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

Abstract: This paper 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 paper 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 $f(V)$ denote the sum of all degree constraints. Several immediate applications of the $f$-factor algorithm are given: The generalized shortest path structure of \cite{GS13}, i.e., the analog of the shortest path 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.