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Three, four, five, six, or the complexity of scheduling with communication delays. (English) Zbl 0816.90083
Summary: A set of unit-time tasks has to be processed on identical parallel processors subject to precedence constraints and unit-time communication delays; does there exist a schedule of length at most \(d\)? The problem has two variants, depending on whether the number of processors is restrictively small or not. For the first variant the question can be answered in polynomial time for \(d= 3\) and is NP-complete for \(d= 4\). The second variant is solvable in polynomial time for \(d= 5\) and NP-complete for \(d= 6\). As a consequence, neither of the corresponding optimization problems has a polynomial approximation scheme, unless \(P= \text{NP}\).

MSC:
90B35 Deterministic scheduling theory in operations research
65Y05 Parallel numerical computation
90C60 Abstract computational complexity for mathematical programming problems
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