an:07179748
Zbl 1439.90018
Abdi, Ahmad; Fukasawa, Ricardo; Sanit??, Laura
Opposite elements in clutters
EN
Math. Oper. Res. 43, No. 2, 428-459 (2018).
00447881
2018
j
90B10 90C27
ideal clutters; set covering polyhedron; packing property; replication conjecture; arborescences
Summary: Let \(E\) be a finite set of elements, and let \(L\) be a clutter over ground set \(E\). We say distinct elements \(e\), \(f\) are opposite if every member and every minimal cover of \(L\) contains at most one of \(e\), \(f\). In this paper, we investigate opposite elements and reveal a rich theory underlying such a seemingly simple restriction. The clutter \(C\) obtained from \(L\) after identifying some opposite elements is called an identification of \(L\); inversely, \(L\) is called a split of \(C\). We will show that splitting preserves three clutter properties, i.e., idealness, the max-flow min-cut property, and the packing property. We will also display several natural examples in which a clutter does not have these properties but a split of them does. We will develop tools for recognizing when splitting is not a useful operation, and as well, we will characterize when identification preserves the three mentioned properties. We will also make connections to spanning arborescences, Steiner trees, comparability graphs, degenerate projective planes, binary clutters, matroids, as well as the results of Menger, Ford and Fulkerson, the replication conjecture, and a conjecture on ideal, minimally nonpacking clutters.