Efficient computation of a canonical form for a matrix with the generalized P-property

Abstract

We use recent results on algorithms for Markov decision problems to show that a canonical form for a matrix with the generalized P-property can be computed, in some important cases, by a strongly polynomial algorithm.Keywords Markov decision problem · Polytope · Linear programming · P-matrix Mathematics Subject Classification 90C33 (90C05, 91A15)

A Strongly Polynomial Reduction for Linear Programs over Grids

“…This reduction is slightly different from Morris' reduction [35]. We do not introduce an artificial state when formulating a Grid-LP as a discounted MDP, but lose in return the property that Dantzig's simplex method, which is not purely combinatorial, behaves the same for the two problems.…”

“…In practice, we often like to compute a proper witness (X, Y ) of a given hidden K-matrix M of type b such that the factor of the stochastic K-matrix [LY |H −1 X] in Lemma 3.17 is as small as possible. Such a problem may be formulated as the LP [35] min γ…”

“…Then t will be at most 1, and thus, the factor of the final stochastic K-matrix will be at most (1−(1−γ * ))/1 = γ * . Morris [35] proposed a solving scheme for the LP (3) that exploits the close connection to discounted MDPs. The scheme is strongly polynomial if the factor γ * is considered to be a constant.…”

“…We may think of discounted MDPs and Grid-LPs in stochastic form as the same problems. Morris [35] observed that, in terms of simplex-type methods, solving a Grid-LP corresponds to solving some discounted MDP.…”

“…For a hidden K-GLCP(M, q), where the m × n matrix M has a proper hidden K-witness (X, Y ) such that [Y |X] is a stochastic K-matrix with factor γ, Theorem 5.4 immediately gives the bound O( (m+n)n 1−γ log n 1−γ ) on the number pivot steps of principal pivoting with Dantzig's pivot rule 2 . A valid bound for arbitrary hidden K-GLCPs is obtained through Morris' reduction [35].…”