Adaptive Sparsification for Linear Programming

2025-10-09

AI Summary

For large-scale linear programs with far more constraints than variables (n ≫ d), this paper proposes an adaptive sparsification framework that decomposes the original problem into a sequence of smaller subproblems solved iteratively. The method integrates a low-violation constraint oracle, multiplicative weight updates, and generalized Grover search, enabling modular acceleration in both classical and quantum settings. Classically, it robustifies and generalizes Clarkson’s algorithm; quantumly, it decouples classical preprocessing from quantum core solving, requiring only Õ(√n d³) row queries—significantly reducing width dependence for mixed packing/covering problems. Experiments demonstrate that the approach achieves state-of-the-art time complexity in both classical and quantum regimes.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Large-scale security measurements

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 $ ilde{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.

Problem

• Solves linear programs with many constraints efficiently via adaptive sparsification
• Accelerates classical LP solving using quantum algorithms and Grover search
• Improves algorithms for mixed packing and covering problems through width reduction

Innovation

• Adaptive sparsification reduces LP constraints via sampling
• Quantum Grover search accelerates violated constraint detection
• Hybrid packing-covering constraints enable width reduction optimization