Dantzig's pivoting rule for shortest paths, deterministic MDPs, and minimum cost to time ratio cycles

Dantzig’s pivoting rule is one of the most studied pivoting rules for the simplex algorithm. Whilethe simplex algorithm with Dantzig’s rule may require an exponential number of pivoting stepson general linear programs, and even on min cost flow problems, Orlin showed that O(mn log n)Dantzig’s pivoting steps suffice to solve shortest paths problems (we denote the number of verticesby n and the number of edges by m). Post and Ye recently showed that the simplex algorithmwith Dantzig’s rule requires only O(mn log n) pivoting steps to solve deterministic MDPs withthe same discount factor for each edge, and only O(mn log n) pivoting steps to solve deterministicMDPs with possibly a distinct discount factor for each edge. We improve Orlin’s bound for shortestpaths and Post and Ye’s bound for deterministic MDPs with the same discount factor by a factor ofn to O(mn log n), and O(mn log n), respectively. We also improve by a factor of n the bound fordeterministic MDPs with varying discounts when all discount factors are close to 1. These boundsfollow from a new proof technique showing that either, after a certain number of steps, many edgescan be excluded from participating in further policies or there is a large decrease in the value. We alsoobtain an Ω(n) lower bound on the number of Dantzig’s pivoting steps required to solve shortestpaths problems, even when m = Θ(n). Finally, we show how to reduce the problem of findinga minimum cost to time ratio cycle to the problem of finding an optimal policy for a discounteddeterministic MDP with varying discount factors that tend to 1. This gives a strongly polynomialtime algorithm for the problem that does not use Megiddo’s parametric search technique. ∗School of Computer Science, Tel Aviv University, Israel. Supported by The Danish Council for Independent Research| Natural Sciences (grant no. 12-126512). E-mail: [email protected].†School of Computer Science, Tel Aviv University, Israel. E-mail: [email protected].‡School of Computer Science, Tel Aviv University, Israel. E-mail: [email protected].