Seymour’s Dyadic Conjecture for ideal clutters

Prove that for every ideal clutter C, the set-covering system T(C)x≥1, x≥0 is totally dual dyadic (TDD).

Background

An ideal clutter is one for which the set-covering polyhedron {x≥0:T(C)x≥1} is integral. The paper defines total dual dyadicity as the property that every admissible integral objective has an optimal dual solution whose entries are dyadic rationals.

The authors identify the statement as Seymour’s Dyadic conjecture and list several classes of ideal clutters for which it is known. They then prove additional sufficient conditions and special cases, indicating that the general conjecture remains unresolved.

References

Seymour \S 79.3e, proposed the following conjecture, % sufficient {\em Let $$ be an ideal clutter, then $T()x\geq1,x\geq0$ is TDD.}

— Generalizations of Total Dual Integrality  (2503.07925 - Guenin et al., 11 Mar 2025) in Section 1, subsection “Totally dyadic systems,” paragraph beginning “The study of TDD systems was initially motivated by a conjecture on ideal clutters”