$\ell_1$-sparsity Approximation Bounds for Packing Integer Programs

TL;DR

This paper develops approximation algorithms for packing integer programs that leverage the -column sparsity of the constraint matrix, providing improved bounds based on width and sparsity, especially for larger widths.

Contribution

It introduces randomized rounding algorithms with alteration for PIPs, achieving approximation ratios based on -sparsity and width, extending prior work focused on -sparsity.

Findings

For width W=1+psilon, approximation ratio is psilon^2/elta_1.

For width W , approximation ratio is (rac{1}{1+elta_1/W})^{1/(W-1)}.

A near-optimal (-psilon) approximation is possible when width is sufficiently large.

Abstract

We consider approximation algorithms for packing integer programs (PIPs) of the form $\max\{\langle c, x\rangle : Ax \le b, x \in \{0,1\}^n\}$ where $c$, $A$, and $b$ are nonnegative. We let $W = \min_{i,j} b_i / A_{i,j}$ denote the width of $A$ which is at least $1$. Previous work by Bansal et al. \cite{bansal-sparse} obtained an $\Omega(\frac{1}{\Delta_0^{1/\lfloor W \rfloor}})$-approximation ratio where $\Delta_0$ is the maximum number of nonzeroes in any column of $A$ (in other words the $\ell_0$-column sparsity of $A$). They raised the question of obtaining approximation ratios based on the $\ell_1$-column sparsity of $A$ (denoted by $\Delta_1$) which can be much smaller than $\Delta_0$. Motivated by recent work on covering integer programs (CIPs) \cite{cq,chs-16} we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al. \cite{bansal-sparse} (but with a twist), yield approximation ratios for PIPs based on $\Delta_1$. First, following an integrality gap example from \cite{bansal-sparse}, we observe that the case of $W=1$ is as hard as maximum independent set even when $\Delta_1 \le 2$. In sharp contrast to this negative result, as soon as width is strictly larger than one, we obtain positive results via the natural LP relaxation. For PIPs with width $W = 1 + \epsilon$ where $\epsilon \in (0,1]$, we obtain an $\Omega(\epsilon^2/\Delta_1)$-approximation. In the large width regime, when $W \ge 2$, we obtain an $\Omega((\frac{1}{1 + \Delta_1/W})^{1/(W-1)})$-approximation. We also obtain a $(1-\epsilon)$-approximation when $W = \Omega(\frac{\log (\Delta_1/\epsilon)}{\epsilon^2})$.

Equations

$$\text{maximize}⟨c,x⟩\text{over}x\in \{0,1{\}}^n\text{s.t.}Ax\le b,$$

$$\text{maximize}⟨x,1⟩\text{over}x\in \{0,1{\}}^n\text{s.t.}Ax\le 1.$$

$$\mathbb{E}[⟨c,x^{\prime \prime }⟩]=\alpha \sum _{j=1}^n{c}_j{x}_j\cdot \mathrm{Pr}[x_j^{\prime \prime }=1|x_j^{\prime }=1].$$

$$\mathrm{Pr}[x_j^{\prime \prime }=0|x_j^{\prime }=1]=\mathrm{Pr}\left[\bigcup _{i\in [m]}{E}_{ij}|x_j^{\prime }=1\right]\le \sum _{i=1}^m\mathrm{Pr}[E_{ij}|x_j^{\prime }=1].$$

$$\mathrm{Pr}[E_{ij}|X_j=1]\le \mathrm{Pr}[Y_{ij}>W-A_{i,j}|X_j=1].$$

$$\mathrm{Pr}[Y_{ij}>W-A_{i,j}|X_j=1]\le \mathrm{Pr}[Y_{ij}\ge W-A_{i,j}].$$

$$\mathbb{E}[Y_{ij}]=\sum _{\ell =1}^{j-1}{A}_{i,\ell }\cdot \mathrm{Pr}[X_{\ell }=1]\le {\alpha }_1\sum _{\ell =1}^n{A}_{i,\ell }{x}_{\ell }\le {\alpha }_{1}W.$$

$$\mathrm{Pr}[Y_{ij}>W-A_{i,j}]\le {\left(\frac{{\alpha }_1{e}^{1-{\alpha }_1}W}{W-A_{i,j}}\right)}^{(W-A_{i,j})/A_{i,j}}.$$

$${\left(\frac{{\alpha }_1{e}^{1-{\alpha }_1}W}{W-A_{i,j}}\right)}^{(W-A_{i,j})/A_{i,j}}\le {\left(2e{\alpha }_1\right)}^{(W-A_{i,j})/A_{i,j}}={\left(\frac{1}{2e^{1/e}{\Delta }_1}\right)}^{(W-A_{i,j})/A_{i,j}}.$$

$${\left(\frac{1}{2{\Delta }_1}\right)}^{(W-1)/A_{i,j}}\le \frac{1}{2{\Delta }_1}.$$

$$(1/e^{1/e}{)}^{(W-A_{i,j})/A_{i,j}}\le (1/e^{1/e}{)}^{1/A_{i,j}}\le A_{i,j}$$

$$\sum _{i=1}^m\mathrm{Pr}[E_{ij}|X_j=1]\le \sum _{i=1}^m\frac{A_{i,j}}{2{\Delta }_1}\le \frac{1}{2}.$$

$$\sum _{i=1}^m\mathrm{Pr}[E_{ij}|X_j=1]\le \sum _{i=1}^m\frac{eϵA_{i,j}}{{\Delta }_1}\le eϵ.$$

$$\mathrm{Pr}[E_{ij}|X_j=1]\le \mathrm{Pr}[\mathcal{E}]\le \sum _{\begin{array}{c}\ell \in B_i\\ \ell \ne j\end{array}}\mathrm{Pr}[X_{\ell }=1]\le {\alpha }_2\sum _{\ell \in B_i}{x}_{\ell },$$

$${\alpha }_2\sum _{\ell \in B_i}{x}_{\ell }\le \frac{{ϵ}^2}{c_3{\Delta }_1}\left(\frac{2}{ϵ}+2\right)\le \frac{4ϵ}{c_3{\Delta }_1}\le \frac{A_{i,j}}{2{\Delta }_1},$$

$$\sum _{i=1}^m\mathrm{Pr}[E_{ij}|X_j=1]\le \sum _{i=1}^m\frac{A_{i,j}}{2{\Delta }_1}\le \frac{1}{2}.$$

