq 0$, $(ii)$ $d(p_{1},p_{2})=d(p_{2},p_{1})$, $(iii)$ $d(p_{1},p_{2})=0$ if and only if $p_{1}=p_{2}$, and $(iv)$ $d(p_{1},p_{2})+d(p_{2},p_{3})\geq d(p_{1},p_{3})$. Condition $(iv)$ is called the triangle inequality. The pair $(\{0,1\}^{m},d_{H})$, binary vectors of lentgh $m$ and the Hamming distance, is a metric space.

Definition 2** ([29]).**

Let $(P,d)$ and $(P^{\prime},d^{\prime})$ be two metric spaces. Let $X,Y\subseteq P$. Let $\delta,\ell,h$ be such that $\delta>0$ and $h>\ell\geq 0$. A mapping $\psi:P\rightarrow P^{\prime}$ is $(\delta,\ell,h)$-distorted on $(X,Y)$ if and only if there exists $\alpha>0$ such that for every $x\in X$ and $y\in Y$, the following conditions hold.

If $d(x,y)<\ell$, then $d(\psi(x),\psi(y))<(1+\delta)\alpha\ell$. 2. 2.

If $d(x,y)>h$, then $d(\psi(x),\psi(y))>(1-\delta)\alpha h$. 3. 3.

If $\ell\leq d(x,y)\leq h$, then $(1-\delta)\alpha d(x,y)\leq d(\psi(x),\psi(y))\leq(1+\delta)\alpha d(x,y)$.

If $X=Y$, then we say that $\psi$ is $(\delta,\ell,h)$-distorted on $X$.

For any $r,r^{\prime}\in{\mathbb{N}}$ and $\varepsilon>0$, ${\cal A}_{r,r^{\prime}}(\varepsilon)$ denotes a distribution over $r^{\prime}\times r$ binary matrices $M\in\{0,1\}^{r^{\prime}\times r}$, where entries are independent, identically distributed, random $0/1$ variables with $\Pr[1]=\varepsilon$.

Proposition 3** ([29]).**

Let $m,\ell\in{\mathbb{N}}$, and let $X\subseteq\{0,1\}^{m}$ be a set of $n$ vectors. For every $0<\epsilon\leq 1/2$, there exists a mapping $\phi:X\rightarrow\{0,1\}^{m^{\prime}}$, where $m^{\prime}={\cal O}(\log n/\epsilon^{4})$, which is $(\epsilon,\ell/4,\ell/2\epsilon)$-distorted on $X$ (with respect to the Hamming distance in both spaces). More precisely, for every $\gamma>0$ there exists $\lambda>0$, such that, setting $m^{\prime}=\lambda\log n/\epsilon^{4}$, the linear map $\mathbf{x}\mapsto A\mathbf{x}$, where $A$ is a random matrix drawn from ${\cal A}_{m,m^{\prime}}(\epsilon^{2}/\ell)$, is $(\epsilon,\ell/4,\ell/2\epsilon)$-distorted on $X$ with probability at least $1-n^{-\gamma}$.

Now we are ready to prove Lemma 1. We restate it for convenience.

See 1

Proof.

Without loss of generality, we may assume ${\sf OPT}(J)>0$. If ${\sf OPT}(J)=0$, there are at most $k$ distinct vectors in $X$, and we trivially construct a single instance of Binary Constrained Partition Center by grouping equal vectors together.

Let $n=|X|$ and $n^{\prime}=n+k$. Let $\lambda=\lambda(\gamma)$ be the constant mentioned in Proposition 3, and $m^{\prime}=\lambda\log n^{\prime}/\epsilon^{4}$. Then, for each $\ell\in[m]$111For an integer $n\in{\mathbb{N}}$, we use $[n]$ as a shorthand for $\{1,\ldots,n\}$., we construct the collection ${\cal I}_{\ell}$ of $n^{{\cal O}(k/\epsilon^{4})}$ Binary Constrained Partition Center instances as follows.

• •

Start with ${\cal I}_{\ell}:=\emptyset$.

• •

Randomly choose a matrix $A^{\ell}$ from the distribution ${\cal A}_{m,m^{\prime}}(\epsilon^{2}/\ell)$.

• •

For each choice of $k$ vectors $\mathbf{c}_{1}^{\prime},\ldots,\mathbf{c}_{k}^{\prime}\in\{0,1\}^{m^{\prime}}$, construct a partition $X_{1}\uplus\ldots\uplus X_{k}$ of $X$ such that for each $\mathbf{x}\in X_{i}$, $\mathbf{c}_{i}^{\prime}$ is one of the closest vectors to $A^{\ell}\mathbf{x}$ among $C^{\prime}=\{\mathbf{c}_{1}^{\prime},\ldots,\mathbf{c}_{k}^{\prime}\}$. Then, add $(k,X=X^{i}_{1}\uplus\ldots X^{i}_{k},{\cal R})$ to ${\cal I}_{\ell}$.

Finally, our algorithm outputs ${\cal I}=\bigcup_{\ell\in[m]}{\cal I}_{\ell}$ as the required collection of Binary Constrained Partition Center instances. Notice that for any $\ell\in[m]$, $|{\cal I}_{\ell}|=2^{m^{\prime}k}=n^{{\cal O}(k/\epsilon^{4})}$. This implies that the cardinality of $\cal I$ is upper bounded by $m\cdot n^{{\cal O}(k/\epsilon^{4})}$, and the construction of ${\cal I}_{\ell}$ takes time $m\cdot n^{{\cal O}(k/\epsilon^{4})}$. Thus, the total running time of the algorithm is $m^{2}\cdot n^{{\cal O}(k/\epsilon^{4})}$.

