The length polyhedron of an interval order $P$ is the convex hull of integer vectors representing the interval lengths in possible interval representations of $P$ in which all intervals have integer endpoints. This polyhedron is an integral translation of a polyhedral cone, with its apex corresponding to the canonical interval representation of $P$ (also known as the minimal endpoint representation). In earlier work, we introduced an arc-weighted directed graph model, termed the key graph, inspired by this canonical representation. We showed that cycles in the key graph correspond, via Fourier-Motzkin elimination, to inequalities that describe supporting hyperplanes of the length polyhedron. These cycle inequalities derived from the key graph form a complete system of linear inequalities defining the length polyhedron. By applying a theorem due to Cook, we establish here that this system of inequalities is totally dual integral (TDI). Leveraging circulations, total dual integrality, and the special structure of the key graph, our main theorem demonstrates that a cycle inequality is a positive linear combination of other cycle inequalities if and only if it is a positive integral linear combination of smaller cycle inequalities (where `smaller' here refers a natural weak ordering among these cycle inequalities). This yields an efficient method to remove redundant cycle inequalities and ultimately construct the unique minimal TDI-system, also known as the Schrijver system, for the length polyhedron. Notably, if the key graph contains a polynomial number of cycles, this gives a polynomial-time algorithm to compute the Schrijver system for the length polyhedron. Lastly, we provide examples of interval orders where the Schrijver system has an exponential size.

翻译：区间序$P$的长度多面体是整数向量的凸包，这些向量代表了$P$在可能的区间表示（其中所有区间具有整数端点）中的区间长度。该多面体是多面体锥的整数平移，其顶点对应于$P$的规范区间表示（也称为最小端点表示）。在先前的工作中，我们受此规范表示的启发，引入了一种弧加权的有向图模型，称为关键图。我们证明了关键图中的环通过Fourier-Motzkin消去法对应于描述长度多面体支撑超平面的不等式。由关键图导出的这些环不等式构成了定义长度多面体的完整线性不等式系统。通过应用Cook的一个定理，我们在此确立了该不等式系统是完全对偶整数（TDI）的。利用环流、完全对偶整数性以及关键图的特殊结构，我们的主要定理证明：一个环不等式是其他环不等式的正线性组合，当且仅当它是更小环不等式（此处“更小”指这些环不等式之间的自然弱序）的正整数线性组合。这提供了一种有效的方法来移除冗余的环不等式，并最终为长度多面体构建唯一的最小TDI系统，也称为Schrijver系统。值得注意的是，如果关键图包含多项式数量的环，这将给出一个计算长度多面体Schrijver系统的多项式时间算法。最后，我们提供了Schrijver系统具有指数规模的区间序示例。