A Faster Deterministic Algorithm for Mader's $\mathcal{S}$-Path Packing

Abstract

Given an undirected graph $G=(V,E)$ with a set of terminals $T\subseteq V$ partitioned into a family $\mathcal{S}$ of disjoint blocks, find the maximum number of vertex-disjoint paths whose endpoints belong to two distinct blocks while no other internal vertex is a terminal. This problem is called Mader's $\mathcal{S}$-path packing. It has been of remarkable interest as a common generalization of the non-bipartite matching and vertex-disjoint $s\text{-}t$ paths problem. This paper presents a new deterministic algorithm for this problem via known reduction to linear matroid parity. The algorithm utilizes the augmenting-path algorithm of Gabow and Stallmann (1986), while replacing costly matrix operations between augmentation steps with a faster algorithm that exploits the original $\mathcal{S}$-path packing instance. The proposed algorithm runs in $O(mnk)$ time, where $n=|V|$, $m=|E|$, and $k=|T|\le n$. This improves on the previous best bound $O(m{n}^{\ddot{I}}‰)$ for deterministic algorithms, where $\ddot{I}‰\ge 2$ denotes the matrix multiplication exponent.