Tighter Bounds for the Randomized Polynomial-Time Simplex Algorithm for Linear Programming
Daniel Gibor
TL;DR
This paper advances randomized polynomial-time simplex methods for linear programming by tightening shadow-size bounds through perturbation-based geometry and refining quasi-convex/quasi-concave analyses. It introduces a higher-probability variant of the Kelner–Spielman algorithm via logarithmic perturbations and log-rounding, coupled with an artificial-start vertex construction to ensure favorable initialization with high probability. The key contributions are explicit, improved bounds on the shadow size for both k-round and non-k-round polytopes and a practical modification that raises the success probability of the method, all while preserving polynomial-time performance in the input size. These results enhance the theoretical understanding and practical efficiency of randomized simplex approaches and open avenues for further progress toward strongly polynomial-time linear programming algorithms.
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 $16 \sqrt{2} πk (1 + λH_n) \sqrt{d} n / 3 λ$. In the non-$k$-round case, we obtain a bound of $26 πt (1 + λH_n) \sqrt{d} n / λρ$. To achieve this, we provide a slightly lower bound of $3 \sqrt{2} λ/ (16 n \sqrt{d})$ 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 $(d - 2) ε^2 / 2$. In the non-$k$-round case, we obtain a quasi-convex bound of $3.4 ε^2 / ρ^2$. 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 $λ= c \log n$ 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 $3/4$.