Short Simplex Paths in Lattice Polytopes

Alberto Del Pia, Carla Michini

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 $$[0,k]^n$$ [ 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 $$O(n^{4} k\, \hbox {log}\, k).$$ 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 $$0,\pm 1$$ 0 , ± 1 constraint matrices, our second contribution is a more sophisticated simplex algorithm which exploits the largest absolute value $$\alpha $$ α of the entries in the constraint matrix. We show that the length of the simplex path generated by this algorithm is in $$O(n^2k\, \hbox {log}\, ({nk} \alpha ))$$ O ( n 2 k log ( nk α ) ) . In particular, if $$\alpha $$ α is bounded by a polynomial in n, k, then the length of the simplex path is in $$O(n^2k\, \hbox {log}\, (nk))$$ O ( n 2 k log ( n k ) ) . 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 $$\hbox {log}\, k$$ log k . If k is polynomially bounded in n and m, the algorithm runs in strongly polynomial time.