Adaptive Sparsification for Linear Programming

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.