$$\mathrm{Pr}[X\ge (1+\delta )\mu ]\le {\left(\frac{e^{\delta }}{(1+\delta {)}^{1+\delta }}\right)}^{\mu /\beta }$$

$$\mathrm{Pr}\left[\sum _i{X}_i>W-\beta \right]\le {\left(\frac{\alpha e^{1-\alpha }W}{W-\beta }\right)}^{(W-\beta )/\beta }.$$

$$\mathrm{Pr}\left[\sum _i{X}_i>W-\beta \right]=\mathrm{Pr}\left[\sum _i{X}_i>(1+\delta )\mu \right]\le {\left(\frac{e^{\delta }}{(1+\delta {)}^{1+\delta }}\right)}^{\mu /\beta }.$$

$${\left(\frac{e^{\delta }}{(1+\delta {)}^{1+\delta }}\right)}^{\mu /\beta }={\left(\frac{e^{W-\beta -\mu }}{((W-\beta )/\mu {)}^{W-\beta }}\right)}^{1/\beta }.$$

$${\left(\frac{e^{W-\beta -\mu }}{((W-\beta )/\mu {)}^{W-\beta }}\right)}^{1/\beta }=\exp \left(\frac{1}{\beta }\left(W-\beta -\mu +(W-\beta )\ln \left(\frac{\mu }{W-\beta }\right)\right)\right)$$

$$\exp \left(\frac{1}{\beta }\left(W-\beta -\mu +(W-\beta )\ln \left(\frac{\mu }{W-\beta }\right)\right)\right)=\exp \left(\frac{1}{\beta }\left((1-\alpha )W-\beta +(W-\beta )\ln \left(\frac{\alpha W}{W-\beta }\right)\right)\right)$$

$$\exp \left(\frac{1}{\beta }\left((1-\alpha )W-\beta -(W-\beta )\ln \left(\frac{W-\beta }{\alpha W}\right)\right)\right)\le {\left(\frac{\alpha e^{1-\alpha }W}{W-\beta }\right)}^{(W-\beta )/\beta }.$$

$$(x/y)\ln (x/y)\ge (1/2)\ln (1/e^{2/e})=-1/e.$$

$$x\ln x\ge ϵ\ln (ϵ/d)$$

$$\mathrm{Pr}[E_{ij}|X_j=1]\le {\left(2e{\alpha }_1\right)}^{(W-A_{i,j})/A_{i,j}}.$$

$${\left(2e{\alpha }_1\right)}^{(W-A_{i,j})/A_{i,j}}={\left(\frac{1}{2e^{2/e}(1+{\Delta }_1/W{)}^{1/(W-1)}}\right)}^{(W-A_{i,j})/A_{i,j}}$$

$${\left(\frac{1}{2(1+{\Delta }_1/W{)}^{1/(W-1)}}\right)}^{(W-A_{i,j})/A_{i,j}}\le \frac{1}{2^{W-1}(1+{\Delta }_1/W)}\le \frac{W}{2{\Delta }_1}.$$

$${\left(\frac{1}{e^{2/e}}\right)}^{(W-A_{i,j})/A_{i,j}}\le {\left(\frac{1}{e^{2/e}}\right)}^{W/2A_{i,j}}\le \frac{A_{i,j}}{W}$$

$$\mathrm{Pr}[E_{ij}|X_j=1]\le \mathrm{Pr}[Y_{ij}>W-A_{i,j}].$$

Full text

1 Introduction

Packing integer programs (abbr. PIPs) are an expressive class of integer programs of the form:

[TABLE]

where $A\in\mathbb{R}_{\geq 0}^{m\times n}$, $b\in\mathbb{R}_{\geq 0}^{m}$ and $c\in\mathbb{R}_{\geq 0}^{n}$ all have nonnegative entries111We can allow the variables to have general integer upper bounds instead of restricting them to be boolean. As observed in [1], one can reduce this more general case to the $\{0,1\}$ case without too much loss in the approximation.. Many important problems in discrete and combinatorial optimization can be cast as special cases of PIPs. These include the maximum independent set in graphs and hypergraphs, set packing, matchings and $b$-matchings, knapsack (when $m=1$), and the multi-dimensional knapsack. The maximum independent set problem (MIS), a special case of PIPs, is NP-hard and unless $P=NP$ there is no $n^{1-\epsilon}$-approximation where $n$ is the number of nodes in the graph [9, 17]. For this reason it is meaningful to consider special cases and other parameters that control the difficulty of PIPs. Motivated by the fact that MIS admits a simple $\frac{1}{\Delta(G)}$-approximation where $\Delta(G)$ is the maximum degree of $G$, previous work considered approximating PIPs based on the maximum number of nonzeroes in any column of $A$ (denoted by $\Delta_{0}$); note that when MIS is written as a PIP, $\Delta_{0}$ coincides with $\Delta(G)$. As another example, when maximum weight matching is written as a PIP, $\Delta_{0}=2$. Bansal et al. [1] obtained a simple and clever algorithm that achieved an $\Omega(1/\Delta_{0})$-approximation for PIPs via the natural LP relaxation; this improved previous work of Pritchard [12, 13] who was the first to obtain an approximation for PIPs only as a function of $\Delta_{0}$. Moreover, the rounding algorithm in [1] can be viewed as a contention resolution scheme which allows one to get similar approximation ratios even when the objective is submodular [1, 5]. It is well-understood that PIPs become easier when the entries in $A$ are small compared to the packing constraints $b$. To make this quantitative we consider the well-studied notion called the width defined as $W:=\min_{i,j:A_{i,j}>0}b_{i}/A_{i,j}$. Bansal et al. obtain an $\Omega((\frac{1}{\Delta_{0}})^{1/\lfloor W\rfloor})$-approximation which improves as $W$ becomes larger. Although they do not state it explicitly, their approach also yields a $(1-\epsilon)$-approximation when $W=\Omega(\frac{1}{\epsilon^{2}}\log(\Delta_{0}/\epsilon))$.

