TY - GEN

T1 - Dantzig's pivoting rule for shortest paths, deterministic MDPs, and minimum cost to time ratio cycles

AU - Hansen, Thomas Dueholm

AU - Kaplan, Haim

AU - Zwick, Uri

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84902089897&partnerID=8YFLogxK

U2 - https://doi.org/10.1137/1.9781611973402.63

DO - https://doi.org/10.1137/1.9781611973402.63

M3 - منشور من مؤتمر

SN - 9781611973389

T3 - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

SP - 847

EP - 860

BT - Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

T2 - 25th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014

Y2 - 5 January 2014 through 7 January 2014

ER -