Improved Bounds for Flow Shop Scheduling
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Approximations for the Two-Machine Cross-Docking Flow Shop Problem
Discrete Applied Mathematics
| Hi-index | 0.00 |
For every $\epsilon 0$, we show that the (acyclic) job shop problemcannot be approximated within ratio $O(\log^{1-\epsilon} lb)$, unless$NP$ has quasi-polynomial Las-Vegas algorithms, and where $lb$ denotesa trivial lower bound on the optimal value. This almost matches thebest known results for acyclic job shops, since an ${O}(\log^{1+\epsilon} lb)$-approximate solution can be obtained inpolynomial time for every $\epsilon0$. Recently, a PTAS was given for the job shop problem, where the numberof machines \emph{and} the number of operations per job are assumed tobe constant. Under $P\not = NP$, and when the number $\mu$ ofoperations per job is a constant, we provide an inapproximabilityresult whose value grows with $\mu$ to infinity. Moreover, we showthat the problem with two machines and the preemptive variant withthree machines have no PTAS, unless $NP$ has quasi-polynomialalgorithms. These results show that the restrictions on the number ofmachines and operations per job are necessary to obtain a PTAS.In summary, the presented results close many gaps in our understandingof the hardness of the job shop problem and resolve (negatively) several openproblems in the literature.