发表机构
Télécom SudParis, Institut Polytechnique de Paris（巴黎综合理工学院电信南方学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究截锥中线性规划所有最优解支撑集的公共交集，证明其为自然簇的横截，引入至多|E|-1层稳定的瓶颈层级，并给出精确上界及多种应用。
AI 中文摘要
设 \\(K\subseteq\mathbb R_+^E\\) 为闭凸锥，\\(P=K\cap[0,1]^E\\)。对于具有正最优值的线性目标，我们研究所有最优解的坐标支撑的交集。我们证明该公共支撑是与锥的正方向相关的自然簇的一个横截，并引入一个递减的瓶颈族层级，该层级在至多 \\(|E|-1\\) 层后稳定。在多面体情形下，第一层恰好由承载可行上界对偶乘子的集合组成，而每个承载最优乘子的集合都属于稳定后的族。由非零顶点的零集在并运算下生成的并半格的宽度给出了与目标无关的通用稳定深度的上界；该界对单纯截断是精确的。每种可能的深度都会出现，且 \\(|E|-1\\) 界是紧的。第二层中的成员资格是 co-NP 完全的，并且在候选瓶颈规模上具有固定参数可处理性。在满足上界整数性和整数分解性质时，层级在第二层坍缩，其稳定成员恰好是承载最优上界乘子的集合。当 \\(P\\) 为整数时，公共最优支撑是所有最优对偶解的支撑的并集。应用包括二分图匹配、最大阶循环子有向图、二元环流、平衡超图匹配、区间超图和最大权闭包。
英文摘要
{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.
