A Strongly Polynomial Reduction for Linear Programs over Grids

TL;DR

This paper introduces a strongly polynomial reduction from linear programs over grids to simpler cube-based problems, impacting the solvability of linear programming and related stochastic games.

Contribution

It presents a novel reduction scheme connecting Grid-LPs, GLCPs, and Cube-LPs, and extends these ideas to stochastic games with perfect information.

Findings

Reduction from Grid-LPs to Cube-LPs is strongly polynomial.

Reduced discounted MDPs to binary MDPs efficiently.

Characterization of two-player stochastic games via GLCPs with P-matrices.

Abstract

We investigate the duality relation between linear programs over grids (Grid-LPs) and generalized linear complementarity problems (GLCPs) with hidden K-matrices. The two problems, moreover, share their combinatorial structure with discounted Markov decision processes (MDPs). Through proposing reduction schemes for the GLCP, we obtain a strongly polynomial reduction from Grid-LPs to linear programs over cubes (Cube-LPs). As an application, we obtain a scheme to reduce discounted MDPs to their binary counterparts. This result also suggests that Cube-LPs are the key problems with respect to solvability of linear programming in strongly polynomial time. We then consider two-player stochastic games with perfect information as a natural generalization of discounted MDPs. We identify the subclass of the GLCPs with P-matrices that corresponds to these games and also provide a characterization in terms of unique-sink orientations. A strongly polynomial reduction from the games to their binary counterparts is obtained through a generalization of our reduction for Grid-LPs.