Random-Facet and Random-Bland require subexponential time even for shortest paths

TL;DR

This paper demonstrates that the Random-Facet and Random-Bland algorithms, used for linear programming, require subexponential time even for shortest path problems, challenging assumptions about their efficiency.

Contribution

The paper extends lower bounds for Random-Facet and Random-Bland algorithms to shortest path instances, showing their subexponential complexity is inherent and not dependent on MDP stochasticity.

Findings

Random-Facet requires subexponential time for shortest path problems.

Lower bounds previously known for MDPs apply to shortest path LPs.

Random-Bland also exhibits subexponential lower bounds.

Abstract

The Random-Facet algorithm of Kalai and of Matousek, Sharir and Welzl is an elegant randomized algorithm for solving linear programs and more general LP-type problems. Its expected subexponential time of $2^{\tilde{O}(\sqrt{m})}$, where $m$ is the number of inequalities, makes it the fastest known combinatorial algorithm for solving linear programs. We previously showed that Random-Facet performs an expected number of $2^{\tilde{\Omega}(\sqrt[3]{m})}$ pivoting steps on some LPs with $m$ inequalities that correspond to $m$-action Markov Decision Processes (MDPs). We also showed that Random-Facet-1P, a one permutation variant of Random-Facet, performs an expected number of $2^{\tilde{O}(\sqrt{m})}$ pivoting steps on these examples. Here we show that the same results can be obtained using LPs that correspond to instances of the classical shortest paths problem. This shows that the stochasticity of the MDPs, which is essential for obtaining lower bounds for Random-Edge, is not needed in order to obtain lower bounds for Random-Facet. We also show that our new $2^{\tilde{\Omega}(\sqrt{m})}$ lower bound applies to Random-Bland, a randomized variant of the classical anti-cycling rule suggested by Bland.