Common Optimal Supports and Bottleneck Hierarchies in Truncated Cones

Abstract

{Let \(K\subseteq\mathbb R_+^E\) be a closed convex cone and \(P=K\cap[0,1]^E\). For a linear objective with positive optimum, we study the intersection of the coordinate supports of all optimal solutions.} We prove that this common support is a transversal of a natural clutter associated with the positive directions of the cone and introduce a decreasing hierarchy of bottleneck families that stabilizes after at most \(|E|-1\) levels. In the polyhedral setting, the first level consists exactly of the sets carrying feasible upper-bound dual multipliers, while every set carrying an optimal multiplier belongs to the stabilized family. {The breadth of the join-semilattice generated under union by the zero sets of the nonzero vertices gives an objective-independent upper bound on the universal stabilization depth;} this bound is exact for simplicial truncations. Every possible depth occurs, and the \(|E|-1\) bound is sharp. Membership in the second level is co-NP-complete and fixed-parameter tractable in the size of the candidate bottleneck. Under upper-box-integrality and the integer decomposition property, the hierarchy collapses at level two, and its stabilized members are exactly the carriers of optimal upper-bound multipliers. When \(P\) is integral, the common optimal support is the union of the supports of all optimal dual solutions. Applications include bipartite matchings, maximum-order cycle subdigraphs, binary circulations, balanced hypergraph matchings, interval hypergraphs, and maximum-weight closures.