Minimizing the Weighted Number of Tardy Jobs is W[1]-hard
Klaus Heeger, Danny Hermelin
TL;DR
The paper addresses the fundamental single-machine scheduling problem $1\mid\mid\sum w_j U_j$ and analyzes its parameterized complexity through the lenses of the number of distinct processing times $p_{\#}$ and the number of distinct weights $w_{\#}$. By constructing a intricate reduction from $k$-Multicolored Clique with a digit-block encoding in base $N$, it introduces vertex-selection, large-edge, and small-edge gadgets to enforce a clique decision within a constrained set of processing times and weights, establishing W[1]-hardness for both $p_{\#}$ and $w_{\#}$. The results, combined with prior bounds, complete the parameterized complexity picture for $d_{\#}$, $p_{\#}$, and $w_{\#}$ and yield ETH-based lower bounds, illustrating near-tight running-time limits under standard hypotheses. The techniques, especially the gadget framework and digit-encoded constructions, offer a blueprint for hardness results in related scheduling problems and open avenues for further refinement of ETH gaps and potential structural containment within the W-hierarchy.
Abstract
We consider the $1||\sum w_J U_j$ problem, the problem of minimizing the weighted number of tardy jobs on a single machine. This problem is one of the most basic and fundamental problems in scheduling theory, with several different applications both in theory and practice. We prove that $1||\sum w_J U_j$ is W[1]-hard with respect to the number $p_{\#}$ of different processing times in the input, as well as with respect to the number $w_{\#}$ of different weights in the input. This, along with previous work, provides a complete picture for $1||\sum w_J U_j$ from the perspective of parameterized complexity, as well as almost tight complexity bounds for the problem under the Exponential Time Hypothesis (ETH).