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

This paper shows the weighted matching problem on general graphs can be solved in timeO(n(m+n log n)) for n andm the number of vertices and edges, respectively. This was previouslyknown only for bipartite graphs. The crux is a data structure for blossom creation. It uses adynamic nearest-common-ancestor algorithm to simplify blossom steps, so they involve onlyback edges rather than arbitrary nontree edges.The rest of the paper presents direct extensions of Edmonds’ blossom algorithm to weightedb-matching and f -factors. Again the time bound is the one previously known for bipartitegraphs: for b-matching the time is O(min{b(V ), n logn}(m+n logn)) and for f -factors the timeis O(min{f(V ),m logn}(m+ n logn)), where b(V ) and f(V ) denote the sum of all degree con-straints. Several immediate applications of the f -factor algorithm are given: The generalizedshortest path structure of [21], i.e., the analog of the shortest path tree for conservative undi-rected graphs, is shown to be a version of the blossom structure for f -factors. This structure isfound in time O(|N |(m + n logn)) for N the set of negative edges (0 < |N | < n). A shortestT -join is found in time O(n(m+n logn)), or O(|T |(m+n logn)) when all costs are nonnegative.These bounds are all slight improvements of previously known ones, and are simply achieved byproper initialization of the f -factor algorithm.