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\documentclass[12pt]{minimal}
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\begin{document}$$[0,k]^n$$\end{document} 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(n4klogk).\documentclass[12pt]{minimal}
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\begin{document}$$O(n^{4} k\, \hbox {log}\, k).$$\end{document} 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\documentclass[12pt]{minimal}
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\begin{document}$$0,\pm 1$$\end{document} constraint matrices, our second contribution is a more sophisticated simplex algorithm which exploits the largest absolute value α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} of the entries in the constraint matrix. We show that the length of the simplex path generated by this algorithm is in O(n2klog(nkα))\documentclass[12pt]{minimal}
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\begin{document}$$O(n^2k\, \hbox {log}\, ({nk} \alpha ))$$\end{document}. In particular, if α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is bounded by a polynomial in n, k, then the length of the simplex path is in O(n2klog(nk))\documentclass[12pt]{minimal}
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\begin{document}$$O(n^2k\, \hbox {log}\, (nk))$$\end{document}. 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 logk\documentclass[12pt]{minimal}
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\begin{document}$$\hbox {log}\, k$$\end{document}. If k is polynomially bounded in n and m, the algorithm runs in strongly polynomial time.
