A Fast Algorithm for Path 2-Packing Problem

Abstract

Let G = (V _G , E _G ) be an undirected graph, $\mathcal{T} = \{T_1, \ldots, T_k\}$ be a collection of disjoint subsets of nodes. Nodes in T ₁ ∪ ... ∪ T _k are called terminals, other nodes are called inner. By a $\mathcal{T}$-path P we mean an undirected path such that P connects terminals from distinct sets in $\mathcal{T}$ and all internal nodes of P are inner. We study the problem of finding a maximum cardinality collection $\mathcal{P}$ of $\mathcal{T}$-paths such that at most two paths in $\mathcal{P}$ pass through any node v ∈ V _G . Our algorithm is purely combinatorial and achieves the time bound of O(mn ²), where n : = |V _G |, m : = |E _G |.