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Integrality Gaps for Random Integer Programs via Discrepancy

Sander Borst, Daniel Dadush, Dan Mikulincer

TL;DR

The gap results imply that for certain random packing and Gaussian IPs B&B trees have size $n^{poly(m)}$ with high probability (i.e., polynomial for fixed $m$), which significantly extends the class of IPs for which B &B is known to be polynometric.

Abstract

We prove new bounds on the additive gap between the value of a random integer program $\max c^Tx,\ Ax\leq b,\ x\in\{0,1\}^n$ with $m$ constraints and that of its linear programming relaxation for a wide range of distributions on $(A,b,c)$ . We are motivated by the work of Dey, Dubey, and Molinaro (SODA '21), who gave a framework for relating the size of Branch-and-Bound (B&B) trees to additive integrality gaps. Dyer and Frieze (MOR '89) and Borst et al. (Mathematical Programming '22), respectively, showed that for certain random packing and Gaussian IPs, where the entries of $A,c$ are independently distributed according to either the uniform distribution on $[0,1]$ or the Gaussian distribution $\mathcal{N}(0,1)$, the integrality gap is bounded by $O_m(\log^2 n / n)$ with probability at least $1-1/n-e^{-Ω_m(1)}$. In this paper, we generalize these results to the case where the entries of $A$ are uniformly distributed on an integer interval (e.g., entries in $\{-1,0,1\}$), and where the columns of $A$ are distributed according to an isotropic logconcave distribution. Second, we substantially improve the success probability to $1-1/poly(n)$, compared to constant probability in prior works (depending on $m$). Leveraging the connection to Branch-and-Bound, our gap results imply that for these IPs B&B trees have size $n^{poly(m)}$ with high probability (i.e., polynomial for fixed $m$), which significantly extends the class of IPs for which B&B is known to be polynomial. Our main technical contribution is a new linear discrepancy theorem for random matrices. Our theorem gives general conditions under which a target vector is equal to or very close to a $\{0,1\}$ combination of the columns of a random matrix $A$ . The proof uses a Fourier analytic approach, building on work of Hoberg and Rothvoss (SODA '19) and Franks and Saks (RSA '20).

Integrality Gaps for Random Integer Programs via Discrepancy

TL;DR

The gap results imply that for certain random packing and Gaussian IPs B&B trees have size with high probability (i.e., polynomial for fixed ), which significantly extends the class of IPs for which B &B is known to be polynometric.

Abstract

We prove new bounds on the additive gap between the value of a random integer program with constraints and that of its linear programming relaxation for a wide range of distributions on . We are motivated by the work of Dey, Dubey, and Molinaro (SODA '21), who gave a framework for relating the size of Branch-and-Bound (B&B) trees to additive integrality gaps. Dyer and Frieze (MOR '89) and Borst et al. (Mathematical Programming '22), respectively, showed that for certain random packing and Gaussian IPs, where the entries of are independently distributed according to either the uniform distribution on or the Gaussian distribution , the integrality gap is bounded by with probability at least . In this paper, we generalize these results to the case where the entries of are uniformly distributed on an integer interval (e.g., entries in ), and where the columns of are distributed according to an isotropic logconcave distribution. Second, we substantially improve the success probability to , compared to constant probability in prior works (depending on ). Leveraging the connection to Branch-and-Bound, our gap results imply that for these IPs B&B trees have size with high probability (i.e., polynomial for fixed ), which significantly extends the class of IPs for which B&B is known to be polynomial. Our main technical contribution is a new linear discrepancy theorem for random matrices. Our theorem gives general conditions under which a target vector is equal to or very close to a combination of the columns of a random matrix . The proof uses a Fourier analytic approach, building on work of Hoberg and Rothvoss (SODA '19) and Franks and Saks (RSA '20).
Paper Structure (29 sections, 50 theorems, 186 equations)

This paper contains 29 sections, 50 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Complexity of Branch-and-Bound
  3. Integrality Gap Bounds and Their Applications
  4. Gap Bounds in Related Settings.
  5. Techniques
  6. Proving Gap Bounds.
  7. Relations to Discrepancy Theory.
  8. Discrepancy
  9. Notation and assumptions:
  10. Main discrepancy results:
  11. Road map for the proof:
  12. Integrality Gap Bounds
  13. Linear Programs and their Duals
  14. The Gap Bound for Centered IPs
  15. Properties of the Optimal Solutions
  16. ...and 14 more sections

Key Result

Theorem 1

Let $n \geq \Omega(m)$, $b \in \mathbb{R}^m$, and $ \in \mathbb{R}^{(m+1) \times n}$ be a matrix whose columns are independent logconcave random vectors with identity covariance. Then, for $G \geq 0$, with probability at least $1-\Pr_{A,c}[{\rm IPGAP}(A,b,c) \geq G]-1/\mathop{\mathrm{\mathrm{poly}}}

Theorems & Definitions (87)

  • Theorem 1: dey_branch-and-bound_2021BDHT22
  • Theorem 2: Gap Bound for Centered IPs
  • Theorem 3: Discrete Packing IPs
  • Corollary 1
  • Theorem 4: Linear Discrepancy Theorem for Random Matrices
  • Definition 1: approximately symmetric distributions
  • Remark 1
  • Definition 2: anti-concentration
  • Theorem 5
  • Theorem 6
  • ...and 77 more