Configuration-LP integrality gap for restricted assignment

Determine the exact integrality gap of the configuration linear program for restricted-assignment makespan minimization, whose currently known bounds are 3/2 and 11/6.

Background

The paper distinguishes the unresolved general restricted-assignment problem from the settled two-weight graph-balancing subclass. For restricted assignment, the configuration linear program is identified as the strongest widely studied relaxation, but its exact integrality gap remains between the lower bound 3/2 and the upper bound 11/6. The paper’s contribution concerns the minimum size and uniqueness of instances attaining the already-known 3/2 gap in two-weight graph balancing; it does not narrow the general restricted-assignment interval.

The unresolved problem is therefore to close the gap between the known lower and upper bounds, either by constructing restricted-assignment instances with a gap exceeding 3/2 or by proving a stronger universal upper bound.

References

Is $N(\varepsilon) = \Theta(1/\varepsilon)$? The explicit constant sharpens the question: what is $\liminf_{\varepsilon \downarrow 0}\varepsilon N(\varepsilon)$, and is the $\tfrac54$ this construction achieves optimal? A lower bound is the missing half.

— Minimum-makespan completion and vertex selection leave the Wang-Sitters constant at 11/6  (2609.10004 - Shavit, 9 Sep 2026) in Section 6, “What remains open”

For restricted assignment the gap is known only to lie in $[3/2, 11/6]$ ---the lower bound realized by Table~1, the upper bound Theorem~1.1, sharpening 's $33/17$---and closing it is a named open problem.

— A minimum witness for the 3/2 configuration-linear-program gap in two-weight graph balancing, unique at its size  (2608.13318 - Shavit, 13 Aug 2026) in Section 1, Introduction

In Section \ref{sec:conifg_lp}, we advance the current knowledge regarding the still unknown integrality gap of the restricted assignment problem with the configuration LP relaxation by using gap-preserving reductions.

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

Their argument is a local search not known to terminate in polynomial time; in their own words it ``does not give a polynomial time approximation algorithm,'' though it does let the optimum be estimated within $1.749 + \epsilon$ in polynomial time.

— The 11/6 supremum of the Wang-Sitters rounding scheme for graph balancing  (2609.03890 - Shavit, 3 Sep 2026) in Section 1, paragraph beginning “In graph balancing each job...”