Next, we prove the correctness of the algorithm. Let $\ell={\sf OPT}(J)$ and $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ be an optimum solution of $J$. Let $Y_{1},\ldots,Y_{k}$ be the clusters corresponding to $C$. Consider the step in the algorithm where we constructed ${\cal I}_{\ell}$. By Proposition 3, the map $\psi\colon\mathbf{x}\mapsto A^{\ell}\mathbf{x}$ is $(\epsilon,\ell/4,\ell/2\epsilon)$-distorted on $X\cup C$ with probability at least $1-n^{-\gamma}$. In the rest of the proof, we assume that this event happened. Let $\mathbf{c}_{1}^{\prime}=A^{\ell}\mathbf{c}_{1},\ldots,\mathbf{c}_{k}^{\prime}=A^{\ell}\mathbf{c}_{k}$. Consider the Binary Constrained Partition Center instance constructed for the choice of vectors $\mathbf{c}_{1}^{\prime},\ldots,\mathbf{c}^{\prime}_{k}$. That is, let $X_{1},\ldots,X_{k}$ be the partition of $X$ such that for each $\mathbf{x}\in X_{i}$, $\mathbf{c}_{i}^{\prime}$ is one of the closest vector to $A^{\ell}\mathbf{x}$ from $C^{\prime}=\{\mathbf{c}_{1}^{\prime},\ldots,\mathbf{c}_{k}^{\prime}\}$. Let $J^{\prime}$ be the instance $(k,X=X^{i}_{1}\uplus\ldots X^{i}_{k},{\cal R})$ of Binary Constrained Partition Center.

Now, we claim that $C$ is a solution to $J^{\prime}$ with cost at most $(1+4\epsilon)\ell=(1+4\epsilon){\sf OPT}(J)$. Since $C$ satisfies ${\cal R}$, $C$ is a solution of $J^{\prime}$. To prove ${\sf OPT}(J^{\prime})\leq(1+4\epsilon)\ell$, it is enough to prove that for each $i\in[k]$ and $\mathbf{x}\in X_{i}$, $d_{H}(\mathbf{x},\mathbf{c}_{i})\leq(1+4\epsilon)\ell$. Fix an index $i\in[k]$ and $\mathbf{x}\in X_{i}$. Suppose $\mathbf{x}\in Y_{i}$. Since $C$ is an optimum solution of $J$ with corresponding clusters $Y_{1},\ldots Y_{k}$, we have that $d_{H}(\mathbf{y},\mathbf{c}_{i})\leq\ell$ for all $\mathbf{y}\in Y_{i}\cap X_{i}$. Thus, $d_{H}(\mathbf{x},\mathbf{c}_{i})\leq\ell$. So, now consider the case $\mathbf{x}\in Y_{j}$ for some $j\neq i$. Notice that if $d_{H}(\mathbf{x},\mathbf{c}_{i})\leq\ell$, then we are done. We have the following two subcases.

Case 1: $d_{H}(\mathbf{x},\mathbf{c}_{i})\leq\frac{\ell}{2\epsilon}$.

We know that the map $\psi\colon\mathbf{x}\mapsto A^{\ell}\mathbf{x}$ is $(\epsilon,\ell/4,\ell/2\epsilon)$-distorted on $X\cup C$, and let $\alpha>0$ be the number such that conditions of Definition 2 hold. Since $\mathbf{x}\in X_{i}$, we have that $(a)$ $d_{H}(\psi(\mathbf{x}),\psi(\mathbf{c}_{i}))\leq d_{H}(\psi(\mathbf{x}),\psi(\mathbf{c}_{j}))$. Since $d_{H}(\mathbf{x},\mathbf{c}_{j})\leq\ell$ (because $\mathbf{x}\in Y_{j}$) and $\psi$ is $(\epsilon,\ell/4,\ell/2\epsilon)$-distorted on $X\cup C$, we have that $(b)$ $d_{H}(\psi(\mathbf{x}),\psi(\mathbf{c}_{j}))\leq(1+\epsilon)\alpha\ell$. Since $\ell<d_{H}(\mathbf{x},\mathbf{c}_{i})\leq\frac{\ell}{2\epsilon}$, and $\psi$ is $(\epsilon,\ell/4,\ell/2\epsilon)$-distorted on $X\cup C$, we have that $(c)$ $(1-\epsilon)\alpha d_{H}(\mathbf{x},\mathbf{c}_{i})\leq d_{H}(\psi(\mathbf{x}),\psi(\mathbf{c}_{i}))$. The statements $(a)$, $(b)$, and $(c)$ imply that

[TABLE]

where the last inequality holds since $\epsilon\leq 1/4$.

Case 2: $d_{H}(\mathbf{x},\mathbf{c}_{i})>\frac{\ell}{2\epsilon}$.

We prove that this case is impossible by showing a contradiction. Since $\epsilon\leq 1/4$, in this case, we have that $d_{H}(\mathbf{x},\mathbf{c}_{i})>2\ell$. Since $\psi$ is $(\epsilon,\ell/4,\ell/2\epsilon)$-distorted on $X\cup C$, $d_{H}(\mathbf{x},\mathbf{c}_{i})>2\ell$, and $d_{H}(\mathbf{x},\mathbf{c}_{j})\leq\ell$, we have that

[TABLE]

Then $2(1-\epsilon)\leq(1+\epsilon)$ and thus $\epsilon\geq 1/3$, which contradicts the assumption that $\epsilon\leq 1/4$.

This completes the proof of the lemma. ∎

3 Proof of Lemma 2

For a set of positions $P\subset[m]$, let us define the Hamming distance restricted to $P$ by

[TABLE]

We use the following lemma in our proof.

Lemma 3**.**

Let $Y=\{\mathbf{y}_{1},\cdots,\mathbf{y}_{l}\}\subset\{0,1\}^{m}$ be a set of vectors and $\mathbf{c}^{*}\in\{0,1\}^{m}$ be a vector. Let $d^{*}={\operatorname{cost}}(Y,\{\mathbf{c}^{*}\})=\max_{\mathbf{y}\in Y}d_{H}(\mathbf{y},\mathbf{c}^{*})$. For any $r\in\mathbb{N}$, $r>2$, there exist indices $i_{1}$, …, $i_{r}$ such that for any $\mathbf{x}\in Y$

[TABLE]

where $P$ is any subset of $Q_{i_{1},\ldots,i_{r}}$ and $Q_{i_{1},\ldots,i_{r}}$ is the set of positions where all of $\mathbf{y}_{i_{1}},\ldots,\mathbf{y}_{i_{r}}$ coincide (i.e., $Q_{i_{1},\ldots,i_{r}}=\{j\in[m]\colon\mathbf{y}_{i_{1}}[j]=\mathbf{y}_{i_{2}}[j]=\ldots=\mathbf{y}_{i_{r}}[j]\}$).

