Improved polynomial algorithms for the shifted objective

Determine whether polynomial-time algorithms can improve on the \(3/2-1/(2N)\) approximation guarantee achieved by the phase-grid assignment algorithm for the shifted objective \(\Phi=F+p\sum_j w_j\) in single-machine total weighted tardiness with release dates and identical processing times.

Background

For the shifted objective $\Phi=F+p\sum_j w_j$, the paper develops a deterministic phase-grid assignment algorithm running in $O(N^5)$ arithmetic operations and proves an approximation ratio of at most $3/2-1/(2N)$. The paper also proves that this bound is tight for the specified phase-grid assignment algorithm. It remains unresolved whether a different polynomial-time algorithm can achieve a better guarantee for the same shifted objective.