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. While the simplex algorithm with Dantzig's rule may require an exponential number of pivoting steps on general linear programs, and even on min cost flow problems, Orlin showed that O(mn² log n) Dantzig's pivoting steps suffice to solve shortest paths problems, where n and m are the number of vertices and edges, respectively, in the graph. Post and Ye recently showed that the simplex algorithm with Dantzig's rule requires only O(m ²n³ log² n) pivoting steps to solve deterministic MDPs with the same discount factor for each edge, and only O(m³n⁵ log² n) pivoting steps to solve deterministic MDPs with possibly a distinct discount factor for each edge. We improve Orlin's bound for shortest paths and Post and Ye's bound for deterministic MDPs with the same discount factor by a factor of n to O(mn log n). and O(m²n² log² n), respectively. We also improve by a factor of n the bound for deterministic MDPs with varying discounts when all discount factors are sufficiently close to 1. These bounds follow from a new proof technique showing that after a certain number of steps, either many edges are excluded from participating in further policies, or there is a large decrease in the value. We also obtain an Ω(n²) lower bound on the number of Dantzig's pivoting steps required to solve shortest paths problems, even when m = Θ(n). Finally, we describe a reduction from the problem of finding a minimum cost to time ratio cycle to the problem of finding an optimal policy for a discounted deterministic MDP with varying discount factors that tend to 1. This gives a strongly polynomial time algorithm for the problem that does not use Megiddo's parametric search technique.