Short Simplex Paths in Lattice Polytopes

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 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,± 1 constraint matrices, our second contribution is a more sophisticated simplex algorithm which exploits the largest absolute value α 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 log (nkα )) . In particular, if α 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, 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 log k . If k is polynomially bounded in n and m , the algorithm runs in strongly polynomial time.