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 $3/4$.

翻译：我们通过应用改进的拟凸性质、对给定多面体的对数舍入及其对数扰动，提出了一种具有更高概率和更紧界的随机多项式时间单纯形算法。我们的工作基于Jonathan A. Kelner和Daniel A. Spielman [KS06] 首次提出的随机多项式时间单纯形方法。我们得到了扰动多面体在二维阴影平面上的投影中期望边数的更强界。在$k$轮情形下，我们得到的界为$16 \sqrt{2} πk (1 + λH_n) \sqrt{d} n / 3 λ$。在非$k$轮情形下，我们得到的界为$26 πt (1 + λH_n) \sqrt{d} n / λρ$。为此，我们给出了阴影中出现的期望边长的略低界$3 \sqrt{2} λ/ (16 n \sqrt{d})$。我们使用的另一个工具是$1$-拟凹最小化和$1$-拟凸最大化的更紧界。在$k$轮情形下，我们得到拟凸界$(d - 2) ε^2 / 2$。在非$k$轮情形下，我们得到拟凸界$3.4 ε^2 / ρ^2$。我们提出了Kelner和Spielman随机单纯形算法（STOC'06）[KS06] 的一种改进，该改进实现了更高的成功概率。为实现此目标，我们应用了更紧的界，其中独立指数分布随机变量的新期望值为$λ= c \log n$，并采用$\log(k)$-舍入。初始化过程中人工顶点构造所产生的期望性质，其成立的概率至少为$1 - (d + 2), e^{-\log n}$，该概率值更高。随机单纯形改进的转轴规则成立的概率至少为$3/4$。