A Unified Approach to Box-Mengerian Hypergraphs

A hypergraph is called box-Mengerian if the linear system Ax ≥ 1, x ≥ 0 is box-totally dual integral (box-TDI), where A is the edge-vertex incidence matrix of the hypergraph. Because it is NP-hard in general to recognize box-Mengerian hypergraphs, a basic theme in combinatorial optimization is to identify such objects associated with various problems. In this paper, we show that the so-called equitably subpartitionable (ESP) property, first introduced by Ding and Zang (Ding, G., W. Zang. 2002. Packing cycles in graphs. J. Combin. Theory Ser. B86 381–407) in their characterization of all graphs with the min-max relation on packing and covering cycles, turns out to be even sufficient for box-Mengerian hypergraphs. We also establish several new classes of box-Mengerian hypergraphs based on ESP property. This approach is of transparent combinatorial nature and is hence fairly easy to work with.