Proof.

For a vector $\mathbf{x}=\mathbf{y}_{\ell^{\prime}}\in Y$ and $P\subseteq Q_{i_{1},\ldots,i_{r}}$, let

[TABLE]

To prove the lemma it is enough to prove that $|J_{P}(\ell^{\prime})|\leq\frac{1}{r-1}d^{*}$. Also, since $J_{P}(\ell^{\prime})\subseteq J(\ell^{\prime})$, to prove the lemma, it is enough to prove that $|J(\ell^{\prime})|\leq\frac{1}{r-1}d^{*}$. Recall that for any $s\in[\ell]$ and $1\leq i_{1},\ldots,i_{s}\leq\ell$, $Q_{i_{1},\ldots,i_{s}}$ is the set of positions where all of $\mathbf{y}_{i_{1}},\ldots,\mathbf{y}_{i_{s}}$ coincide. For any $2\leq s\leq r+1$ and $1\leq i_{1},\ldots,i_{s}\leq\ell$, let $p_{i_{1},\ldots,i_{s}}$ be the number of mismatches between $\mathbf{y}_{i_{1}}$ and $\mathbf{c}^{*}$ at the positions in $Q_{i_{1},\ldots,i_{s}}$. Let

[TABLE]

Notice that for any $2\leq s\leq r+1$, $\rho_{s}\leq 1$.

Claim 1** (Claim 2.2 [21]).**

222We remark that Claim 2.2 in [21] is stated for a vector $\mathbf{c}$ such that $d^{*}={\operatorname{cost}}(Y,\{\mathbf{c}\})=\min_{\mathbf{c}^{\prime}}{\operatorname{cost}}(Y,\{\mathbf{c}^{\prime}\})$. But the steps of the same proof work in our case as well.

For any $s$ such that $2\leq s\leq r$, there are indices $1\leq i_{1},i_{2},\ldots,i_{r}\leq\ell$ such that for any $\mathbf{x}=\mathbf{y}_{\ell^{\prime}}\in Y$, $|J(\ell^{\prime})|\leq(\rho_{s}-\rho_{s+1})d^{*}$.

Proof.

Consider indices $1\leq i_{1},\ldots,i_{s}\leq\ell$ such that $p_{i_{1},\ldots,i_{s}}=\rho_{s}\cdot d^{*}$. Next arbitrarily pick $r-s$ indices $i_{s+1},i_{s+2},\ldots,i_{r}$ from $[\ell]\setminus\{i_{1},\ldots,i_{s}\}$. Next we prove that $i_{1},i_{2},\ldots,i_{r}$ are the required set of indices. Towards that, fix $\mathbf{x}=\mathbf{y}_{\ell^{\prime}}\in Y$,

[TABLE]

The equality (3) holds since $Q_{i_{1},\ldots,i_{s}}\supseteq Q_{i_{1},\ldots,i_{s},\ell^{\prime}}$. The inequality (4) holds because $p_{i_{1},\ldots,i_{s}}=\rho_{s}\cdot d^{*}$ by the choice of $i_{1}$, …, $i_{s}$, and $\rho_{s+1}d^{*}\leq p_{i_{1},\ldots,i_{s},\ell^{\prime}}$ by definition. This completes the proof of the claim. ∎

Notice that $(\rho_{2}-\rho_{3})+(\rho_{3}-\rho_{4})+\ldots+(\rho_{r}-\rho_{r+1})=(\rho_{2}-\rho_{r+1})\leq\rho_{2}\leq 1.$ Thus, one of $(\rho_{2}-\rho_{3}),(\rho_{3}-\rho_{4}),\ldots,(\rho_{r}-\rho_{r+1})$ is at most $1/(r-1)$. This completes the proof of the lemma. ∎

Consider the instance $J=(k,X=X^{i}_{1}\uplus\ldots X^{i}_{k},{\cal R})$ of Binary Constrained Partition Center. Let $C^{*}=(\mathbf{c}^{*}_{1},\cdots,\mathbf{c}^{*}_{k})\subset\{0,1\}^{m}$ be an optimum solution to $J$. Let $d_{opt}={\sf OPT}(J)=\max_{i\in[k],\mathbf{x}\in X_{i}}d_{H}(\mathbf{x},\mathbf{c}^{*}_{i})$. For each $i\in[k]$ and $r\geq 2$, by Lemma 3, there exist $r$ elements $\mathbf{x}^{(1)}_{i}$, …, $\mathbf{x}^{(r)}_{i}$ of $X_{i}$ such that for any $\mathbf{x}\in X_{i}$,

[TABLE]

where $P$ is any subset of $Q_{i}$, and $Q_{i}$ is the set of coordinates on which $\mathbf{x}^{(1)}_{i}$, …, $\mathbf{x}^{(r)}_{i}$ agree. Let us denote as $Q$ the intersection of all $Q_{i}$ from which the positions not satisfying $\mathcal{R}$ are removed. That is,

[TABLE]

Because of (5), there is an approximate solution where the coordinates $j\in Q$ are identified using $\mathbf{x}^{(1)}_{1},\ldots,\mathbf{x}^{(1)}_{k}$. Let $\overline{Q}=[m]\setminus Q$. Now the idea is to solve the problem restricted to $\overline{Q}$ separately, and then complement the solution on $Q$ by the values of $\mathbf{x}^{(1)}_{i}$. We prove that for the ‘subproblem’ restricted on $\overline{Q}$, the optimum value is large. Towards that we first prove the following lemma.

