On the diameter of partition polytopes and vertex-disjoint cycle cover

We study the combinatorial diameter of partition polytopes, a special class of transportation polytopes. They are associated to partitions of a set X = {x1, . . . , xn} of items into clusters C1, . . . , Ck of prescribed sizes κ1 ≥ · · · ≥ κk. We derive upper bounds on the diameter in the form of κ1 + κ2, n − κ1 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lfloor \frac{n}{2} \rfloor}$$\end{document}. This is a direct generalization of the diameter-2 result for the Birkhoff polytope. The bounds are established using a constructive, graph-theoretical approach where we show that special sets of vertices in graphs that decompose into cycles can be covered by a set of vertex-disjoint cycles. Further, we give exact diameters for partition polytopes with k = 2 or k = 3 and prove that, for all k ≥ 4 and all κ1, κ2, there are cluster sizes κ3, . . . , κk such that the diameter of the corresponding partition polytope is at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\lceil \frac{4}{3} \kappa_2 \rceil}$$\end{document}. Finally, we provide an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${O(n(\kappa_1 + \kappa_2(\sqrt{k} - 1)))}$$\end{document} algorithm for an edge-walk connecting two given vertices of a partition polytope that also adheres to our diameter bounds.