Half-integrality of feasible configuration-LP solutions

Establish that, for every restricted assignment scheduling instance P, the polyhedron CLP(P,LP(P)) contains a feasible solution whose configuration variables all belong to {0, 1/2, 1}.

Background

The paper identifies the lack of a sufficiently strong structural understanding of the vertices of the configuration-LP polytope as the main obstacle to extending its gap-preserving reductions. It proposes a half-integrality property as a possible structural substitute.

The conjecture concerns the existence of a half-integral feasible solution at the minimum feasible makespan LP(P), not half-integrality of every vertex. The stronger assertion for every vertex was experimentally disproved, whereas the stated existence conjecture survived the authors’ small-scale experiments. If true, each job would occur in at most two configurations, allowing a reduction to instances in which every job is assignable to at most two machines and consequently connecting the general configuration-LP gap to the graph-balancing gap.

References

This gave motivation to further study the properties of the CLP polytope, leading us to the following conjecture, which we experimentally asserted for several small-scale instances.

For every instance $P$ of the restricted assignment problem, the polyhedron $CLP(P,LP(P))$ contains a half-integral feasible solution; that is, a vector $x$ such that $x_{i,C}\in {0, \frac{1}{2}, 1}$ for all $i \in [m]$ and $C \in C_{i}(P,LP(P))$.

— Integrality gap preserving reductions  (2609.19967 - Encz et al., 17 Sep 2026) in Conjecture 1, Section 6, “Conclusion and open problems”