Short Simplex Paths in Lattice Polytopes

Abstract

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.