Equitable subpartitionability does not fully describe box-tdi polyhedra

Equitable Subpartitionability Does Not Characterize Box-TDI Polyhedra

Fri 9 Oct 2026Discrete Mathematics

The paper deals with a mathematical condition called equitable subpartitionability (ESP), which was known to guarantee a certain property called box-totally dual integrality (box-TDI) in systems of linear inequalities. The authors show that while ESP ensures box-TDI, the reverse is not always true—even if you rewrite the system describing a shape in different ways. They also provide a new way to understand when a shape can be described by ESP based on geometric concepts, and they give an example of a shape that cannot be described using ESP even though it has related nice properties. This answers a question they asked in earlier work.

What this means in practice

For optimization software developers: Avoid relying solely on ESP condition when designing algorithms for integer linear programs tied to box-TDI polyhedra.
For operations research analysts: Use geometric criteria from normal cones to identify polyhedra suitable for certain integral optimization methods without assuming ESP.

A theory result. No direct application yet.

Authors

Emiliano Lancini, Francesco Pisanu

Abstract

Equitable subpartitionability (ESP) is a sufficient condition for an integral system to be box-totally dual integral. We show that the converse fails even when alternative descriptions of the same polyhedron are allowed. We provide a geometrical characterization of polyhedra that can be described by ESP systems in terms of normal cones. We construct a four-dimensional integer polytope described by a totally equimodular matrix that admits no integral ESP description. This answers negatively a previously posed question by the same authors.

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Equitable subpartitionability does not fully describe box-tdi polyhedra

Summary

What this means in practice

Authors

Abstract