$\Delta_{0}$ is a natural measure for combinatorial applications such as MIS and matchings where the underlying matrix $A$ has entries from $\{0,1\}$. However, in some applications of PIPs such as knapsack and its multi-dimensional generalization which are more common in resource-allocation problems, the entries of $A$ are arbitrary rational numbers (which can be assumed to be from the interval $[0,1]$ after scaling). In such applications it is natural to consider another measure of column-sparsity which is based on the $\ell_{1}$ norm. Specifically we consider $\Delta_{1}$, the maximum column sum of $A$. Unlike $\Delta_{0}$, $\Delta_{1}$ is not scale invariant so one needs to be careful in understanding the parameter and its relationship to the width $W$. For this purpose we normalize the constraints $Ax\leq b$ as follows. Let $W=\min_{i,j:A_{i,j}>0}b_{i}/A_{i,j}$ denote the width as before (we can assume without loss of generality that $W\geq 1$ since we are interested in integer solutions). We can then scale each row $A_{i}$ of $A$ separately such that, after scaling, the $i$’th constraint reads as $A_{i}x\leq W$. After scaling all rows in this fashion, entries of $A$ are in the interval $[0,1]$, and the maximum entry of $A$ is equal to $1$. Note that this scaling process does not alter the original width. We let $\Delta_{1}$ denote the maximum column sum of $A$ after this normalization and observe that $1\leq\Delta_{1}\leq\Delta_{0}$. In many settings of interest $\Delta_{1}\ll\Delta_{0}$. We also observe that $\Delta_{1}$ is a more robust measure than $\Delta_{0}$; small perturbations of the entries of $A$ can dramatically change $\Delta_{0}$ while $\Delta_{1}$ changes minimally.

Bansal et al. raised the question of obtaining an approximation ratio for PIPs as a function of only $\Delta_{1}$. They observed that this is not feasible via the natural LP relaxation by describing a simple example where the integrality gap of the LP is $\Omega(n)$ while $\Delta_{1}$ is a constant. In fact their example essentially shows the existence of a simple approximation preserving reduction from MIS to PIPs such that the resulting instances have $\Delta_{1}\leq 2$; thus no approximation ratio that depends only on $\Delta_{1}$ is feasible for PIPs unless $P=NP$. These negative results seem to suggest that pursuing bounds based on $\Delta_{1}$ is futile, at least in the worst case. However, the starting point of this paper is the observation that both the integrality gap example and the hardness result are based on instances where the width $W$ of the instance is arbitrarily close to $1$. We demonstrate that these examples are rather brittle and obtain several positive results when we consider $W\geq(1+\epsilon)$ for any fixed $\epsilon>0$.

1.1 Our results

Our first result is on the hardness of approximation for PIPs that we already referred to. The hardness result suggests that one should consider instances with $W>1$. Recall that after normalization we have $\Delta_{1}\geq 1$ and $W\geq 1$ and the maximum entry of $A$ is $1$. We consider three regimes of $W$ and obtain the following results, all via the natural LP relaxation, which also establish corresponding upper bounds on the integrality gap.

• (i)

$1<W\leq 2$. For $W=1+\epsilon$ where $\epsilon\in(0,1]$ we obtain an $\Omega(\frac{\epsilon^{2}}{\Delta_{1}})$-approximation.

• (ii)

$W\geq 2$. We obtain an $\Omega((\frac{1}{1+\frac{\Delta_{1}}{W}})^{1/(W-1)})$-approximation which can be simplified to $\Omega((\frac{1}{1+\Delta_{1}})^{1/(W-1)})$ since $W\geq 1$.

• (iii)

A $(1-\epsilon)$-approximation when $W=\Omega(\frac{1}{\epsilon^{2}}\log(\Delta_{1}/\epsilon))$.

Our results establish approximation bounds based on $\Delta_{1}$ that are essentially the same as those based on $\Delta_{0}$ as long as the width is not too close to $1$. We describe randomized algorithms which can be derandomized via standard techniques. The algorithms can be viewed as contention resolution schemes, and via known techniques [1, 5], the results yield corresponding approximations for submodular objectives; we omit these extensions in this version.

All our algorithms are based on a simple randomized rounding plus alteration framework that has been successful for both packing and covering problems. Our scheme is similar to that of Bansal et al. at a high level but we make a simple but important change in the algorithm and its analysis. This is inspired by recent work on covering integer programs [4] where $\ell_{1}$-sparsity based approximation bounds from [6] were simplified.

1.2 Other related work

We note that PIPs are equivalent to the multi-dmensional knapsack problem. When $m=1$ we have the classical knapsack problem which admits a very efficient FPTAS (see [2]). There is a PTAS for any fixed $m$ [7] but unless $P=NP$ an FPTAS does not exist for $m=2$.

Approximation algorithms for PIPs in their general form were considered initially by Raghavan and Thompson [14] and refined substantially by Srinivasan [15]. Srinivasan obtained approximation ratios of the form $\Omega(1/n^{W})$ when $A$ had entries from $\{0,1\}$, and a ratio of the form $\Omega(1/n^{1/\lfloor W\rfloor})$ when $A$ had entries from $[0,1]$. Pritchard [12] was the first to obtain a bound for PIPs based solely on the column sparsity parameter $\Delta_{0}$. He used iterated rounding and his initial bound was improved in [13] to $\Omega(1/\Delta_{0}^{2})$. The current state of the art is due to Bansal et al. [1]. Previously we ignored constant factors when describing the ratio. In fact [1] obtains a ratio of $(1-o(1)\frac{e-1}{e^{2}\Delta_{0}})$ by strengthening the basic LP relaxation.

In terms of hardness of approximation, PIPs generalize MIS and hence one cannot obtain a ratio better than $n^{1-\epsilon}$ unless $P=NP$ [9, 17]. Building on MIS, [3] shows that PIPs are hard to approximate within a $n^{\Omega(1/W)}$ factor for any constant width $W$. Hardness of MIS in bounded degree graphs [16] and hardness for $k$-set-packing [10] imply that PIPs are hard to approximate to within $\Omega(1/\Delta_{0}^{1-\epsilon})$ and to within $\Omega((\log\Delta_{0})/\Delta_{0})$ when $\Delta_{0}$ is a sufficiently large constant. These hardness results are based on $\{0,1\}$ matrices for which $\Delta_{0}$ and $\Delta_{1}$ coincide.

There is a large literature on deterministic and randomized rounding algorithms for packing and covering integer programs and connections to several topics and applications including discrepancy theory. $\ell_{1}$-sparsity guarantees for covering integer programs were first obtained by Chen, Harris and Srinivasan [6] partly inspired by [8].

2 Hardness of approximating PIPs as a function of $\Delta_{1}$

Bansal et al. [1] showed that the integrality gap of the natural LP relaxation for PIPs is $\Omega(n)$ even when $\Delta_{1}$ is a constant. One can use essentially the same construction to show the following theorem.

