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Conforti–Cornuéjols packing conjecture for clutters

Determine whether the Conforti–Cornuéjols packing conjecture for clutters holds; equivalently, ascertain whether ordinary and symbolic powers of edge ideals coincide for clutters, thereby establishing the equality I(C)^{(n)} = I(C)^n for all integers n ≥ 1 when the packing property is satisfied.

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Background

The packing problem of Conforti–Cornuéjols is a central conjecture in combinatorial optimization concerning clutters. Prior work established a deep link between this conjecture and commutative algebra by showing its equivalence to the equality of symbolic and ordinary powers of edge ideals.

The authors state that, to the best of their knowledge, the conjecture remains unsolved, highlighting its significance to both optimization and algebraic approaches to edge ideals.

References

A famous conjecture of Conforti–Cornu´ejols [7] from combinatorial optimization, known as the packing problem for clutters, was shown to be equivalent to the equality of ordinary and symbolic powers of edge ideals [22, Conjecture 3.10], [21, Theorem 4.6]. To the best of our knowledge the conjecture is still unsolved.

— Symbolic powers: Simis and weighted monomial ideals (2402.08833 - Méndez et al., 13 Feb 2024) in Section 6 (Symbolic powers of squarefree monomial ideals)