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Data Structures for Weighted Matching and Extensions to b-matching and f-factors

Author:

Harold N. GabowAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 14, Issue 3

Article No.: 39, Pages 1 - 80

https://doi.org/10.1145/3183369

Published: 22 June 2018 Publication History

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Abstract

This article shows the weighted matching problem on general graphs can be solved in time O(n(m + n log n)) for n and m the number of vertices and edges, respectively. This was previously known only for bipartite graphs. The crux is a data structure for blossom creation. It uses a dynamic nearest-common-ancestor algorithm to simplify blossom steps, so they involve only back edges rather than arbitrary nontree edges.

The rest of the article presents direct extensions of Edmonds’ blossom algorithm to weighted b-matching and f-factors. Again, the time bound is the one previously known for bipartite graphs: for b-matching the time is O(min {b(V),n log n}(m + n log n)), and for f-factors the time is O(min { f(V), m log n}(m + n log n)), where b(V) and f(V) both denote the sum of all degree constraints. Several immediate applications of the f-factor algorithm are given: The generalized shortest path structure of Reference [19], i.e., the analog of the shortest-paths tree for conservative undirected graphs, is shown to be a version of the blossom structure for f-factors. This structure is found in time O(|N|(m+n log n)) for N, the set of negative edges (0 < |N| < n). A shortest T-join is found in time O(n(m + n log n)) or O(|T|(m + n log n)) when all costs are nonnegative. These bounds are all slight improvements of previously known ones, and are simply achieved by proper initialization of the f-factor algorithm.

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Index Terms

Data Structures for Weighted Matching and Extensions to b-matching and f-factors
1. Theory of computation
  1. Design and analysis of algorithms
    1. Data structures design and analysis
    2. Graph algorithms analysis

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 14, Issue 3

Special Issue on SODA’16 and Regular Papers

July 2018

393 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3233176

Editor:
Aravind Srinivasan
University of Maryland, USA

Issue’s Table of Contents

Copyright © 2018 ACM.

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Publication History

Published: 22 June 2018

Accepted: 01 January 2018

Revised: 01 January 2018

Received: 01 August 2016

Published in TALG Volume 14, Issue 3

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Gabow H(2023)A Weight-Scaling Algorithm for f-Factors of MultigraphsAlgorithmica10.1007/s00453-023-01127-x85:10(3214-3289)Online publication date: 18-May-2023
https://dl.acm.org/doi/10.1007/s00453-023-01127-x
Gabow H(2023)Blocking Trails for f-factors of MultigraphsAlgorithmica10.1007/s00453-023-01126-y85:10(3168-3213)Online publication date: 13-May-2023
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