Improved Decomposition Bounds for Partition Polytopes and Odd-Covers

Abstract: The assignments of a set of $m$ items into $n$ clusters of prescribed sizes $\kappa_1,\dots,\kappa_n$ can be encoded as the vertices of the \emph{partition polytope} $\mathrm{PP}(\kappa_1,\dots,\kappa_n)$. We prove that, if $K = \max{\kappa_1,\dots,\kappa_n}$, then the combinatorial diameter of $\mathrm{PP}(\kappa_1,\dots,\kappa_n)$ is at most $\lceil 3K/2\rceil$. This improves the previously known upper bound of $2K$. A \emph{cycle} (resp.~\emph{path}) \emph{odd-cover} of a graph $G$ is a set $\mathcal{H}$ of cycles (resp.~paths) such that $G$ is the symmetric difference of the graphs in $\mathcal{H}$. We prove that every Eulerian graph $G$ with maximum degree $\Delta$ admits a cycle odd-cover and a path odd-cover, each of size at most $\lceil 3\Delta/4\rceil$. This improves the previously known upper bound of $\Delta$. The two proofs share many similarities and are both based on the proof of Akiyama, Exoo, and Harary that every graph with maximum degree 4 has linear arboricity at most 3.