Theorem 1**.**

There is an approximation preserving reduction from MIS to instances of PIPs with $\Delta_{1}\leq 2$.

Proof.

Let $G=(V,E)$ be an undirected graph without self-loops and let $n=\left|V\right|$. Let $A\in[0,1]^{n\times n}$ be indexed by $V$. For all $v\in V$, let $A_{v,v}=1$. For all $uv\in E$, let $A_{u,v}=A_{v,u}=1/n$. For all the remaining entries in $A$ that have not yet been defined, set these entries to [math]. Consider the following PIP:

[TABLE]

Let $S$ be the set of all feasible integral solutions of (1) and $\mathcal{I}$ be the set of independent sets of $G$. Define $g:S\to\mathcal{I}$ where $g(x)=\{v:x_{v}=1\}$. To show $g$ is surjective, consider a set $I\in\mathcal{I}$. Let $y$ be the characteristic vector of $I$. That is, $y_{v}$ is $1$ if $v\in I$ and [math] otherwise. Consider the row in $A$ corresponding to an arbitrary vertex $u$ where $y_{u}=1$. For all $v\in V$ such that $v$ is a neighbor to $u$, $y_{v}=0$ as $I$ is an independent set. Thus, as the nonzero entries in $A$ of the row corresponding to $u$ are, by construction, the neighbors of $u$, it follows that the constraint corresponding to $u$ is satisfied in (1). As $u$ is an arbitrary vertex, it follows that $y$ is a feasible integral solution to (1) and as $I=\{v:y_{v}=1\}$, $g(y)=I$.

Define $h:S\to\mathbb{N}_{0}$ such that $h(x)=\left|g(x)\right|$. It is clear that $\max_{x\in S}h(x)$ is equal to the optimal value of (1). Let $I_{max}$ be a maximum independent set of $G$. As $g$ is surjective, there exists $z\in S$ such that $g(z)=I_{max}$. Thus, $\max_{x\in S}h(x)\geq\left|I_{max}\right|$. As $\max_{x\in S}h(x)$ is equal to the optimum value of (1), it follows that a $\beta$-approximation for PIPs implies a $\beta$-approximation for maximum independent set.

Furthermore, we note that for this PIP, $\Delta_{1}\leq 2$, thus concluding the proof. ∎

Unless $P=NP$, MIS does not admit a $n^{1-\epsilon}$-approximation for any fixed $\epsilon>0$ [9, 17]. Hence the preceding theorem implies that unless $P=NP$ one cannot obtain an approximation ratio for PIPs solely as a function of $\Delta_{1}$.

3 Round and alter framework

The algorithms in this paper have the same high-level structure. The algorithms first scale down the fractional solution $x$ by some factor $\alpha$, and then randomly round each coordinate independently. The rounded solution $x^{\prime}$ may not be feasible for the constraints. The algorithm alters $x^{\prime}$ to a feasible $x^{\prime\prime}$ by considering each constraint separately in an arbitrary order; if $x^{\prime}$ is not feasible for constraint $i$ some subset $S$ of variables are chosen to be set to [math]. Each constraint corresponds to a knapsack problem and the framework (which is adapted from [1]) views the problem as the intersection of several knapsack constraints. A formal template is given in Figure 1. To make the framework into a formal algorithm, one must define $\alpha$ and how to choose $S$ in the for loop. These parts will depend on the regime of interest.

For an algorithm that follows the round-and-alter framework, the expected output of the algorithm is $\mathbb{E}\left[\langle c,x^{\prime\prime}\rangle\right]=\sum_{j=1}^{n}c_{j}\cdot\Pr[x_{j}^{\prime\prime}=1]$. Independent of how $\alpha$ is defined or how $S$ is chosen, $\Pr[x_{j}^{\prime\prime}=1]=\Pr[x_{j}^{\prime\prime}=1|x_{j}^{\prime}=1]\cdot\Pr[x_{j}^{\prime}=1]$ since $x_{j}^{\prime\prime}\leq x_{j}^{\prime}$. Then we have

[TABLE]

Let $E_{ij}$ be the event that $x_{j}^{\prime\prime}$ is set to [math] when ensuring constraint $i$ is satisfied in the for loop. As $x_{j}^{\prime\prime}$ is only set to [math] if at least one constraint sets $x_{j}^{\prime\prime}$ to [math], we have

[TABLE]

Combining these two observations, we have the following lemma, which applies to all of our subsequent algorithms.

Lemma 2**.**

Let $\mathcal{A}$ be a randomized rounding algorithm that follows the round-and-alter framework given in Figure 1. Let $x^{\prime}$ be the rounded solution obtained with scaling factor $\alpha$. Let $E_{ij}$ be the event that $x_{j}^{\prime\prime}$ is set to [math] by constraint $i$. If for all $j\in[n]$ we have $\sum_{i=1}^{m}\Pr[E_{ij}|x_{j}^{\prime}=1]\leq\gamma,$ then $\mathcal{A}$ is an $\alpha(1-\gamma)$-approximation for PIPs.

We will refer to the quantity $\Pr[E_{ij}|x_{j}^{\prime}=1]$ as the rejection probability of item $j$ in constraint $i$. We will also say that constraint $i$ rejects item $j$ if $x_{j}^{\prime\prime}$ is set to [math] in constraint $i$.

4 The large width regime: $W\geq 2$

In this section, we consider PIPs with width $W\geq 2$. Recall that we assume $A\in[0,1]^{m\times n}$ and $b_{i}=W$ for all $i\in[m]$. Therefore we have $A_{i,j}\leq W/2$ for all $i,j$ and from a knapsack point of view all items are “small”. We apply the round-and-alter framework in a simple fashion where in each constraint $i$ the coordinates are sorted by the coefficents in that row and the algorithm chooses the largest prefix of coordinates that fit in the capacity $W$ and the rest are discarded. We emphasize that this sorting step is crucial for the analysis and differs from the scheme in [1]. Figure 2 describes the formal algorithm.

The key property for the analysis:

The analysis relies on obtaining a bound on the rejection probability of coordinate $j$ by constraint $i$. Let $X_{j}$ be the indicator variable for $j$ being chosen in the first step. We show that $\Pr[E_{ij}\mid X_{j}=1]\leq cA_{ij}$ for some $c$ that depends on the scaling factor $\alpha$. Thus coordinates with smaller coefficients are less likely to be rejected. The total rejection probability of $j$, $\sum_{i=1}^{m}\Pr[E_{ij}\mid X_{j}=1]$, is proportional to the column sum of coordinate $j$ which is at most $\Delta_{1}$.

