Tighter Bounds for the Randomized Polynomial-Time Simplex Algorithm for Linear Programming

TL;DR

This paper introduces a refined randomized polynomial-time simplex algorithm for linear programming, achieving tighter bounds and higher success probability through improved quasi-convex properties and probabilistic analysis.

Contribution

It provides new bounds for the expected number of edges in the shadow projection and modifies the Kelner-Spielman algorithm for better success probability.

Findings

Stronger bounds for the expected edges in the polytope shadow.

A modified algorithm with higher success probability.

Tighter bounds for quasi-convex and quasi-concave optimization.

Abstract

We present a randomized polynomial-time simplex algorithm with higher probability and tighter bounds for linear programming by applying improved quasi-convex properties, a logarithmic rounding on a given polytope and its logarithmic perturbation. We base our work on the first randomized polynomial-time simplex method by Jonathan A. Kelner and Daniel A. Spielman [KS06]. We obtain stronger bounds for the expected number of edges in the projection of a perturbed polytope onto a two-dimensional shadow plane. In the $k$-round case, we obtain a bound of $16 \sqrt{2} \pi k (1 + \lambda H_n) \sqrt{d} n / 3 \lambda$. In the non-$k$-round case, we obtain a bound of $26 \pi t (1 + \lambda H_n) \sqrt{d} n / \lambda \rho$. To achieve this, we provide a slightly lower bound of $3 \sqrt{2} \lambda / (16 n \sqrt{d})$ on the expected edge length that appears in the shadow. Another tool we employ is a tighter bound for $1$-quasi-concave minimization and $1$-quasi-convex maximization. In the $k$-round case, we obtain a quasi-convex bound of $(d - 2) \epsilon^2 / 2$. In the non-$k$-round case, we obtain a quasi-convex bound of $3.4 \epsilon^2 / \rho^2$. We propose a modification of the Kelner and Spielman randomized simplex algorithm (STOC'06) [KS06] that achieves a higher success probability. To accomplish this, we apply our tighter bounds with a new expected value of $\lambda = c \log n$ for independent exponentially distributed random variables and with $\log(k)$-rounding. The desired properties resulting from the construction of an artificial vertex during initialization hold with a higher probability of at least $1 - (d + 2), e^{-\log n}$. The pivot rule of the randomized simplex modification holds with a probability of at least $3/4$.