# b-Matchings and T-Joins

Part of the Algorithms and Combinatorics book series (AC, volume 21)

## Abstract

In this chapter we introduce two more combinatorial optimization problems, the Minimum Weight b-Matching Problem in Section 12.1 and the Minimum Weight T-Join Problem in Section 12.2. Both can be regarded as generalizations of the Minimum Weight Perfect Matching Problem and also include other important problems. On the other hand, both problems can be reduced to the Minimum Weight Perfect Matching Problem. They have combinatorial polynomial-time algorithms as well as polyhedral descriptions. Since in both cases the Separation Problem turns out to be solvable in polynomial time, we obtain another polynomial-time algorithm for these generalized matching problems (using the Ellipsoid Method; see Section 4.6). In fact, the Separation Problem can be reduced to finding a minimum capacity T-cut in both cases; see Sections 12.3 and 12.4. This problem, finding a minimum capacity cut δ(X) such that |XT| is odd for a specified vertex set T, can be solved with network flow techniques.

## Keywords

Undirected Graph Minimum Weight Short Path Problem Separation Problem Minimum Cardinality
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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