Lemma 4**.**

Let $J=(k,X=X^{i}_{1}\uplus\ldots X^{i}_{k},{\cal R})$ be an instance of Binary Constrained Partition Center. Let $({\mathbf{c}}^{*}_{1},\ldots,{\mathbf{c}}^{*}_{k})$ be an optimal solution for $J$, and $r\geq 2$ be an integer. Then, there exist $\{\mathbf{x}^{(1)}_{1}$, …, $\mathbf{x}^{(r)}_{1}\}\subset X_{1},\ldots,\{\mathbf{x}^{(1)}_{k}$, …, $\mathbf{x}^{(r)}_{k}\}\subset X_{k}$ with the following properties. For each $i\in[k]$, let $Q_{i}$ be the set of coordinates on which $\mathbf{x}^{(1)}_{i}$, …, $\mathbf{x}^{(r)}_{i}$ agree, $Q=\left\{j\in\bigcap_{i\in[k]}Q_{i}\colon(\mathbf{x}^{(1)}_{1}[j],\mathbf{x}^{(1)}_{2}[j],\ldots,\mathbf{x}^{(1)}_{k}[j])\in R_{j}\right\}$, and $\overline{Q}=[m]\setminus Q$.

• •

For any $i\in[k]$ and $\mathbf{x}\in X_{i}$, $d_{H}^{Q}(\mathbf{x},\mathbf{x}^{(1)}_{i})-d_{H}^{Q}(\mathbf{x},\mathbf{c}_{i}^{*})\leq\frac{1}{r-1}{\sf OPT}(J)$, and

• •

$|\overline{Q}|\leq rk\cdot{\sf OPT}(J)$.

Proof.

Fix an integer $i\in[k]$. By substituting $Y=X_{i}$ and $\mathbf{c}^{*}=\mathbf{c}_{i}^{*}$ in Lemma 3, we get $\{\mathbf{x}^{(1)}_{i}$, …, $\mathbf{x}^{(r)}_{i}\}\subset X_{i}$ such that for any $\mathbf{x}\in X_{i}$, $d_{H}^{Q}(\mathbf{x},\mathbf{x}^{(1)}_{i})-d_{H}^{Q}(\mathbf{x},\mathbf{c}_{i}^{*})\leq\frac{1}{r-1}{\sf OPT}(J)$. That is, we have proved the first condition in the lemma. The second condition is proved in the following claim.

Claim 2**.**

$|\overline{Q}|\leq rk\cdot{\sf OPT}(J)$.

Proof.

We claim that for each position $j\in\overline{Q}$ there exist $i\in[k]$ and $s\in[r]$ such that $\mathbf{x}^{(s)}_{i}[j]\neq{\mathbf{c}}^{*}_{i}[j]$. There are two kinds of positions in $\overline{Q}$. First, positions, where for some $i\in[k]$ vectors $\mathbf{x}^{(1)}_{i}$, …, $\mathbf{x}^{(r)}_{i}$ do not agree, in this case certainly one of them does not agree with the corresponding position in $\mathbf{c}^{*}_{i}$. Second, positions $j$ where for any $i\in[k]$, $\mathbf{x}^{(1)}_{i}[j]=\mathbf{x}^{(2)}_{i}[j]=\cdots=\mathbf{x}^{(r)}_{i}[j]$, but $(\mathbf{x}^{(1)}_{1}[j],\ldots,\mathbf{x}^{(1)}_{k}[j])\notin R_{j}$. Then, there exists $i\in[k]$ such that $\mathbf{x}^{(1)}_{i}[j]\neq{\mathbf{c}}^{*}_{i}[j]$ because $({\mathbf{c}}^{*}_{i}[j],\ldots,{\mathbf{c}}^{*}_{k}[j])\in R_{j}$.

Now, for any $i\in[k]$ and $s\in[r]$, $\mathbf{x}^{(s)}_{i}$ contributes at most ${\sf OPT}(J)$ positions to $\overline{Q}$, since $d_{H}(\mathbf{x}^{(s)}_{i},{\mathbf{c}}^{*}_{i})\leq{\sf OPT}(J)$. Thus, in total there are at most $rk\cdot{\sf OPT}(J)$ positions in $\overline{Q}$. ∎

This completes the proof of the lemma. ∎

As mentioned earlier, we fix the entries of our solution in positions $j$ of $Q$ with values in $\mathbf{x}^{(1)}_{1}[j],\ldots,\mathbf{x}^{(1)}_{k}[j]$. Towards finding the entries of our solution in positions of $\overline{Q}$, we define the following problem and solve it.

Binary Constrained Partition Center*⋆*

Input: A positive integer $k$, a set $X\subseteq\{0,1\}^{m}$ of $n$ vectors partitioned into $X_{1}\uplus\ldots\uplus X_{k}$, a tuple of $k$-ary relations $\mathcal{R}=(R_{1},\dots,R_{m})$, and for all $\mathbf{x}\in X$, $d_{\mathbf{x}}\in{\mathbb{N}}$

Task: Among all tuples $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ of vectors from $\{0,1\}^{m}$ satisfying $\mathcal{R}$, find a tuple $C$ that minimizes the integer $d$ such that for all $i\in[k]$ and $\mathbf{x}\in X_{i}$, $d_{H}(\mathbf{x},\mathbf{c}_{i})\leq d-d_{\mathbf{x}}$.

Lemma 5**.**

Let $J^{\prime}=(k,X=X_{1}\uplus\ldots X_{k},{\cal R},(d_{\mathbf{x}})_{\mathbf{x}\in X})$ be an instance of Binary Constrained Partition Center⋆, ${\sf OPT}(J^{\prime})\geq\frac{m}{c}$ for some integer $c$, and $0<\delta<1/c$. Then, there is an algorithm which runs in time $m^{{\cal O}(1)}n^{{\cal O}(c^{2}k/\delta^{2})}$, and outputs a solution $C$ of $J^{\prime}$, of cost at most $(1+\delta){\sf OPT}(J^{\prime})$ with probability at least $1-n^{-2}$.

Before proving Lemma 5, we explain how all these results puts together to form a proof of Lemma 2. We restate Lemma 2 for the convenience of the reader.

See 2

Proof.

