Circuit and Graver Walks and Linear and Integer Programming

TL;DR

This paper demonstrates polynomial-time algorithms for circuit and Graver walks in linear and integer programming, respectively, using linear algebra and oracles, with implications for sparse integer programming.

Contribution

It introduces polynomial-time methods for computing circuit and Graver walks, leveraging linear algebra and oracles, and extends results to sparse integer programming.

Findings

Circuit walks can be computed in polynomial time using linear algebra.

Graver walks are polynomial-time computable with an integer programming oracle.

Polynomial-time computation of Graver walks over matrices with bounded properties.

Abstract

We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.