A new arXiv preprint gives researchers a structural way to identify coordinates that are positive in every optimal solution of linear optimization over a unit-box truncation of a nonnegative cone. It links that common support to positive cone directions, a hierarchy of bottleneck tests and dual solutions.
The target is the part of a solution shared across all optima. That focus distinguishes a common-support question from the search for one best point, since different optimal solutions may exist while some coordinates remain positive in all of them.
A support problem with a set-theory answer
The first structural result turns the common support into a set-intersection problem. It is a transversal of the family formed by inclusion-minimal supports of positive cone directions. A transversal is simply a set that meets every member of that family, giving the common support a combinatorial description.
A higher-order theorem extends the same idea. If a collection of feasible points has a total objective value greater than the optimum multiplied by one less than the number of points, those points share a coordinate that is present in every optimal solution.
Where the hierarchy settles
The tests are organized into a bottleneck hierarchy with increasingly strong requirements. For a finite ground set, the infinity-level condition is reached by level one less than the number of elements, so the hierarchy stabilizes by that finite level rather than continuing indefinitely.
For every polyhedral truncation, the universal stabilization depth is bounded by the smaller of two quantities: one less than the size of the ground set, and the breadth of the zero-set semilattice generated by taking unions of zero sets from nonzero vertices. The second term brings the structure of the truncation into the depth bound.
In the simplicial case, the semilattice breadth is not merely an upper bound. It equals the universal stabilization depth, making the relationship exact for that class of truncations.
The paper also gives constructions for every integer depth from 1 through m using rational simplicial cones, where m is the number of nonzero vertices in the construction. A further construction shows that the general one-less-than-the-ground-set bound is sharp even for a simplex: the bottleneck families shrink strictly from level one through m, and level m is already the infinity-level family.
A second language: duality
The hierarchy has a parallel description through linear-programming duality. For rational polyhedral cones, the optimal dual carriers sit inside the stabilized bottleneck family. That family sits inside the first bottleneck family, which is exactly the family of feasible dual carriers. A carrier is the set of coordinates carrying an upper-bound multiplier.
Under upper-box-integrality and the integer-decomposition property, the hierarchy collapses from level two onward. Every bottleneck family from the second level onward equals the stabilized family, and these families are precisely the carriers of optimal upper-bound multipliers. If the truncation is integral, the common optimal support is the union of the supports of all optimal dual solutions.
Hard to recognize, sometimes easy to test
The framework also sets a formal complexity boundary. For the stated rational polyhedral encoding, deciding membership in the second bottleneck family is co-NP-complete, a classification that marks the recognition problem as computationally difficult in this model. The result still holds with a unit-vector objective with alpha equal to 1 and a polynomial number of homogeneous-equation and nonnegativity constraints.
There is a complementary tractable regime. If optimization over the truncation with prescribed zero coordinates can be done in polynomial time, fixed-level membership uses at most the bottleneck level multiplied by the candidate-set size in linear programs. Under that assumption, both fixed-level and infinity-level recognition are fixed-parameter tractable when candidate-bottleneck size is the parameter.
What the assumptions buy
The strongest conclusions are conditional. The dual-carrier nesting requires a rational polyhedral cone, the semilattice depth bound applies to polyhedral truncations, and the exact breadth result applies to simplicial truncations. The level-two collapse requires upper-box-integrality and the integer-decomposition property, while the dual-core identity requires an integral truncation.
The supplied record identifies the work as an arXiv version 1 preprint dated 29 September 2026. Its contribution is a structural bridge between common supports, bottleneck hierarchies and dual carriers, with integrality determining when those descriptions coincide exactly.
Paper data and sources
Original title: Common Optimal Supports and Bottleneck Hierarchies in Truncated Cones
Authors: Walid Ben-Ameur, Alessandro Maddaloni
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-09-29
DOI: Not available
Original paper · Full text