The analysis relies on the Chernoff bound, and depending on the parameters, one needs to adjust the analysis. In order to highlight the main ideas we provide a detailed proof for the simplest case and include the proofs of the other cases in the appendix.

4.1 An $\Omega(1/\Delta_{1})$-approximation algorithm

We show that $\mathsf{round}\mathchar 45\relax\mathsf{and}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{by}\mathchar 45\relax\mathsf{sorting}$ yields an $\Omega(1/\Delta_{1})$-approximation if we set the scaling factor $\alpha_{1}=\frac{1}{c_{1}\Delta_{1}}$ where $c_{1}=4e^{1+1/e}$.

The rejection probability is captured by the following main lemma.

Lemma 3**.**

Let $\alpha_{1}=\frac{1}{c_{1}\Delta_{1}}$ for $c_{1}=4e^{1+1/e}$. Let $i\in[m]$ and $j\in[n]$. Then we have $\Pr[E_{ij}|X_{j}=1]\leq\frac{A_{i,j}}{2\Delta_{1}}$ in the algorithm $\mathsf{round}\mathchar 45\relax\mathsf{and}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{by}\mathchar 45\relax\mathsf{sorting}(A,b,\alpha_{1})$.

Proof.

At iteration $i$ of $\mathsf{round}\mathchar 45\relax\mathsf{and}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{by}\mathchar 45\relax\mathsf{sorting}$, after the set $\{A_{i,1},\ldots,A_{i,n}\}$ is sorted, the indices are renumbered so that $A_{i,1}\leq\cdots\leq A_{i,n}$. Note that $j$ may now be a different index $j^{\prime}$, but for simplicity of notation we will refer to $j^{\prime}$ as $j$. Let $\xi_{\ell}=1$ if $x_{\ell}^{\prime}=1$ and [math] otherwise. Let $Y_{ij}=\sum_{\ell=1}^{j-1}A_{i,\ell}\xi_{\ell}$.

If $E_{ij}$ occurs, then $Y_{ij}>W-A_{i,j}$, since $x_{j}^{\prime\prime}$ would not have been set to zero by constraint $i$ otherwise. That is,

[TABLE]

The event $Y_{ij}>W-A_{i,j}$ does not depend on $x_{j}^{\prime}$. Therefore,

[TABLE]

To upper bound $\mathbb{E}[Y_{ij}]$, we have

[TABLE]

As $A_{i,j}\leq 1$, $W\geq 2$, and $\alpha_{1}<1/2$, we have $\frac{(1-\alpha_{1})W}{A_{i,j}}>1$. Using the fact that $A_{i,j}$ is at least as large as all entries $A_{i,j^{\prime}}$ for $j^{\prime}<j$, we satisfy the conditions to apply the Chernoff bound in Theorem 13. This implies

[TABLE]

Note that $\frac{W}{W-A_{i,j}}\leq 2$ as $W\geq 2$. Because $e^{1-\alpha_{1}}\leq e$ and by the choice of $\alpha_{1}$, we have

[TABLE]

Then we prove the final inequality in two parts. First, we see that $W\geq 2$ and $A_{i,j}\leq 1$ imply that $\frac{W-A_{i,j}}{A_{i,j}}\geq 1$. This implies

[TABLE]

Second, we see that

[TABLE]

for $A_{i,j}\leq 1$, where the first inequality holds because $W-A_{i,j}\geq 1$ and the second inequality holds by Lemma 14. This concludes the proof. ∎

Theorem 4**.**

When setting $\alpha_{1}=\frac{1}{c_{1}\Delta_{1}}$ where $c_{1}=4e^{1+1/e}$, $\mathsf{round}\mathchar 45\relax\mathsf{and}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{by}\mathchar 45\relax\mathsf{sorting}(A,b,\alpha_{1})$ is a randomized $(\alpha_{1}/2)$-approximation algorithm for PIPs with width $W\geq 2$.

Proof.

Fix $j\in[n]$. By Lemma 3 and the definition of $\Delta_{1}$, we have

[TABLE]

By Lemma 2, which shows that upper bounding the sum of the rejection probabilities by $\gamma$ for every item leads to an $\alpha_{1}(1-\gamma)$-approximation, we get the desired result. ∎

4.2 An $\Omega(\frac{1}{(1+\Delta_{1}/W)^{1/(W-1)}})$-approximation

We improve the bound from the previous section by setting $\alpha_{1}=\frac{1}{c_{2}(1+\Delta_{1}/W)^{1/(W-1)}}$ where $c_{2}=4e^{1+2/e}$. Note that the scaling factor becomes larger as $W$ increases. The analysis of the following lemma is similar to that of Lemma 3 and is therefore left for the appendix.

Lemma 5**.**

Let $\alpha_{1}=\frac{1}{c_{2}(1+\Delta_{1}/W)^{1/(W-1)}}$ for $c_{2}=4e^{1+2/e}$. Let $i\in[m]$ and $j\in[n]$. Then in the algorithm $\mathsf{round}\mathchar 45\relax\mathsf{and}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{by}\mathchar 45\relax\mathsf{sorting}(A,b,\alpha_{1})$, we have $\Pr[E_{ij}|X_{j}=1]\leq\frac{A_{i,j}}{2\Delta_{1}}$.

If we replace Lemma 3 with Lemma 5 in the proof of Theorem 4, we obtain the following stronger guarantee.

Theorem 6**.**

When setting $\alpha_{1}=\frac{1}{c_{2}(1+\Delta_{1}/W)^{1/(W-1)}}$ where $c_{2}=4e^{1+2/e}$, for PIPs with width $W\geq 2$, $\mathsf{round}\mathchar 45\relax\mathsf{and}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{by}\mathchar 45\relax\mathsf{sorting}(A,b,\alpha_{1})$ is a randomized $(\alpha_{1}/2)$-approximation.

4.3 A $(1-O(\epsilon))$-approximation when $W\geq\Omega(\frac{1}{\epsilon^{2}}\ln(\frac{\Delta_{1}}{\epsilon}))$

In this section, we give a randomized $(1-O(\epsilon))$-approximation for the case when $W\geq\Omega(\frac{1}{\epsilon^{2}}\ln(\frac{\Delta_{1}}{\epsilon}))$. We use the algorithm $\mathsf{round}\mathchar 45\relax\mathsf{and}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{by}\mathchar 45\relax\mathsf{sorting}$ in Figure 2 with the scaling factor $\alpha_{1}=1-\epsilon$. The analysis follows the same structure as the analyses for the lemmas bounding the rejection probabilities from the previous sections. The proof can be found in the appendix.