Let $J=(k,X=X^{i}_{1}\uplus\ldots X^{i}_{k},{\cal R})$ be the input instance of Binary Constrained Partition Center, and $0<\epsilon<\frac{1}{2}$ be the error parameter. Let $({\mathbf{c}}^{*}_{1},\ldots,{\mathbf{c}}^{*}_{k})$ be an optimal solution for $J$. Let $r\geq 2$ be an integer which we fix later. First, for each $i\in[k]$ we obtain $r$ vectors $\mathbf{x}^{(1)}_{i}$, …, $\mathbf{x}^{(r)}_{i}\in X_{i}$ which satisfy the conditions of Lemma 4. Their existence is guaranteed by Lemma 4, and we guess them in time $n^{{\cal O}(rk)}$ over all $i\in[k]$. For each $i\in[k]$, let $Q_{i}$ be the set of coordinates on which $\mathbf{x}^{(1)}_{i}$, …, $\mathbf{x}^{(r)}_{i}$ agree, $Q=\left\{j\in\bigcap_{i\in[k]}Q_{i}\colon(\mathbf{x}^{(1)}_{1}[j],\mathbf{x}^{(1)}_{2}[j],\ldots,\mathbf{x}^{(1)}_{k}[j])\in R_{j}\right\}$, and $\overline{Q}=[m]\setminus Q$. Next, we construct a solution $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ as follows. For each $i\in[k]$ and $j\in Q$, we set $\mathbf{c}_{i}[j]=\mathbf{x}_{i}^{(1)}[j]$.

Towards finding the entries of vectors $\mathbf{c}_{1},\ldots,\mathbf{c}_{k}$ at the coordinates in $\overline{Q}$, we use Lemma 5. Let $J^{\prime}$ be the instance of Binary Constrained Partition Center*⋆*, which is a natural restriction of $J$ to $\overline{Q}$. That is, $J^{\prime}=(k,X^{\prime}=X^{\prime}_{1}\uplus\ldots X^{\prime}_{k},{\cal R}|_{\overline{Q}},(d_{\mathbf{x}|_{\overline{Q}}})_{\mathbf{x}\in X^{\prime}})$, where for each $i\in[k]$, $X_{i}^{\prime}=\{\mathbf{x}|_{\overline{Q}}\colon\mathbf{x}\in X_{i}\}$ and for each $\mathbf{x}\in X_{i}$, $d_{\mathbf{x}|_{\overline{Q}}}=d_{H}^{Q}(\mathbf{x},\mathbf{x}^{(1)}_{i})$. By the second condition in Lemma 4, we have that $|\overline{Q}|\leq rk\cdot{\sf OPT}(J)$.

Claim 3**.**

${\sf OPT}(J)\leq{\sf OPT}(J^{\prime})\leq\left(1+\frac{1}{r-1}\right){\sf OPT}(J).$**

Proof.

First, we prove that ${\sf OPT}(J)\leq{\sf OPT}(J^{\prime})$. Towards that we show that we can transform a solution $C^{\prime}=(\mathbf{c}^{\prime}_{1},\cdots,\mathbf{c}^{\prime}_{k})$ of $J^{\prime}$ with the objective value $d$ to a solution $C$ of $J$ with the same objective value. For each $i\in[k]$, consider $\widehat{\mathbf{c}}_{i}$ which is equal to $\mathbf{x}^{(1)}_{i}$ restricted to $Q$, and to $\mathbf{c}^{\prime}_{i}$ restricted to $\overline{Q}$, and the solution $\widehat{C}=(\widehat{\mathbf{c}}_{1},\cdots,\widehat{\mathbf{c}}_{k})$. Clearly, $\widehat{C}$ satisfies ${\cal R}$ since on $\overline{Q}$ it is guaranteed by $C^{\prime}$ being a solution to $J^{\prime}$, and on $Q$ by construction of $Q$. The objective value of $C$ is

[TABLE]

Thus, ${\sf OPT}(J)\leq{\sf OPT}(J^{\prime})$.

Next, we prove that ${\sf OPT}(J^{\prime})\leq\left(1+\frac{1}{r-1}\right){\sf OPT}(J)$. Recall that $({\mathbf{c}}^{*}_{1},\ldots,{\mathbf{c}}^{*}_{k})$ is an optimal solution for $J$. Then, $({\mathbf{e}}^{*}_{1},\ldots,{\mathbf{e}}^{*}_{k})$, where each $\mathbf{e}^{*}_{i}$ is the restriction of $\mathbf{c}^{*}_{i}$ on $\overline{Q}$, is a solution for $J^{\prime}$. For each $i\in[k]$ and $\mathbf{x}\in X_{i}$,

[TABLE]

This completes the proof of the claim. ∎

Since $|\overline{Q}|\leq rk\cdot{\sf OPT}(J)$ and by Claim 3, we have that ${\sf OPT}(J^{\prime})\geq\frac{|\overline{Q}|}{rk}=\frac{|\overline{Q}|}{c}$, where $c=rk$. Let $0<\delta<\frac{1}{c}$ be a number which we fix later.

Now we apply Lemma 5 on the input $J^{\prime}$ and $\delta$, and let $C^{\prime}=(\mathbf{c}^{\prime}_{1}$, …, $\mathbf{c}^{\prime}_{k})$ be the solution for $J^{\prime}$ obtained. We know that the cost $d^{\prime}$ of $\mathbf{c}^{\prime}$ is at most $(1+\delta){\sf OPT}(J^{\prime})$ with probability at least $1-n^{-2}$. For the rest of the proof we assume that the cost $d^{\prime}\leq(1+\delta){\sf OPT}(J^{\prime})$. Recall that we have partially computed the entries of the solution $\mathbf{c}=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ for the instance $J$. That is, for each $j\in Q$ and $i\in[k]$, we have already set the value of $\mathbf{c}_{i}[j]$. Notice that $C^{\prime}\subseteq\{0,1\}^{|\overline{Q}|}$. Since $J^{\prime}$ is obtained from $J$ by restricting to $\overline{Q}$, there is a natural bijection $f$ from $\overline{Q}$ to $[|\overline{Q}|]$ such that for each $\mathbf{x}\in X$ and $j\in{\overline{Q}}$, $\mathbf{x}[j]=\mathbf{y}[f(j)]$, where $\mathbf{y}=\mathbf{x}|_{\overline{Q}}$. Now for each $i\in[k]$ and $j\in\overline{Q}$, we set $\mathbf{c}_{i}[j]=\mathbf{c}^{\prime}_{i}[f(j)]$.

