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Quantum Speedups for Group Relaxations of Integer Linear Programs

Brandon Augustino, Dylan Herman, Guneykan Ozgul, Jacob Watkins, Atithi Acharya, Enrico Fontana, Junhyung Lyle Kim, Shouvanik Chakrabarti

TL;DR

A competitive feasibility-preserving classical local-search algorithm for the group relaxation, and a corresponding quantum algorithm that, under reasonable technical conditions, achieves a super-quadratic speedup for Gomory's group relaxation are presented.

Abstract

Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to obtain because classical algorithms for many-constraint ILPs are global and exhaustive, whereas quantum frameworks that offer super-quadratic speedup exploit local structure of the objective and feasible set. We address this via quantum algorithms for Gomory's group relaxation. The group relaxation of an ILP is obtained by dropping nonnegativity on variables that are positive in the optimal solution of the linear programming (LP) relaxation, while retaining integrality of the decision variables. We present a competitive feasibility-preserving classical local-search algorithm for the group relaxation, and a corresponding quantum algorithm that, under reasonable technical conditions, achieves a super-quadratic speedup. When the group relaxation satisfies a nondegeneracy condition analogous to, but stronger than, LP non-degeneracy, our approach yields the optimal solution to the original ILP. Otherwise, the group relaxation tightens bounds on the optimal objective value of the ILP, and can improve downstream branch-and-cut by reducing the integrality gap; we numerically observe this on several practically relevant ILPs. To achieve these results, we derive efficiently constructible constraint-preserving mixers for the group relaxation with favorable spectral properties, which are of independent interest.

Quantum Speedups for Group Relaxations of Integer Linear Programs

TL;DR

A competitive feasibility-preserving classical local-search algorithm for the group relaxation, and a corresponding quantum algorithm that, under reasonable technical conditions, achieves a super-quadratic speedup for Gomory's group relaxation are presented.

Abstract

Integer Linear Programs (ILPs) are a flexible and ubiquitous model for discrete optimization problems. Solving ILPs is \textsf{NP-Hard} yet of great practical importance. Super-quadratic quantum speedups for ILPs have been difficult to obtain because classical algorithms for many-constraint ILPs are global and exhaustive, whereas quantum frameworks that offer super-quadratic speedup exploit local structure of the objective and feasible set. We address this via quantum algorithms for Gomory's group relaxation. The group relaxation of an ILP is obtained by dropping nonnegativity on variables that are positive in the optimal solution of the linear programming (LP) relaxation, while retaining integrality of the decision variables. We present a competitive feasibility-preserving classical local-search algorithm for the group relaxation, and a corresponding quantum algorithm that, under reasonable technical conditions, achieves a super-quadratic speedup. When the group relaxation satisfies a nondegeneracy condition analogous to, but stronger than, LP non-degeneracy, our approach yields the optimal solution to the original ILP. Otherwise, the group relaxation tightens bounds on the optimal objective value of the ILP, and can improve downstream branch-and-cut by reducing the integrality gap; we numerically observe this on several practically relevant ILPs. To achieve these results, we derive efficiently constructible constraint-preserving mixers for the group relaxation with favorable spectral properties, which are of independent interest.
Paper Structure (34 sections, 15 theorems, 135 equations, 1 figure, 1 table, 6 algorithms)

This paper contains 34 sections, 15 theorems, 135 equations, 1 figure, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Relaxations of ILPs
  4. Contributions
  5. Solving the group relaxation with null space walks
  6. Generalized Short Path algorithm
  7. Discussion
  8. Related work
  9. Integer linear programming
  10. Quantum algorithms for integer and combinatorial optimization
  11. The group relaxation and the corner polyhedron
  12. Preliminaries
  13. Abelian groups and lattices
  14. Basics of finite Markov chains
  15. Background on linear programming
  16. ...and 19 more sections

Key Result

Theorem 1

Let ${\cal B}$ be an optimal basis for the LP relaxation of e:ILP. Define the set of optimal solutions to the (finite) group relaxation associated with this basis to be We construct a quantum algorithm that, with high probability, finds an optimal solution $x^{\star}_{{\cal N}} \in {\cal P}_{\cal B}^\star$ with runtime where $\alpha (n) > 0$ and satisfies $\alpha (n) = {\cal O}_n(1)$. Under mild

Figures (1)

  • Figure 1: Histogram of $R_{\%} = 100 \cdot (\textup{OPT}_{\cal B} - \textup{OPT}_{\mathrm{LP}})/(\textup{OPT} - \textup{OPT}_{\mathrm{LP}})$ over the cutting‑stock benchmark. Dashed line at $100\%$ marks full gap closure by the group relaxation.

Theorems & Definitions (29)

  • Theorem 1: Main Result, informal statement of Theorems \ref{['thm:main-speedup']} & \ref{['thm:main-speedup-poincare']}
  • Definition 1: Cyclic metric
  • Definition 2: Log-Sobolev inequality
  • Definition 3: Discriminant matrix
  • Definition 4: $P$-pseudo Lipschitz norm
  • Definition 5: $\gamma$-Spectral Density -- Definition 3.2 chakrabarti2024generalized
  • Definition 6: $\Delta_P$-stability -- Definition 3.1 chakrabarti2024generalized
  • Theorem 2: Log-Sobolev Version chakrabarti2024generalized
  • Theorem 3: Spectral Gap Version chakrabarti2024generalized
  • Corollary 1: Super-Quadratic Condition chakrabarti2024generalized
  • ...and 19 more