Polynomial-time implementation of the makespan-minimizing matching oracle

Determine whether the makespan-minimizing valid matching selection used in the companion modification of Wang and Sitters’ rounding scheme can be implemented in polynomial time.

Background

The Wang–Sitters scheme permits any valid Shmoys–Tardos slot matching in Step 3. The companion paper instead considers a modified Step 3 that selects a valid matching minimizing the resulting makespan. Although exhaustive enumeration provides such a matching computationally, the paper explicitly leaves unresolved whether this optimization can be performed in polynomial time.

References

Can the two choices be made together in polynomial time? Theorem~\ref{thm:step3hard} decides the subproblem for a supplied $x$ and no more: Step~1 is not adversarial, and a different Step-1 output on the same instance may make Step~3 trivial. Is there a polynomial-time map taking an instance and a feasible threshold to a feasible relaxation solution together with a minimum-makespan valid matching for it? If none exists, everything here about the oracle concerns a procedure of no direct algorithmic use.

— 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”

What does completion cost at a structured $x$? The same question with Step~1 pinned: is minimum-makespan completion polynomial-time solvable when $x$ is a vertex returned by a specified polynomial-time linear-programming algorithm? And since Theorem~\ref{thm:step3hard} is weak NP-completeness, from Partition, nothing here excludes a pseudo-polynomial algorithm for job sizes with a bounded common denominator, or a fixed-parameter algorithm in the number of machines or of slots per machine.

— 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”

The companion paper instead studies a modified Step~3 that returns a makespan-minimizing valid matching; we do not know whether that selection can be made in polynomial time; it is implemented there by exhaustive enumeration, and no polynomial-time implementation is claimed for it.

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

What remains open is outside the threshold: whether a different relaxation, or a rounding rule that is not Shmoys--Tardos slot matching, moves the constant at all.

— The 11/6 supremum of the Wang-Sitters rounding scheme for graph balancing  (2609.03890 - Shavit, 3 Sep 2026) in Section “What this note establishes,” paragraph “Open”