Blocking Trails for f-factors of Multigraphs
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Blocking flows, introduced by Dinic [2] for network flow, have been used to speed up many augmenting-path type algorithms, especially matching algorithms e.g., [18, 23, 16]. We present an O(m) time algorithm for blocking trails for f-factors of general multigraphs. This improves a previous algorithm by a factor of α(m,n). This speeds up a number of efficient algorithms for f-factors, e.g., the algorithm of [11] for maximum weight f-matching improves by the aforementioned α(m,n) factor to running time O( √(ΦlogΦ) m log (Φ W) ) for Φ =∑ _v∈ V f(v), W the maximum edge weight. This time bound is within a factor √(logΦ) of the bound for bipartite multigraphs. The technical difficulty for this work stems from the fact that previous algorithms for both matching and f-matching use vertex contractions to form blossoms, but our dfs-based approach necessitates using edge contractions. As an example difficulty, edge contractions introduce a new form of blossoms we call "skew blossoms". These are configurations that must be reorganized in order to become valid blossoms.
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