In Claim 3, we have proven that the solution $C$ of $J$ obtained in this way has cost at most $d^{\prime}$. By Lemma 5, we know that $d^{\prime}\leq(1+\delta){\sf OPT}(J^{\prime})$. By Claim 3, ${\sf OPT}(J^{\prime})\leq(1+\frac{1}{r-1}){\sf OPT}(J)$. Thus, we have that the cost of the solution $C$ of $J$ is at most $(1+\delta)(1+\frac{1}{r-1}){\sf OPT}(J)$. Now we fix $r=(1+\frac{4}{\epsilon})$ and $\delta=\frac{\epsilon}{(2\epsilon+8)k}$. Then the cost of $C$ is at most $(1+\epsilon){\sf OPT}(J)$.

Running time analysis.

The number of choices for $\{\mathbf{x}^{(1)}_{1}$, …, $\mathbf{x}^{(r)}_{1}\}\subset X_{1},\ldots,\{\mathbf{x}^{(1)}_{k}$, …, $\mathbf{x}^{(r)}_{k}\}\subset X_{k}$ is at most $n^{{\cal O}(rk)}=n^{{\cal O}(k/\epsilon)}$. For each such choice, we run the algorithm of Lemma 5 which takes time at most $m^{{\cal O}(1)}n^{{\cal O}(c^{2}k/\delta^{2})}=m^{{\cal O}(1)}n^{{\cal O}((k/\epsilon)^{4})}$. Thus, the total running time is $m^{{\cal O}(1)}n^{{\cal O}((k/\epsilon)^{4})}$. ∎

Now the only piece left is the proof of Lemma 5. We use the following tail inequality (a variation of Chernoff bound) in the proof of Lemma 5.

Proposition 4** (Lemma 1.2 [21]).**

Let $X_{1},\ldots,X_{n}$ be $n$ independent [math]-$1$ random variables, $X=\sum_{i=1}^{n}X_{i}$, and $0<\epsilon\leq 1$. Then, $\Pr[X>E[X]+\epsilon n]\leq e^{-\frac{1}{3}n\epsilon^{2}}.$

Finally, we prove Lemma 5.

Proof of Lemma 5.

First, assume that $m<9c^{2}\log n/\delta^{2}$. If this is the case, we enumerate all possible solutions for $J^{\prime}$ and output the best solution. The number of solutions is at most $2^{k\cdot m}=n^{{\cal O}(c^{2}k/\delta^{2})}$. Thus, in this case the algorithm is exact and deterministic, and the running time bound holds. For the rest of the proof we assume that $m\geq 9c^{2}\log n/\delta^{2}$.

Binary Constrained Partition Center*⋆* can be formulated as a [math]-$1$ optimization problem as explained below. For each $j\in[m]$ and tuple $t\in R_{j}$, we use a [math]-$1$ variable $y_{j,t}$ to indicate whether the $j^{th}$ entries of a solution form a tuple $t\in R_{j}$ or not. For any $i\in[k]$, $\mathbf{x}\in X_{i}$, $j\in[m]$ and $t\in R_{j}$, denote $\chi_{i}(\mathbf{x}[j],t)=0$ if $\mathbf{x}[j]=t[i]$ and $\chi_{i}(\mathbf{x}[j],t)=1$ if $\mathbf{x}[j]\neq t[i]$. Now Binary Constrained Partition Center*⋆* can be defined as the following [math]-$1$ optimization problem.

[TABLE]

Any solution $y_{j,t}$ ($j\in[m]$ and $t\in R_{j}$) to (6) corresponds to the solution $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ where for all $j\in[m]$ and $t\in R_{j}$ such that $y_{j,t}=1$, we have $(\mathbf{c}_{1}[j],\ldots,\mathbf{c}_{k}[j])=t$.

Now, we solve the above optimization problem using linear programming relaxation and obtain a fractional solution $y^{\star}_{j,t}$ ($j\in[m]$ and $t\in R_{j}$) with cost $d^{\prime}$. Clearly, $d^{\prime}\leq d_{opt}={\sf OPT}(J^{\prime})$. Now, for each $j\in[m]$, independently with probability $y^{\star}_{j,t}$, we set $y^{\prime}_{j,t}=1$ and $y^{\prime}_{j,t^{\prime}}=0$, for any $t^{\prime}\in R_{j}\setminus\{t\}$. Then $y^{\prime}_{j,t}$ ($j\in[m]$ and $t\in R_{j}$) form a solution to (6). Next we construct the solution $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ to Binary Constrained Partition Center*⋆*, corresponding to $y^{\prime}_{j,t}$ ($j\in[m]$ and $t\in R_{j}$). That is, for all $j\in[m]$ and $t\in R_{j}$ such that $y_{j,t}=1$, we have $(\mathbf{c}_{1}[j],\ldots,\mathbf{c}_{k}[j])=t$.

For the running time analysis, notice that solving the linear program and performing the random rounding takes polynomial time in the size of the problem (6). And the size of (6) is polynomial in the size of $J^{\prime}$, so the running time bound is satisfied. It remains to show that the constructed solution has cost at most $(1+\delta){\sf OPT}(J^{\prime})$ with probability at least $1-n^{-2}$.

For any $j\in[m]$, the above random rounding procedure ensures that there is exactly one tuple $t\in R_{j}$ such that $y^{\prime}_{j,t}=1$. This implies that for any $j\in[m]$, $i\in[k]$ and $\mathbf{x}\in X_{i}$, $\sum_{t\in R_{j}}\chi_{i}(\mathbf{x}[j],t)\cdot y^{\prime}_{j,t}$ is a [math]-$1$ random variable. Since for each $j\in[m]$ the rounding procedure is independent, we have that for any $i\in[k]$ and $\mathbf{x}\in X_{i}$ the random variables $(\sum_{t\in R_{1}}\chi_{i}(\mathbf{x}[1],t)\cdot y^{\prime}_{1,t}),\ldots,(\sum_{t\in R_{m}}\chi_{i}(\mathbf{x}[m],t)\cdot y^{\prime}_{j,t})$ are independent. Hence, for any $i\in[k]$ and $\mathbf{x}\in X_{i}$, the Hamming distance between $\mathbf{x}$ and $\mathbf{c}_{i}$, $d_{H}(\mathbf{x},\mathbf{c}_{i})=\sum_{j\in[m]}\sum_{t\in R_{j}}\chi_{i}(\mathbf{x}[j],t)\cdot y^{\prime}_{j,t}$, is the sum of $m$ independent [math]-$1$ random variables. For each $i\in[k]$ and $\mathbf{x}\in X_{i}$, we upper bound the expected value of $d_{H}(\mathbf{x},\mathbf{c}_{i})$ as follows.

