Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The length polyhedron of an interval order

Csaba Biró, André E. Kézdy, Jenő Lehel

Abstract

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.

The length polyhedron of an interval order

Abstract

The length polyhedron of an interval order is the convex hull of integral vectors representing the interval lengths in interval representations of . This polyhedron has been studied by various authors, including Fishburn and Isaak. Notably, 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 . 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.

Paper Structure

This paper contains 13 sections, 23 theorems, 16 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Preliminaries
  4. Minimal representations of an interval order
  5. Slack
  6. The full representation polyhedron
  7. The location polyhedron
  8. The length polyhedron
  9. A unique minimal Hilbert basis
  10. How to compute the minimal Hilbert basis for $Q^*_P$
  11. Hilbert extender graphs are perfect
  12. Exponential-sized Hilbert basis
  13. Conclusions

Key Result

Theorem 1.1

A $\{0,\pm 1\}$-matrix $A$ is totally unimodular if and only if every submatrix of $A$ has an equitable bicoloring.

Figures (10)

  • Figure 1: Collapsing an interval $[i,j]$
  • Figure 2: Hasse diagram and the canonical representation of the interval order $P=[0,1,0,1,3,1,3]$ that has magnitude $m=5$
  • Figure 3: $x$ is minimal in $P$ and $-\ell_x> 0$
  • Figure 4: $\ell^*_y=r_x^*+1$ and $-\ell_y+r_x > -1$
  • Figure 5: $\ell^*_x=r_y^*$ and $\ell_x-r_y> 0$,
  • ...and 5 more figures

Theorems & Definitions (44)

  • Theorem 1.1: Ghouila-Houri G-H (1962)
  • Theorem 1.2: Hoffman and Kruskal HoffmanKruskal(1956)
  • Lemma 2.1: Greenough Greenough
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • ...and 34 more