$\ell _1$-sparsity Approximation Bounds for Packing Integer Programs

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. [1] obtained an $\varOmega (\frac{1}{\varDelta _0^{1/\lfloor W \rfloor }})$-approximation ratio where $\varDelta _0$ 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 $\varDelta _1$) which can be much smaller than $\varDelta _0$. Motivated by recent work on covering integer programs (CIPs) [4, 7] we show that simple algorithms based on randomized rounding followed by alteration, similar to those of Bansal et al. [1] (but with a twist), yield approximation ratios for PIPs based on $\varDelta _1$. First, following an integrality gap example from [1], we observe that the case of $W=1$ is as hard as maximum independent set even when $\varDelta _1 \le 2$. 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 $\varOmega (\epsilon ^2/\varDelta _1)$-approximation. In the large width regime, when $W \ge 2$, we obtain an $\varOmega ((\frac{1}{1 + \varDelta _1/W})^{1/(W-1)})$-approximation. We also obtain a $(1-\epsilon )$-approximation when $W = \varOmega (\frac{\log (\varDelta _1/\epsilon )}{\epsilon ^2})$.

C. Chekuri and K. Quanrud supported in part by NSF grant CCF-1526799. M. Torres supported in part by fellowships from NSF and the Sloan Foundation.

Appendices

Appendix

A Chernoff Bounds and Useful Inequalities

The following standard Chernoff bound is used to obtain a more convenient Chernoff bound in Theorem 7. The proof of Theorem 7 follows directly from choosing $\delta$ such that $(1 + \delta )\mu = W - \beta$ and applying Theorem 6.

Theorem 6

([12]). Let $X_1,\ldots , X_n$ be independent random variables where $X_i$ is defined on $\{0, \beta _i\}$, where $0 < \beta _i \le \beta \le 1$ for some $\beta$. Let $X = \sum _i X_i$ and denote $\mathbb {E}[X]$ as $\mu$. Then for any $\delta > 0$,

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

Theorem 7

Let $X_1,\ldots , X_n \in [0,\beta ]$ be independent random variables for some $0 < \beta \le 1$. Suppose $\mu = \mathbb {E}[\sum _i X_i] \le \alpha W$ for some $0< \alpha < 1$ and $W \ge 1$ where $(1 - \alpha )W > \beta$. Then

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

Lemma 7

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

Lemma 8

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

B Skipped Proofs

1.1 B.1 Proof of Theorem 1

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 0. Consider the following PIP:

$$\begin{aligned} \text {maximize}\ \langle x, \mathbf {1}\rangle \ \text {over} \ x \in \{0,1\}^n \ \text {s.t.} \ Ax \le 1. \end{aligned}$$

(1)

Let S be the set of all feasible integral solutions of (1) and $\mathcal {I}$ be the set of independent sets of G. Define $g : S \rightarrow \mathcal {I}$ 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 0 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 $h : S \rightarrow \mathbb {N}_0$ 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, $\max _{x\in S} h(x) \ge \left| I_{max}\right|$. 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, $\varDelta _1 \le 2$, thus concluding the proof.

1.2 B.2 Proof of Lemma 3

Proof

The proof proceeds similarly to the proof of Lemma 2. 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 $j \in [n]$,

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

By choice of $\alpha _1$, we have

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

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

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

Second, we see that

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

for $A_{i,j} \le 1$, where the first inequality holds because $W \ge 2$ and the second inequality holds by Lemma 8.