The Schrijver system of the length polyhedron of an interval order

TL;DR

This paper characterizes the Schrijver system for the length polyhedron of interval orders, providing a polynomial-time method to identify minimal TDI-systems and illustrating cases with exponential complexity.

Contribution

It introduces an efficient approach to construct the minimal TDI-system for the length polyhedron using key graph cycle inequalities and circulations.

Findings

Cycle inequalities form a complete system for the length polyhedron.

The minimal TDI-system can be computed in polynomial time if the key graph has polynomial cycles.

Examples show the Schrijver system can be exponentially large.

Abstract

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.