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Computer Science > Data Structures and Algorithms

arXiv:2003.07589 (cs)
[Submitted on 17 Mar 2020]

Title:A Scaling Algorithm for Weighted $f$-Factors in General Graphs

Authors:Ran Duan, Haoqing He, Tianyi Zhang
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Abstract:We study the maximum weight perfect $f$-factor problem on any general simple graph $G=(V,E,w)$ with positive integral edge weights $w$, and $n=|V|$, $m=|E|$. When we have a function $f:V\rightarrow \mathbb{N}_+$ on vertices, a perfect $f$-factor is a generalized matching so that every vertex $u$ is matched to $f(u)$ different edges. The previous best algorithms on this problem have running time $O(m f(V))$ [Gabow 2018] or $\tilde{O}(W(f(V))^{2.373}))$ [Gabow and Sankowski 2013], where $W$ is the maximum edge weight, and $f(V)=\sum_{u\in V}f(u)$. In this paper, we present a scaling algorithm for this problem with running time $\tilde{O}(mn^{2/3}\log W)$. Previously this bound is only known for bipartite graphs [Gabow and Tarjan 1989]. The running time of our algorithm is independent of $f(V)$, and consequently it first breaks the $\Omega(mn)$ barrier for large $f(V)$ even for the unweighted $f$-factor problem in general graphs.
Comments: 35 pages
Subjects: Data Structures and Algorithms (cs.DS)
Cite as: arXiv:2003.07589 [cs.DS]
  (or arXiv:2003.07589v1 [cs.DS] for this version)
  https://doi.org/10.48550/arXiv.2003.07589

Submission history

From: Ran Duan [view email]
[v1] Tue, 17 Mar 2020 09:05:16 UTC (30 KB)
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