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

TL;DR

This paper introduces a faster deterministic algorithm for Mader's $\

Contribution

It presents a new $O(mnk)$ time algorithm for Mader's $\

Findings

Runs in $O(mnk)$ time, improving previous bounds.

Utilizes a novel approach combining known reductions and augmenting-path algorithms.

Provides a more efficient solution for a generalization of matching and disjoint path problems.

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 $O(mn^{\omega})$ for deterministic algorithms, where $\omega\ge2$ denotes the matrix multiplication exponent.