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Generalizations of Total Dual Integrality

Bertrand Guenin, Levent Tunçel

TL;DR

The paper develops a unifying framework for variants of total dual integrality (TDI) by introducing total duality in a heavy set $L$ and a key tilt constraint that governs how faces of the feasible region behave under lattice shifts.A central result provides a geometric, generating‑set based criterion: a system is TD in $L$ exactly when the implicit equalities form an $L$‑generating conical set and every admissible weight admits a tilt solution in $L$, tying density arguments to polyhedral integrality.The authors derive practical sufficient conditions for TD and even totally dyadic duality (TDD) from notions of resiliency and braces, and they connect these to existing conjectures, notably Seymour's Dyadic Conjecture for ideal clutters.A precise geometric characterization for non‑degenerate TDI is given: resilience of the polyhedron together with a Hilbert cone structure on the tight constraints characterizes TDI, enriching the classical integrality criteria with a robust geometric viewpoint.Together, the results offer a versatile toolkit for certifying integrality properties of polyhedra across diverse number systems (integers, dyadics, p‑adics) and provide concrete pathways to verify or refute TD and TDI in nonstandard arithmetic settings.

Abstract

We design new tools to study variants of Total Dual Integrality. As an application, we obtain a geometric characterization of Total Dual Integrality for the case where the associated polyhedron is non-degenerate. We also give sufficient conditions for a system to be Totally Dual Dyadic, and prove new special cases of Seymour's Dyadic conjecture on ideal clutters.

Generalizations of Total Dual Integrality

TL;DR

The paper develops a unifying framework for variants of total dual integrality (TDI) by introducing total duality in a heavy set $L$ and a key tilt constraint that governs how faces of the feasible region behave under lattice shifts.A central result provides a geometric, generating‑set based criterion: a system is TD in $L$ exactly when the implicit equalities form an $L$‑generating conical set and every admissible weight admits a tilt solution in $L$, tying density arguments to polyhedral integrality.The authors derive practical sufficient conditions for TD and even totally dyadic duality (TDD) from notions of resiliency and braces, and they connect these to existing conjectures, notably Seymour's Dyadic Conjecture for ideal clutters.A precise geometric characterization for non‑degenerate TDI is given: resilience of the polyhedron together with a Hilbert cone structure on the tight constraints characterizes TDI, enriching the classical integrality criteria with a robust geometric viewpoint.Together, the results offer a versatile toolkit for certifying integrality properties of polyhedra across diverse number systems (integers, dyadics, p‑adics) and provide concrete pathways to verify or refute TD and TDI in nonstandard arithmetic settings.

Abstract

We design new tools to study variants of Total Dual Integrality. As an application, we obtain a geometric characterization of Total Dual Integrality for the case where the associated polyhedron is non-degenerate. We also give sufficient conditions for a system to be Totally Dual Dyadic, and prove new special cases of Seymour's Dyadic conjecture on ideal clutters.

Paper Structure

This paper contains 22 sections, 30 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Totally dual in $\mathbb L$ systems
  3. A geometric characterization of non-degenerate TDI systems
  4. Totally dyadic systems
  5. Organization of the paper
  6. The key result
  7. The tilt constraint
  8. The characterization
  9. Applications of \ref{['main-char']}
  10. From braces to solutions of tilt constraints
  11. From resiliency to braces
  12. Totally Dual in $\mathbb L$ - sufficient conditions
  13. Total Dual Integrality
  14. Sufficiency for TDI
  15. Necessity for TDI
  16. ...and 7 more sections

Key Result

Theorem 1.2

Let $Mx\leq b$ be a system where $\{x:Mx\leq b\}$ is a full-dimensional polyhedron. For $i=1,\ldots,k$, let $\mathcal{S}_i$ be a set of primes and suppose that $Mx\leq b$ is TD in $\mathbb L(\mathcal{S}_i)$. If $\cap_{i\in[k]}\mathcal{S}_i=\emptyset$ then $Mx\leq b$ is TD in $\mathbb L_q$ for every

Figures (2)

  • Figure 1: Hierarchy of linear systems of inequalites.
  • Figure 2: Left $Q$ and a shifted hyperplane, right $Q'$ and a shifted hyperplane.

Theorems & Definitions (74)

  • Remark 1.1
  • proof
  • Theorem 1.2
  • Proposition 1.3
  • Remark 1.4
  • proof
  • Theorem 1.5
  • Proposition 1.6
  • Remark 1.7
  • proof : Proof of \ref{['geometric']}
  • ...and 64 more