Polynomial time approximation algorithms for machine scheduling: Ten open problems

We discuss what we consider to be the ten most vexing open questions in the area of polynomial time approximation algorithms for NP-hard deterministic machine scheduling problems. We summarize what is known on these problems, we discuss related results, and we provide pointers to the literature. Prologue In the early days of scheduling, the work of the theoretical research branch mainly consisted of classifying scheduling problems into easy (i.e., polynomially solvable) and hard (i.e., NP-hard) ones. Nowadays, researchers have become interested in a better understanding and a much ner classiication of the hard problems; one very active branch of research classiies hard scheduling problems according to their approximability. In the sequel, we summarize and discuss what in our opinion are the most outstanding open questions in the approximation of scheduling problems. Throughout the paper, we use the standard three-eld scheduling notation (see e. Shmoys 41]). All scheduling problems discussed are minimization problems, where the goal is to nd a feasible schedule with minimum possible cost. We consider only scheduling problems that are NP-hard (cf. Garey & Johnson 19]). We distinguish between positive results, which establish the existence of some approximation algorithm, and negative results, which disprove the existence of certain approximation results under the assumption P 6 = NP. Positive (approximability) results. A standard way of dealing with NP-hard problems is not to search for an optimal solution, but to go for near-optimal solutions. An algorithm that returns near-optimal solutions is called an approximation algorithm; if it does this in polynomial time, then it is called a polynomial time approximation algorithm. An approximation algorithm that always returns a near-optimal solution with cost at most a factor above the optimal cost (where > 1 is some xed real number) is called a-approximation algorithm, and the value