SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 485-503, March 2024.
Abstract. The investigation of combinatorial diameters of polyhedra is a classical topic in linear programming due to its connection with the possibility of an efficient pivot rule for the simplex method. We are interested in the diameters of polyhedra formed from the so-called parallel or series connection of oriented matroids. Oriented matroids are the natural way to connect representable matroid theory with the combinatorics of linear programming, and these connections are fundamental operations for the construction of more complicated matroids from elementary matroid blocks. We prove that, for polyhedra whose combinatorial diameter satisfies the Hirsch-conjecture bound regardless of the right-hand sides in a standard-form description, the diameters of their parallel or series connections remain small in the Hirsch-conjecture bound. These results are a substantial step toward devising a diameter bound for all polyhedra defined through totally unimodular matrices based on Seymour’s famous decomposition theorem. Our proof techniques and results exhibit a number of interesting features. While the parallel connection leads to a bound that adds just a constant, for the series connection one has to linearly take into account the maximal value in a specific coordinate of any vertex. Our proofs also require a careful treatment of non-revisiting edge walks in degenerate polyhedra as well as the construction of edge walks that may take a “detour" to facets that satisfy the non-revisiting conjecture when the underlying polyhedron may not.

中文翻译：

---

关于并联和串联的组合直径

SIAM 离散数学杂志，第 38 卷，第 1 期，第 485-503 页，2024 年 3 月。
摘要。多面体组合直径的研究是线性规划中的一个经典主题，因为它与单纯形法的有效主元规则的可能性有关。我们对由所谓的定向拟阵并联或串联连接形成的多面体的直径感兴趣。有向拟阵是将可表示拟阵理论与线性规划组合数学连接起来的自然方式，并且这些连接是从基本拟阵块构造更复杂拟阵的基本操作。我们证明，对于组合直径满足赫希猜想界限的多面体，无论标准形式描述中的右侧如何，其并联或串联连接的直径在赫希猜想界限中仍然很小。这些结果是朝着根据 Seymour 著名的分解定理设计通过完全幺模矩阵定义的所有多面体的直径界限迈出的重要一步。我们的证明技术和结果展示了许多有趣的特征。虽然并联连接导致仅添加一个常数的界限，但对于串联连接，必须线性地考虑任何顶点的特定坐标中的最大值。我们的证明还需要仔细处理简并多面体中的非重访边游走，以及构建边游走，这些边游走可能会“绕道”到满足非重游猜想的面，而底层多面体可能不满足。