André E. Kézdy

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.

The interval count problem, a classical question in the study of interval orders, was introduced by Ronald Graham in the 1980s. This problem asks: given an interval order $P$, what is the minimum number of distinct interval lengths required to construct an interval representation of $P$? Interval orders that can be represented with just one interval length are known as semiorders, and their characterizations are well known. However, the characterization of interval orders that require at most $k$ interval lengths -- termed $k$-count interval orders -- remains an open and challenging problem for $k\geq 2$. Our investigation into $2$-count interval orders led us naturally to consider a related problem, interval representations of permutations, which we introduce in this paper. Specifically, we characterize permutations that have a $2$-count interval representation. We prove that a permutation admits a $2$-count interval representation if and only if its longest decreasing subsequences have length at most $2$. For larger values of $k$, however, a similar characterization does not hold. There are permutations that do not permit a $3$-count interval representation despite having decreasing subsequences of length at most $3$. Characterizing $k$-count permutations remains open for $k \geq 3$. The $k$-count permutation representation problem appears to capture essential aspects of the broader problem of characterizing $k$-count interval orders. To support this connection, we apply our findings on interval representations of permutations to demonstrate that a height-$3$ interval order is $2$-count if and only if it has depth at most $2$, where the depth of an interval order refers to the length of the longest nested chain of intervals required in any interval representation of the order.

The length polyhedron $Q_P$ of an interval order $P$ is the convex hull of integral vectors representing the interval lengths in interval representations of $P$. This polyhedron has been studied by various authors, including Fishburn and Isaak. Notably, $Q_P$ forms a pointed affine cone, a property inherited from being a projection of the representation polyhedron, a structure explored also by Doignon and Pauwels. The apex of the length polyhedron corresponds to Greenough's minimal endpoint representation, which is, in fact, the length vector of the canonical interval representation -- an interval representation that minimizes the sum of the interval lengths. Building on a combinatorial perspective of canonical representations, we refine Isaak's graph-theoretical model by introducing a new and simpler directed graph. From directed cycles of this key digraph, we extract a linear system of inequalities that precisely characterizes the length polyhedron $Q_P$. This combinatorial approach also reveals the unique Hilbert basis of the polyhedron. We prove that the intersection graph of the sets corresponding to these binary rays are Berge graphs; therefore they are perfect graphs. As a result, for interval orders with bounded width, the length polyhedron has a polynomial-sized Hilbert basis, which can be computed in polynomial time. We also provide an example of interval orders with a Hilbert basis of exponential size. In a companion paper we determine the Schrijver system for the length polyhedron. We conclude with open problems.