A Weight-Scaling Algorithm for f-Factors of Multigraphs

• Main Author: Gabow, H.N (Author)
• Format: Article
• Language: English
• Published: Springer 2023
• Subjects: Automata theory; Bipartite graphs; Directed graphs; F factor; f-Factor; f-Matching; F-matching; Graph; Graphic methods; Matching; Matchings; Multigraph; Multigraphs; Pattern matching; Scaling; Scalings; Simple++; Time bound
• Online Access: View Fulltext in Publisher; View in Scopus
• ISBN: 01784617 (ISSN)
• DOI: 10.1007/s00453-023-01127-x

The challenge for graph matching algorithms is to extend known time bounds for bipartite graphs to general graphs. We discuss combinatorial algorithms for finding a maximum weight f-factor on an arbitrary multigraph, for given integral weights of magnitude at most W. (An f-factor is a subgraph whose degree function is the given function f: V→ N .) For simple bipartite graphs the best-known time bound for combinatorial algorithms is O(n2/3mlognW) [Gabow and Tarjan, SIAM J Comput 18(5):1013–1036, 1989; n and m are respectively the number of vertices and edges.] A recent algorithm of Duan et al. [in: Proc. of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), 2020] for f-factors of simple general graphs comes within logarithmic factors of this bound, O~(n2/3mlogW) . The best-known bound for bipartite multigraphs is O(ΦmlogΦW) (Φ ≤ m is the size of the f-factor, Φ = ∑ v∈Vf(v) / 2). This bound is more general than the restriction to simple graphs, and is even superior on “small” simple graphs, i.e., Φ = o(n4 / 3) . We present an algorithm that comes within a logΦ factor of this bound, i.e., O(ΦlogΦmlogΦW) . The algorithm is a direct generalization of the algorithm of Gabow and Tarjan [J ACM 38(4):815–853, 1991] for the special case of ordinary matching (f≡ 1). We present that algorithm first. Our analysis is a simplified and more concrete version of Gabow and Tarjan [J ACM 38(4):815–853, 1991] and has independent interest. Furthermore the algorithm and analysis are both incorporated, without modification, into the f-factor algorithm. To extend these ideas to f-factors, the first step is “expanding” edges (i.e., replacing an edge by a length 3 alternating path). Duan et al. [in: Proc. of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), 2020] uses a one-time expansion of the entire graph. In contrast, our algorithm keeps the graph small by only expanding selected edges (edges incident to blossoms, in “I(B) sets”). Expanded edges get “compressed” back to their source when no longer needed. Expansion necessitates using an alternate graph model for blossoms (we call them “e-blossoms”). Compression requires coordinating e-blossoms with standard blossoms. © 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.