[TABLE]

Fix $\epsilon=\frac{\delta}{c}$. Then, by Proposition 4, for all $i\in[k]$, and $\mathbf{x}\in X_{i}$,

[TABLE]

Therefore, by the union bound,

[TABLE]

We remind that $m\geq 9c^{2}\log n/\delta^{2}=9\log n/\epsilon^{2}$ and so $n\cdot e^{-\frac{1}{3}m\epsilon^{2}}\leq n^{-2}$. Thus, by (7),

[TABLE]

Since $d^{\prime}\leq{\sf OPT}(J^{\prime})$ and ${\sf OPT}(J^{\prime})\geq m/c$, $d^{\prime}+\epsilon m\leq(1+c\epsilon){\sf OPT}(J^{\prime})$. Then, the probability that there exist $i\in[k]$ and $\mathbf{x}\in X_{i}$ such that $d_{H}(\mathbf{x},\mathbf{c}_{i})>(1+c\epsilon){\sf OPT}(J^{\prime})-d_{\mathbf{x}}$ is at most $n^{-2}$ by (8). Since $c\epsilon=\delta$, the proof is complete. ∎

4 Applications

In this section we explain the impact of Theorem 2 about Binary Constrained $k$-Center to other problems around low-rank matrix approximation. We would like to mention that Binary Constrained $k$-Center is very similar to the Binary Constrained Clustering problem from [9]. In Binary Constrained $k$-Center we want to minimize the maximum distance of a vector from the input set of vectors to the closest center, whereas in Binary Constrained Clustering the sum of distances is minimized. While these problems are different, the reduction we explain here, except a few details, are identical to the ones described in [9]. For reader’s convenience, we give one reduction (Lemma 6) in full details and skip all other reductions, which are similar.

In the following lemma we show that $\ell_{1}$-Rank-$r$ Approximation over GF(2) is a special case of Binary Constrained $k$-Center.

Lemma 6**.**

*There is an algorithm that given an instance $(\mathbf{A},r)$ of $\ell_{1}$-Rank-$r$ Approximation over GF(2), where $\mathbf{A}$ is an $m\times n$-matrix and $r$ is an integer, runs in time ${\cal O}(m+n+2^{2r})$, and outputs an instance $J=(X,k=2^{r},\mathcal{R})$ of Binary Constrained $k$-Center  with the following property. Given any $\alpha$-approximate solution $C$ to $J$, an $\alpha$-approximate solution $\mathbf{B}$ to $(\mathbf{A},r)$ can be constructed in time ${\cal O}(rmn)$ and vice versa. *

Proof.

Notice that if ${\rm{GF}}(2)\text{{\rm-rank}}(\mathbf{B})\leq r$, then $\mathbf{B}$ has at most $2^{r}$ distinct columns, because each column is a linear combination of at most $r$ vectors of a basis of the column space of $\mathbf{B}$. Moreover, $\ell_{1}$-Rank-$r$ Approximation over GF(2) can also be stated as follows: find vectors $\mathbf{s}_{1},\ldots,\mathbf{s}_{r}\in\{0,1\}^{m}$ such that $\max_{i\in[n]}d_{H}(\mathbf{a}_{i},S)$ is minimum, where $\mathbf{a}_{1},\ldots,\mathbf{a}_{n}$ are the columns of $\mathbf{A}$ and $S=\{\mathbf{s}\in\{0,1\}^{m}\colon\mathbf{s}~{}\text{is a linear combination of}~{}\mathbf{s}_{1},\ldots,\mathbf{s}_{r}~{}\text{over}~{}{GF}(2){}\}$.

To encode an instance of $\ell_{1}$-Rank-$r$ Approximation over GF(2) as an instance of Binary Constrained $k$-Center, we construct the following relation $R$. Set $k=2^{r}$. Let $\Lambda=(\mathbf{\lambda}_{1},\ldots,\mathbf{\lambda}_{k})$ be the $k$-tuple composed of all distinct vectors in $\{0,1\}^{r}$. Thus, each element $\lambda_{i}\in\Lambda$ is a binary $r$-vector. We define $R=\{(x^{\intercal}\mathbf{\lambda}_{1},\ldots,x^{\intercal}\mathbf{\lambda}_{k})\mid x\in\{0,1\}^{r}\}.$ Thus, $R$ consists of $k=2^{r}$ $k$-tuples and every $k$-tuple in $R$ is a row of the matrix $\Lambda^{\intercal}\cdot\Lambda$. Now we define $X$ to be the set of columns of $\mathbf{A}$ and for each $i\in[m]$, $R_{i}=R$. Our algorithm outputs the instance $J=(X,k,{\cal R}=(R_{1},\ldots,R_{m}))$.

To show that the instance $(\mathbf{A},r)$ of $\ell_{1}$-Rank-$r$ Approximation over GF(2) is equivalent to the constructed instance $J$, assume first that the vectors $\mathbf{s}_{1},\ldots,\mathbf{s}_{r}\in\{0,1\}^{m}$ compose an (approximate) solution of $\ell_{1}$-Rank-$r$ Approximation over GF(2). For every $i\in[k]$ define the vector

[TABLE]

where $\mathbf{\lambda}_{i}^{\intercal}=(\lambda_{i}[1],\ldots,\lambda_{i}[r])$, $\oplus$ denotes the sum over GF(2), and define the tuple $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$. That is, $C$ contains all linear combinations of $\mathbf{s}_{1},\ldots,\mathbf{s}_{r}$. For every $i\in[k]$ and $j\in[m]$, we have that $\mathbf{c}_{i}[j]=(\mathbf{s}_{1}[j],\ldots,\mathbf{s}_{r}[j])\mathbf{\lambda}_{i}$. Therefore, $(\mathbf{c}_{1}[j],\ldots,\mathbf{c}_{k}[j])\in R$ for all $j\in[m]$. Thus, $C$ is a solution to $J$ of cost $\max_{\mathbf{x}\in X}d_{H}(\mathbf{x},C)$.

