A Fast Algorithm for the Path 2-Packing Problem

Let G be an undirected graph and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{T}=\{T_{1},\ldots,T_{k}\}$\end{document} be a collection of disjoint subsets of nodes. Nodes in T1∪⋅⋅⋅∪Tk are called terminals, other nodes are called inner. By a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{T}$\end{document}-path we mean a path P such that P connects terminals from distinct sets in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{T}$\end{document} and all internal nodes of P are inner. We study the problem of finding a maximum cardinality collection ℘ of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{T}$\end{document} -paths such that at most two paths in ℘ pass through any node. Our algorithm is purely combinatorial and has the time complexity O(mn2), where n and m denote the numbers of nodes and edges in G, respectively.