Increasing interest has recently been shown in analyzing the worst-case behavior of local search algorithms. In particular, the quality of local optima and the time needed to find the local optima by the simplest form of local search has been studied. This paper deals with worst-case performance of local search algorithms for makespan minimization on parallel machines. We analyze the quality of the local optima obtained by iterative improvement over the jump, swap, multi-exchange, and the newly defined push neighborhoods. Finally, for the jump neighborhood we provide bounds on the number of local search steps required to find a local optimum.

We provide several non-approximability results for deterministic scheduling problems whose objective is to minimize the total job completion time. Unless 𝒫 =𝒩𝒫, none of the problems under consideration can be approximated in polynomial time within arbitrarily good precision. Most of our results are derived by APX-hardness proofs. We show that, whereas scheduling on unrelated machines with unit weights is polynomially solvable, the problem becomes APX-hard if release dates or weights are added. We further show APX-hardness for scheduling in flow shops, job shops, and open shops. We also investigate the problems of scheduling on parallel machines with precedence constraints and unit processing times, and two variants of the latter problem with unit communication delays; for these problems we provide lower bounds on the worst-case behavior of any polynomial-time approximation algorithm through the gap-reduction technique.