The goal of this paper is to design a simplex algorithm for linear programs on lattice polytopes that traces “short” simplex paths from any given vertex to an optimal one. We consider a lattice polytope P contained in \([0,k]^n\) and defined via m linear inequalities. Our first contribution is a simplex algorithm that reaches an optimal vertex by tracing a path along the edges of P of length in \(O(n^{4} k\, \hbox {log}\, k).\) The length of this path is independent from m and it is the best possible up to a polynomial function. In fact, it is only polynomially far from the worst-case diameter, which roughly grows as nk. Motivated by the fact that most known lattice polytopes are defined via \(0,\pm 1\) constraint matrices, our second contribution is a more sophisticated simplex algorithm which exploits the largest absolute value \(\alpha \) of the entries in the constraint matrix. We show that the length of the simplex path generated by this algorithm is in \(O(n^2k\, \hbox {log}\, ({nk} \alpha ))\). In particular, if \(\alpha \) is bounded by a polynomial in n, k, then the length of the simplex path is in \(O(n^2k\, \hbox {log}\, (nk))\). For both algorithms, if P is “well described”, then the number of arithmetic operations needed to compute the next vertex in the path is polynomial in n, m, and \(\hbox {log}\, k\). If k is polynomially bounded in n and m, the algorithm runs in strongly polynomial time.

本文的目的是设计一种用于格型多边形上线性程序的单纯形算法，该算法可跟踪从任何给定顶点到最佳顶点的“短”单纯形路径。我们考虑包含在\（[0，k] ^ n \）中并通过m个 线性不等式定义的晶格多面体P。我们的第一个贡献是一个单纯形算法到达由沿着边缘跟踪的路径的最佳顶点P的长度的\（O（N ^ {4} K和\，\ hbox中{日志} \，K）。\）的长度这条路径的绝对值与m无关，并且最大可能是多项式函数。实际上，它离最坏情况的直径只有几项之遥，后者随着nk的增长而增大 。通过大多数已知的晶格多面体经由限定的事实动机\（0，\下午1 \）约束矩阵，我们的第二个贡献是一个更复杂的单纯形算法其利用最大绝对值 \（\阿尔法\）在条目约束矩阵。我们证明了此算法生成的单纯形路径的长度在\（O（n ^ 2k \，\ hbox {log} \，（{nk} \ alpha））\）中。特别是，如果\（\ alpha \）被n，  k中的多项式所限制，则单纯形路径的长度为\（O（n ^ 2k \，\ hbox {log} \，（nk））\ ）。对于两种算法，如果P如果“描述得很好”，则计算路径中下一个顶点所需的算术运算次数就是n，m和\（\ hbox {log} \，k \）中的多项式。如果k是n和 m的多项式边界，则该算法将在强多项式时间内运行。