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Adaptive Sparsification for Linear Programming

Étienne Objois, Adrian Vladu

TL;DR

This work develops an adaptive sparsification framework for linear programs with many constraints ($n \gg d$), enabling reduction to small, adaptively sampled subproblems via a multiplicative-weights-based loop and a central low-violation oracle. It delivers a quantum variant of Clarkson's exact LP solver achieving $\tilde{O}(\sqrt{n}\,d^3)$ row-queries, along with classical and quantum results for low-precision solvers and for mixed packing/covering problems that benefit from width-reduction and discretization techniques. The framework is modular, yielding new state-of-the-art algorithms for mixed packing and covering with tall constraint matrices and enabling substantial quantum speedups through Grover-based constraint search. By combining constraint sparsification, MWU analysis, and quantum sampling, the paper provides a unified approach to accelerate LP solvers across exact, approximate, and mixed-constraint regimes, with practical impact in large-scale optimization and quantum-enhanced computation.

Abstract

We introduce a generic framework for solving linear programs (LPs) with many constraints $(n \gg d)$ via adaptive sparsification. Our approach provides a principled generalization of the techniques of [Assadi '23] from matching problems to general LPs and robustifies [Clarkson's '95] celebrated algorithm for the exact setting. The framework reduces LP solving to a sequence of calls to a ``low-violation oracle'' on small, adaptively sampled subproblems, which we analyze through the lens of the multiplicative weight update method. Our main results demonstrate the versatility of this paradigm. First, we present a quantum version of Clarkson's algorithm that finds an exact solution to an LP using $\tilde{O}(\sqrt{n} d^3)$ row-queries to the constraint matrix. This is achieved by accelerating the classical bottleneck (the search for violated constraints) with a generalization of Grover search, decoupling the quantum component from the classical solver. Second, our framework yields new state-of-the-art algorithms for mixed packing and covering problems when the packing constraints are ``simple''. By retaining all packing constraints while sampling only from the covering constraints, we achieve a significant width reduction, leading to faster solvers in both the classical and quantum query models. Our work provides a modular and powerful approach for accelerating LP solvers.

Adaptive Sparsification for Linear Programming

TL;DR

This work develops an adaptive sparsification framework for linear programs with many constraints (), enabling reduction to small, adaptively sampled subproblems via a multiplicative-weights-based loop and a central low-violation oracle. It delivers a quantum variant of Clarkson's exact LP solver achieving row-queries, along with classical and quantum results for low-precision solvers and for mixed packing/covering problems that benefit from width-reduction and discretization techniques. The framework is modular, yielding new state-of-the-art algorithms for mixed packing and covering with tall constraint matrices and enabling substantial quantum speedups through Grover-based constraint search. By combining constraint sparsification, MWU analysis, and quantum sampling, the paper provides a unified approach to accelerate LP solvers across exact, approximate, and mixed-constraint regimes, with practical impact in large-scale optimization and quantum-enhanced computation.

Abstract

We introduce a generic framework for solving linear programs (LPs) with many constraints via adaptive sparsification. Our approach provides a principled generalization of the techniques of [Assadi '23] from matching problems to general LPs and robustifies [Clarkson's '95] celebrated algorithm for the exact setting. The framework reduces LP solving to a sequence of calls to a ``low-violation oracle'' on small, adaptively sampled subproblems, which we analyze through the lens of the multiplicative weight update method. Our main results demonstrate the versatility of this paradigm. First, we present a quantum version of Clarkson's algorithm that finds an exact solution to an LP using row-queries to the constraint matrix. This is achieved by accelerating the classical bottleneck (the search for violated constraints) with a generalization of Grover search, decoupling the quantum component from the classical solver. Second, our framework yields new state-of-the-art algorithms for mixed packing and covering problems when the packing constraints are ``simple''. By retaining all packing constraints while sampling only from the covering constraints, we achieve a significant width reduction, leading to faster solvers in both the classical and quantum query models. Our work provides a modular and powerful approach for accelerating LP solvers.

Paper Structure

This paper contains 55 sections, 18 theorems, 52 equations, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Low-Precision.
  4. Mixed Packing and Covering Problems.
  5. Overview of Approach
  6. Constraint sparsification for low-precision algorithms.
  7. A unified framework for constraint sparsification.
  8. Exploiting quantum speedups via Grover search.
  9. Organization
  10. Preliminaries
  11. Notation
  12. Classes of Linear Programs
  13. General LPs.
  14. Positive LPs.
  15. Quantum Complexity Model
  16. ...and 40 more sections

Key Result

Theorem 1

There is a randomized quantum algorithm that finds an exact solution of eq:intro-lp using $\widetilde{O}(\sqrt{n} d^3)$ row-queries to $\boldsymbol{\mathit{A}}$. Furthermore, this algorithm runs in expected time where $r$ is the row-sparsity of $\boldsymbol{\mathit{A}}$, and $\mathcal{T}_{\textnormal{exact}}(d,n)$ is the time required to compute an exact solution to a linear program with $d$ vari

Theorems & Definitions (43)

  • Theorem 1
  • Remark
  • Remark
  • Theorem 2: Short version of \ref{['thm:main-theorem']}
  • Theorem 3
  • Claim 1
  • Theorem 4
  • Definition 1
  • proof : Proof of \ref{['thm:main-theorem']}
  • Lemma 5: Corollary of clarksonVegasAlgorithmsLinear1995
  • ...and 33 more