A cost-scaling algorithm for computing the degree of determinants

Abstract

In this paper, we address computation of the degree deg {rm Det} A of Dieudonné determinant {rm Det} A ofA=∑k=1mAkxktck,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}A = \\sum_{k=1}^m A_k x_k t^{c_k},\\end{aligned}$$\\end{document} where A_k are n times n matrices over a field mathbb{K}, x_k are noncommutative variables,t is a variable commuting with x_k, c_k are integers,and the degree is considered for t.This problem generalizes noncommutative Edmonds' problem andfundamental combinatorial optimization problemsincluding the weighted linear matroid intersection problem.It was shown that deg {rm Det} A is obtained bya discrete convex optimization on a Euclidean building (Hirai 2019).We extend this framework by incorporating a cost-scaling techniqueand show that deg {rm Det} Acan be computed in time polynomial of n,m,log_2 C, where C:= max_k |c_k|.We give a polyhedral interpretation of deg {rm Det},which says that deg {rm Det}A is given by linear optimizationover an integral polytope with respect to objective vector c = (c_k).Based on it, we show that our algorithm becomes a strongly polynomial one.We also apply our result to an algebraic combinatorial optimization problemarising from a symbolic matrix having 2 times 2-submatrix structure.