Balanced Network Flows. II. Simple Augmentation Algorithms Christian Fremuth-Paeger, Dieter Jungnickel Lehrstuhl fu ¨ r Diskrete Mathematik, Optimierung und Operations Research, University of Augsburg, D-86135 Augsburg, Germany Received 8 December 1997; accepted 29 April 1998 Abstract: In previous papers, we discussed the fundamental theory of matching problems and algo- rithms in terms of a network flow model. In this paper, we present explicit augmentation procedures which apply to the wide range of capacitated matching problems and which are highly efficient for k-factor problems and the f-factor problem.  1999 John Wiley & Sons, Inc. Networks 33: 29–41, 1999 Keywords: capacitated matching problems; network flows; balanced flow networks; skew-symmetric graphs; antisymmetrical digraphs; augmenting a matching; matching heuristics 13. INTRODUCTION OF to the known matching algorithms and which are differ- THE PSEUDOCODE FORMALISM ent. At least, the search strategy and the disjoint set union mechanism used belong to the latter class. Hence, these components should be encapsulated for better replace- Balanced networks can be considered as a network flow ments. Many applications of network flow algorithms are description of matching problems. Such networks are de- reductions of other optimization problems to network fined on skew-symmetric graphs where arc labelings satisfy flows. A certain part of an algorithm may be best per- a certain symmetry constraint. A comprehensive introduc- forming in the general problem setting, but should be tion into the terminology and theory of balanced network replaced in special cases. Hence, an inheritance mecha- flows was given in [5], where the reader can find problem nism is useful. reduction mechanisms and explicit examples. As an example for the formalism, consider the follow- Instead of repeating this general setup, we present ing class declarations: The class OBJECT represents the some object-oriented pseudocode for the methods associ- universe of all available data objects. The class SET repre- ated with balanced networks, especially for solving the sents set objects. The expression SET ( OBJECT ) indicates maximum balanced flow problem. Our pseudocode is the subclass relationship between SET and OBJECT . In object-oriented for the following reasons: a way, the declarations of Procedure 1 are dummies since We want to point out which techniques are common there are no implementations given. A set S can be allocated in computer storage and initial- Correspondence to: D. Jungnickel; e-mail: jungnickel@math. ized by the constructor S.MAKE and can be disallocated uni-augsburg.de by the destructor S.FREE. Note that these procedures The results of this paper form part of the first author’s doctoral thesis are not associated with class SET but with class OBJECT which was written under the supervision of the second author. AMS subject classification: 05C70, 90B10, 90C35 in the declaration above, since any dynamic data object  1999 John Wiley & Sons, Inc. CCC 0028-3045/99/010029-13 29 8U26 840 / 8U26$$0840 11-12-98 11:08:01 netwa W: Networks