For the opposite direction, assume that $C=(\mathbf{c}_{1},\ldots,\mathbf{c}_{k})$ is an (approximate) solution to $J$. We construct the vectors $\mathbf{s}_{1},\ldots,\mathbf{s}_{r}$ as follows. Let $j\in[m]$. We have that $(\mathbf{c}_{1}[j],\ldots,\mathbf{c}_{k}[j])\in R$. Therefore, there is $\mathbf{x}\in\{0,1\}^{r}$ such that $(\mathbf{c}_{1}[j],\ldots,\mathbf{c}_{k}[j])=(\mathbf{x}^{\intercal}\mathbf{\lambda}_{1},\ldots,\mathbf{x}^{\intercal}\mathbf{\lambda}_{k})$. We set $\mathbf{s}_{i}[j]=\mathbf{x}[i]$ for $i\in[r]$. Observe that vectors in $C$ are linear combinations of the vectors $\mathbf{s}_{1},\ldots,\mathbf{s}_{r}$. This immediately implies that for any $\alpha$-approximate solution $C$ of $J$ an $\alpha$-approximate solution $\mathbf{B}$ of $(\mathbf{A},r)$ can be constructed in time ${\cal O}(rmn)$. ∎

Thus, Theorem 1 follows from Theorem 2 and Lemma 6.

Low Boolean-Rank Approximation.

Let $\mathbf{A}$ be a binary $m\times n$ matrix. Now we consider the elements of $\mathbf{A}$ to be Boolean variables. The Boolean rank of $\mathbf{A}$ is the minimum $r$ such that $\mathbf{A}=\mathbf{U}\wedge\mathbf{V}$ for a Boolean $m\times r$ matrix $\mathbf{U}$ and a Boolean $r\times n$ matrix $\mathbf{V}$, where the product is Boolean, that is, the logical $\wedge$ plays the role of multiplication and $\vee$ the role of sum. Here $0\wedge 0=0$, $0\wedge 1=0$, $1\wedge 1=1$ , $0\vee 0=0$, $0\vee 1=1$, and $1\vee 1=1$. Thus the matrix product is over the Boolean semi-ring $({0,1},\wedge,\vee)$. This can be equivalently expressed as the normal matrix product with addition defined as $1+1=1$. Binary matrices equipped with such algebra are called Boolean matrices.

In Boolean $\ell_{1}$-Rank-$r$ Approximation, we are given an $m\times n$ binary data matrix $\mathbf{A}$ and a positive integer $r$, and we seek a binary matrix $\mathbf{B}$ optimizing

[TABLE]

Here, by the rank of binary matrix $\mathbf{B}$ we mean its Boolean rank, and norm $\|\cdot\|_{1}$ is the column sum norm. Similar to Lemma 6, one can prove that Boolean $\ell_{1}$-Rank-$r$ Approximation is a special case of Binary Constrained $k$-Center, where $k=2^{r}$. Thus, we get the following corollary from Theorem 2.

Corollary 1**.**

There is an algorithm for Boolean $\ell_{1}$-Rank-$r$ Approximation that given an instance $I=(\mathbf{A},r)$ and $0<\varepsilon<1$, runs in time $m^{{\cal O}(1)}n^{{\cal O}(2^{4r}/\varepsilon^{4})}$, and outputs a $(1+\varepsilon)$-approximate solution with probability at least $1-2n^{-2}$.

Projective $k$-center.

The Binary Projective $k$-Center problem is a variation of the Binary $k$-Center problem, where the centers of clusters are linear subspaces of bounded dimension $r$. (For $r=1$ this is Binary $k$-Center and for $k=1$ this is $\ell_{1}$-Rank-$r$ Approximation over GF(2).) Formally, in Binary Projective $k$-Center we are given a set $X\subseteq\{0,1\}^{m}$ of $n$ vectors and positive integers $k$ and $r$. The objective is to find a family of $r$-dimensional linear subspaces $C=\{C_{1},\ldots,C_{k}\}$ over GF(2) minimizing $\max_{\mathbf{x}\in X}d_{H}(\mathbf{x},\bigcup_{i=1}^{k}C).$

To see that Binary Projective $k$-Center is a special case of Binary Constrained $k$-Center, we observe that the condition that $C_{i}$ is an $r$-dimensional subspace over GF(2) can be encoded (as in Lemma 6) by $2^{r}$ constraints. This observation leads to the following lemma.

Lemma 7**.**

There is an algorithm that given an instance $(X,r,k)$ of Binary Projective $k$-Center, runs in time ${\cal O}(m+n+2^{{\cal O}(rk)})$, and outputs an instance $J=(X,k^{\prime}=2^{kr},\mathcal{R})$ of Binary Constrained $k$-Center  with the following property. Given any $\alpha$-approximate solution $C$ to $J$, an $\alpha$-approximate solution $C^{\prime}$ to $(X,r,k)$ can be constructed in time ${\cal O}(rkmn)$ and vice versa.

Combining Theorem 2 and Lemma 7 together, we get the following corollary.

Corollary 2**.**

There is an algorithm for Binary Projective $k$-Center that given an instance $I=(X,r,k)$ and $0<\varepsilon<1$, where $X\subseteq\{0,1\}^{m}$ is a set of $n$ vectors and $r,k\in{\mathbb{N}}$, runs in time $m^{{\cal O}(1)}n^{{\cal O}(2^{4kr}/\varepsilon^{4})}$, and outputs a $(1+\varepsilon)$-approximate solution with probability at least $1-2n^{-2}$.