Lemma 7**.**

Let $0<\epsilon<\frac{1}{e}$, $\alpha_{1}=1-\epsilon$, and $W=\frac{2}{\epsilon^{2}}\ln(\frac{\Delta_{1}}{\epsilon})+1$. Let $i\in[m]$ and $j\in[n]$. Then in $\mathsf{round}\mathchar 45\relax\mathsf{and}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{by}\mathchar 45\relax\mathsf{sorting}(A,b,\alpha_{1})$, we have $\Pr[E_{ij}|X_{j}=1]\leq e\cdot\frac{\epsilon A_{i,j}}{\Delta_{1}}$.

Lemma 7 implies that we can upper bound the sum of the rejection probabilities for any item $j$ by $e\epsilon$, leading to the following theorem.

Theorem 8**.**

Let $0<\epsilon<\frac{1}{e}$ and $W=\frac{2}{\epsilon^{2}}\ln(\frac{\Delta_{1}}{\epsilon})+1$. When setting $\alpha_{1}=1-\epsilon$ and $c=e+1$, $\mathsf{round}\mathchar 45\relax\mathsf{and}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{by}\mathchar 45\relax\mathsf{sorting}(A,b,\alpha_{1})$ is a randomized $(1-c\epsilon)$-approximation algorithm.

Proof.

Fix $j\in[n]$. By Lemma 7 and the definition of $\Delta_{1}$,

[TABLE]

By Lemma 2, which shows that an upper bound on the rejection probabilities of $\gamma$ leads to an $\alpha_{1}(1-\gamma)$-approximation, we have an $\alpha_{1}(1-e\epsilon)$-approximation. Then note that $\alpha_{1}(1-e\epsilon)=(1-\epsilon)(1-e\epsilon)\geq 1-(e+1)\epsilon$. This concludes the proof. ∎

5 The small width regime: $W=(1+\epsilon)$

We now consider the regime when the width is small. Let $W=1+\epsilon$ for some $\epsilon\in(0,1]$. We cannot apply the simple sorting based scheme that we used for the large width regime. We borrow the idea from [1] in splitting the coordinates into big and small in each constraint; now the definition is more refined and depends on $\epsilon$. Moreover, the small coordinates and the big coordinates have their own reserved capacity in the constraint. This is crucial for the analysis. We provide more formal details below.

We set $\alpha_{2}$ to be $\frac{\epsilon^{2}}{c_{3}\Delta_{1}}$ where $c_{3}=8e^{1+2/e}$. The alteration step differentiates between “small” and “big” coordinates as follows. For each $i\in[m]$, let $S_{i}=\{j:A_{i,j}\leq\epsilon/2\}$ and $B_{i}=\{j:A_{i,j}>\epsilon/2\}$. We say that an index $j$ is small for constraint $i$ if $j\in S_{i}$. Otherwise we say it is big for constraint $i$ when $j\in B_{i}$. For each constraint, the algorithm is allowed to pack a total of $1+\epsilon$ into that constraint. The algorithm separately packs small indices and big indices. In an $\epsilon$ amount of space, small indices that were chosen in the rounding step are sorted in increasing order of size and greedily packed until the constraint is no longer satisfied. The big indices are packed by arbitrarily choosing one and packing it into the remaining space of $1$. The rest of the indices are removed to ensure feasibility. Figure 3 gives pseudocode for the randomized algorithm $\mathsf{round}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{small}\mathchar 45\relax\mathsf{width}$ which yields an $\Omega(\epsilon^{2}/\Delta_{1})$-approximation.

It remains to bound the rejection probabilities. Recall that for $j\in[n]$, we define $X_{j}$ to be the indicator random variable $\mathds{1}(x_{j}^{\prime}=1)$ and $E_{ij}$ is the event that $j$ was rejected by constraint $i$.

We first consider the case when index $j$ is big for constraint $i$. Note that it is possible that there may not exist any big indices for a given constraint. The same holds true for small indices.

Lemma 9**.**

Let $\epsilon\in(0,1]$ and $\alpha_{2}=\frac{\epsilon^{2}}{c_{3}\Delta_{1}}$ where $c_{3}=8e^{1+2/e}$. Let $i\in[m]$ and $j\in B_{i}$. Then in $\mathsf{round}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{small}\mathchar 45\relax\mathsf{width}(A,b,\epsilon,\alpha_{2})$, we have $\Pr[E_{ij}|X_{j}=1]\leq\frac{A_{i,j}}{2\Delta_{1}}$.

Proof.

Let $\mathcal{E}$ be the event that there exists $j^{\prime}\in B_{i}$ such that $j^{\prime}\neq j$ and $X_{j^{\prime}}=1$. Observe that if $E_{ij}$ occurs and $X_{j}=1$, then it must be the case that at least one other element of $B_{i}$ was chosen in the rounding step. Thus,

[TABLE]

where the second inequality follows by the union bound. Observe that for all $\ell\in B_{i}$, we have $A_{i,\ell}>\epsilon/2$. By the LP constraints, we have $1+\epsilon\geq\sum_{\ell\in B_{i}}A_{i,\ell}x_{\ell}>\frac{\epsilon}{2}\cdot\sum_{\ell\in B_{i}}x_{\ell}$. Thus, $\sum_{\ell\in B_{i}}x_{\ell}\leq\frac{1+\epsilon}{\epsilon/2}=2/\epsilon+2$.

Using this upper bound for $\sum_{\ell\in B_{i}}x_{\ell}$, we have

[TABLE]

where the second inequality utilizes the fact that $\epsilon\leq 1$ and the third inequality holds because $c_{3}\geq 16$ and $A_{i,j}>\epsilon/2$. ∎

Next we consider the case when index $j$ is small for constraint $i$. The analysis here is similar to that in the preceding section with width at least $2$. The proof is left for the appendix.

