Minmax relations for cyclically ordered digraphs

Abstract

We prove a range of minmax theorems about cycle packing and covering in digraphs whose vertices are cyclically ordered, a notion promoted by Bessy and Thomassé in their beautiful proof of the following conjecture of Gallai: the vertices of a strongly connected digraph can be covered by at most as many cycles as the stability number. The results presented here provide relations between cycle packing and covering and various objects in graphs such as stable sets, their unions, or feedback vertex- and arc-sets. They contain the results of Bessy and Thomassé with simple algorithmic proofs, including polynomial algorithms for weighted variants, classical results on posets extending Greene and Kleitman's theorem (that in turn contains Dilworth's theorem), and a common generalization of these. The most general minmax results concern the maximum number of vertices of a k-chromatic subgraph ( k ∈ N ) —as a consequence, this number is greater than or equal to the minimum of | X | + k | C | , running on subsets of vertices X and families of cycles C covering all vertices not in X, in strongly connected digraphs. This is the “circuit cover” version of a conjecture of Linial (like Gallai's conjecture is the circuit cover version of the Gallai–Milgram theorem, meaning that path partitions are replaced by circuit covers under strong connectivity); we also deduce the circuit cover version of a conjecture of Berge on path partitions; these conjectures remain open, but the proven statements also bound the maximum size of a k-chromatic subgraph, contain Gallai's conjecture, and Bessy and Thomassé's theorems. All presented minmax relations are proved using cyclic orders and a unique elementary argument based on network flows—algorithmically only shortest paths and potentials in conservative digraphs—varying the parameters of the network flow model. In this way antiblocking and blocking relations can be established, leading to a general polyhedral phenomenon—a combination of “integer decomposition, integer rounding” and “dual integrality”—that also contains the matroid partition theorem and Dilworth's theorem. We provide a common reason for all these minmax equalities to hold and some possible other ones which satisfy the same abstract properties.