Near-optimal bounded-degree spanning trees

Random costsC(i, j) are assigned to the arcs of a complete directed graph onn labeled vertices. Given the cost matrixCn =(C(i, j)), letT*k =T*k (Cn ) be the spanning tree that has minimum cost among spanning trees with in-degree less than or equal tok. Since it is NP-hard to findT*k , we instead consider an efficient algorithm that finds a near-optimal spanning treeTka. If the edge costs are independent, with a common exponential(I) distribution, then, asn → ∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(Cost(T_k^a {\text{)) = }}E(Cost(T_k^* {\text{)) + }}o\left( 1 \right).$$\end{document} Upper and lower bounds forE(Cost(T*k )) are also obtained fork≥2.