Lemma 10**.**

Let $\epsilon\in(0,1]$ and $\alpha_{2}=\frac{\epsilon^{2}}{c_{3}\Delta_{1}}$ where $c_{3}=8e^{1+2/e}$. Let $i\in[m]$ and $j\in S_{i}$. Then in $\mathsf{round}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{small}\mathchar 45\relax\mathsf{width}(A,b,\epsilon,\alpha_{2})$, we have $\Pr[E_{ij}|X_{j}=1]\leq\frac{A_{i,j}}{2\Delta_{1}}$.

As Lemma 10 shows that the rejection probability is small, we can prove the following approximation guarantee much like in Theorems 4 and 6.

Theorem 11**.**

Let $\epsilon\in(0,1]$. When setting $\alpha_{2}=\frac{\epsilon^{2}}{c_{3}\Delta_{1}}$ for $c_{3}=8e^{1+2/e}$, for PIPs with width $W=1+\epsilon$, $\mathsf{round}\mathchar 45\relax\mathsf{alter}\mathchar 45\relax\mathsf{small}\mathchar 45\relax\mathsf{width}(A,b,\epsilon,\alpha_{2})$ is a randomized $(\alpha_{2}/2)$-approximation algorithm .

Proof.

Fix $j\in[n]$. Then by Lemmas 9 and 10 and the definition of $\Delta_{1}$, we have

[TABLE]

Recall that Lemma 2 gives an $\alpha_{2}(1-\gamma)$-approximation where $\gamma$ is an upper bound on the sum of the rejection probabilities for any item. This concludes the proof. ∎

Appendix

Appendix A Chernoff Bounds and Useful Inequalities

The following standard Chernoff bound is used to obtain a more convenient Chernoff bound in Theorem 13. The proof of Theorem 13 follows directly from choosing $\delta$ such that $(1+\delta)\mu=W-\beta$ and applying Theorem 12. We include the proof for convenience.

Theorem 12** ([11]).**

Let $X_{1},\ldots,X_{n}$ be independent random variables where $X_{i}$ is defined on $\{0,\beta_{i}\}$, where $0<\beta_{i}\leq\beta\leq 1$ for some $\beta$. Let $X=\sum_{i}X_{i}$ and denote $\mathbb{E}[X]$ as $\mu$. Then for any $\delta>0$,

[TABLE]

Theorem 13**.**

Let $X_{1},\ldots,X_{n}\in[0,\beta]$ be independent random variables for some $0<\beta\leq 1$. Suppose $\mu=\mathbb{E}[\sum_{i}X_{i}]\leq\alpha W$ for some $0<\alpha<1$ and $W\geq 1$ where $(1-\alpha)W>\beta$. Then

[TABLE]

Proof.

Since the right-hand side is increasing in $\alpha$, it suffices to assume $\mu=\alpha W$. Choose $\delta$ such that $(1+\delta)\mu=W-\beta$. Then $\delta=(W-\beta-\mu)/\mu$. Because $\mu=\alpha W$ and since $(1-\alpha)W>\beta$, we have $\delta=((1-\alpha)W-\beta)/\mu>0$. We apply the standard Chernoff bound in Theorem 13 to obtain

[TABLE]

Because $1+\delta=(W-\beta)/\mu$ and $\delta=(W-\beta-\mu)/\mu$,

[TABLE]

Exponentiating the denominator,

[TABLE]

As $\mu=\alpha W$,

[TABLE]

We can rewrite the exponent to show that

[TABLE]

∎

The following three lemmas are used in the proofs bounding the rejection probabilities for different regimes of width. The inequalities are easily verified via calculus. The proofs are included for the sake of completeness.

Lemma 14**.**

Let $x\in(0,1]$. Then $(1/e^{1/e})^{1/x}\leq x$.

Proof.

Taking logs of both sides of the stated inequality and rearranging, it suffices to show that $\ln(1/e^{1/e})\leq x\ln x$ for $x>0$. $x\ln x$ is convex and its minimum is $-1/e$ at $x=1/e$. Since $\ln(1/e^{1/e})=-1/e$, the inequality holds. ∎

Lemma 15**.**

Let $y\geq 2$ and $x\in(0,1]$ . Then $x/y\geq(1/e^{2/e})^{y/2x}$.

Proof.

We start with a simple rewriting of the statement. After taking logs and rearranging, it is sufficient to show

[TABLE]

Replacing $x/y$ with $z$, we see that it suffices the prove $z\ln z\geq-1/e$ for $0<z\leq 1/2$. We note that $x\ln x$ is convex and its minimum is $-1/e$ at $x=1/e$. Thus, $z\ln z\geq-1/e$. This concludes the proof. ∎

Lemma 16**.**

Let $0<\epsilon\leq 1$ and $x\in(0,1]$. Then $\epsilon x/2\geq(\epsilon/e^{2/e})^{1/x}$.

Proof.

To start, let $d=e^{2/e}/2$ and observe that $d>1$. We first do a change of variables, replacing $\epsilon/2$ with $\epsilon$ and $x$ with $x/\epsilon$. If we take a $\log$ of both sides, then our reformulated goal is to show that

[TABLE]

for $0<\epsilon\leq 1/2$ and $x\in(0,\epsilon]$. Letting $f(y)=y\ln y$ and $g(y)=y\ln(y/d)$, we want to show that $f(x)\geq g(\epsilon)$. We will proceed by cases.

First, suppose $0<\epsilon\leq d/e$. It is easy to show that $f$ is decreasing on $(0,1/e]$ and increasing on $[1/e,\infty)$ and that $g$ is decreasing on $(0,d/e]$ and increasing on $[d/e,\infty)$. As $f$ is decreasing on $(0,1/e]$, for $0<\epsilon\leq 1/e$, we have $f(x)\geq f(\epsilon)$ as $x\leq\epsilon$. As $d>1$, it follows that $f(\epsilon)\geq g(\epsilon)$. Therefore, $f(x)\geq g(\epsilon)$ for $0<\epsilon\leq 1/e$. Furthermore, as $g$ is decreasing on $[1/e,d/e]$ and $f$ is increasing on $[1/e,d/e]$, we have $f(x)\geq g(\epsilon)$ for $0<\epsilon\leq d/e$.

For the second case, suppose $d/e<\epsilon\leq 1/2$. Note that the minimum of $f$ on the interval $(0,1/2]$ is $f(1/e)=-1/e$. Thus, it would suffice to show that $g(\epsilon)\leq-1/e$. As we noted previously that $g$ is increasing on $[d/e,1/2]$, it would suffice to show that $g(1/2)\leq-1/e$. By definition of $g$, we see $g(1/2)=-1/e$. Therefore, $f(x)\geq g(\epsilon)$. This concludes the proof. ∎

Appendix B Skipped Proofs

B.1 Proof of Lemma 5

Proof.

