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 $O(n^4 k\log(nk)$. 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 a linear function in $n$ and $k$. Motivated by the fact that most known lattice polytopes are defined via $0,\pm 1$ constraint matrices, our second contribution is an iterative 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 the iterative algorithm is in $O(n^2k \log(nk\alpha))$. In particular, if $\alpha$ is bounded by a polynomial in $n, k$, then the length of the simplex path is in $O(n^2k \log(nk))$. For both algorithms, the number of arithmetic operations needed to compute the next vertex in the path is polynomial in $n$, $m$ and $\log k$. If $k$ is polynomially bounded by $n$ and $m$, the algorithm runs in strongly polynomial time.