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arXiv · 2411.18292

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

Satoru Iwata·Hirota Kinoshita
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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.

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BibTeXRIS

Satoru Iwata, Hirota Kinoshita. 2024-11-27. A Faster Deterministic Algorithm for Mader's $\mathcal{S}$-Path Packing. https://arxiv.org/abs/2411.18292

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