The proof proceeds similarly to the proof of Lemma 3. Since $\alpha_{1}<1/2$, everything up to and including the application of the Chernoff bound there applies. This gives that for each $i\in[m]$ and $j\in[n]$,

[TABLE]

By choice of $\alpha_{1}$, we have

[TABLE]

We prove the final inequality in two parts. First, note that $\frac{W-A_{i,j}}{A_{i,j}}\geq W-1$ since $A_{i,j}\leq 1$. Thus,

[TABLE]

Second, we see that

[TABLE]

for $A_{i,j}\leq 1$, where the first inequality holds because $W\geq 2$ and the second inequality holds by Lemma 15. ∎

B.2 Proof of Lemma 7

Proof.

Renumber indices so that $A_{i,1}\leq\cdots\leq A_{i,n}$ and if the index of $j$ changes to $j^{\prime}$, we still refer to $j^{\prime}$ as $j$. Let $Y_{ij}=\sum_{\ell=1}^{j-1}A_{i,\ell}\xi_{\ell}$ where $\xi_{\ell}=1$ if $x_{\ell}^{\prime}=1$ and [math] otherwise. We first note that

[TABLE]

By the choice of $\alpha_{1}$ and the fact that $A_{i,j}\leq 1$ and $W=\frac{2}{\epsilon^{2}}\ln(\frac{\Delta_{1}}{\epsilon})+1$, we have $((1-\alpha_{1})W)/A_{i,j}\geq\epsilon W=\frac{2}{\epsilon}\ln(\frac{\Delta_{1}}{\epsilon})+\epsilon$. A direct argument via calculus shows $\frac{2}{\epsilon}\ln(\frac{\Delta_{1}}{\epsilon})+\epsilon>1$ for $\epsilon\in(0,\frac{1}{e})$. Thus, $(1-\alpha_{1})W>A_{i,j}$.

By the LP constraints, $\mathbb{E}[Y_{ij}]\leq\alpha_{1}W$. Then as $A_{i,j^{\prime}}\leq A_{i,j}$ for all $j^{\prime}<j$, we can apply the Chernoff bound in Theorem 13 to obtain

[TABLE]

As $A_{i,j}\leq 1$,

[TABLE]

where the last inequality follows from the fact that $(1-1/z)^{z-1}\geq 1/e$ for all $z\geq 1$. Then

[TABLE]

By the choice of $\alpha_{1}$,

[TABLE]

For $0<\epsilon<\frac{1}{e}$, we have $1-\epsilon\leq\exp(-\epsilon-\frac{\epsilon^{2}}{2})$. As $W=\frac{2}{\epsilon^{2}}\ln(\frac{\Delta_{1}}{\epsilon})+1$ and $A_{i,j}\leq 1$,

[TABLE]

Observe that $\frac{1}{A_{i,j}}-\ln(\frac{e}{A_{i,j}})\geq 0$. For $A_{i,j}\in[0,1]$, a direct argument shows $\frac{\ln(t)}{A_{i,j}}-\ln(\frac{t}{A_{i,j}})$ is increasing in $t$ for $t\geq e$. As $\Delta_{1}/\epsilon>e$, we have $\frac{\ln(\frac{\Delta_{1}}{\epsilon})}{A_{i,j}}\geq\ln(\frac{\Delta_{1}}{\epsilon A_{i,j}})$. Therefore,

[TABLE]

This concludes the proof. ∎

B.3 Proof of Lemma 10

Proof.

Renumber the indices so that $A_{i,1}\leq\cdots\leq A_{i,n}$. Note that the index $j$ might have changed to $j^{\prime}$ but for simplicity we refer to $j^{\prime}$ as $j$. Let $\xi_{\ell}=1$ if $x_{\ell}^{\prime}=1$ and [math] otherwise. Let $Y_{ij}=\sum_{\ell=1}^{j-1}A_{i,\ell}\xi_{\ell}$. We have

[TABLE]

Let $A_{i,\ell}^{\prime}=\frac{2}{\epsilon}\cdot A_{i,\ell}$ for $\ell\in[j]$. As $A_{i,\ell}\leq\epsilon/2$ for all $\ell\in[j]$, we have $A_{i,\ell}^{\prime}\in[0,1]$. Let $Y_{ij}^{\prime}=\sum_{\ell=1}^{j-1}A_{i,\ell}^{\prime}\xi_{\ell}$. Then

[TABLE]

To upper bound $\mathbb{E}[Y_{ij}^{\prime}]$, we have

[TABLE]

Let $\alpha_{2}^{\prime}=\frac{2\epsilon}{c_{3}\Delta_{1}}$ and $W=2$. Then $\mathbb{E}[Y_{ij}^{\prime}]\leq\alpha_{2}^{\prime}W$. As $\alpha_{2}^{\prime}<1/2$ and $A_{i,j}^{\prime}\leq 1$, we see that $((1-\alpha)W)/A_{i,j}^{\prime}>1$. Therefore, as $A_{i,\ell}^{\prime}\leq A_{i,j}^{\prime}$ for all $\ell<j$, we can apply the Chernoff bound in Theorem 13 to obtain

[TABLE]

Observe that $e^{1-\alpha_{2}^{\prime}}\leq e$ and $\frac{W}{W-A_{i,j}^{\prime}}\leq 2$ since $W=2$ and $A_{i,j}^{\prime}\leq 1$. By our choice of $\alpha_{2}^{\prime}$,

[TABLE]

We prove the final inequality in two parts. First, we note that $\frac{W-A_{i,j}^{\prime}}{A_{i,j}^{\prime}}\geq 1$ since $W=2$ and $A_{i,j}^{\prime}\leq 1$. Then

[TABLE]

Second, we observe $\frac{W-A_{i,j}^{\prime}}{A_{i,j}^{\prime}}\geq 1/A_{i,j}^{\prime}$ since $W=2$ and $A_{i,j}^{\prime}\leq 1$. Then we can apply Lemma 16 to obtain

[TABLE]

We have shown $\Pr[E_{ij}|X_{j}=1]\leq\frac{\epsilon A_{i,j}^{\prime}}{4\Delta_{1}}$. Since $A_{i,j}^{\prime}=A_{i,j}\cdot\frac{2}{\epsilon}$, the result follows. ∎