Short simplex paths in lattice polytopes

TL;DR

This paper introduces new simplex algorithms for lattice polytopes that find optimal vertices along short paths with polynomial bounds, improving efficiency in linear programming on such polytopes.

Contribution

The paper presents two novel simplex algorithms with polynomial path length bounds for lattice polytopes, one independent of the number of inequalities and the other exploiting constraint matrix entries.

Findings

Path length in $O(n^4 k \, \log(nk))$ for the first algorithm.

Path length in $O(n^2 k \, \log(nk \alpha))$ for the iterative algorithm.

Algorithms run in strongly polynomial time if $k$ is polynomially bounded by $n$ and $m$.

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.