Advances in packing directed joins

Several of the finest unclaimed prizes in directed graph theory involve the packing of directed joins. One difficulty in claiming these prizes is that the broad conjecture posed by Edmonds and Giles, whether the maximum number of disjoint directed joins equals the smallest weight of a directed cut in every weighted directed graph, is not true in general. This is despite the fact that the conjecture is true in several special cases, and is also true if the roles of directed joins and directed cuts are reversed. Another difficulty is that the known minimal counterexamples, one found by Schrijver and two found by Cornuéjols and Guenin during a computer search, are mysterious in nature. We dispel some of this mystery by providing a framework for understanding the known counterexamples. We then use this framework to construct several new counterexamples, and to prove that every “smallest” minimal counterexample has now been found. Finally, we temper these advances by introducing an NP-completeness result for a more difficult problem.