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NP-complete problems form an extensive equivalence class of combinatorial problems for which no nonenumerative algorithms are known. Our first result shows that determining a shortest-length schedule in an m-machine flowshop is NP-complete for m ≥ 3. (For m = 2, there is an efficient algorithm for finding such schedules.) The second result shows that determining a minimum mean-flow-time schedule in an m-machine flowshop is NP-complete for every m ≥ 2. Finally we show that the shortest-length schedule problem for an m-machine jobshop is NP-complete for every m ≥ 2. Our results are strong in that they hold whether the problem size is measured by number of tasks, number of bits required to express the task lengths, or by the sum of the task lengths.

We consider economical representations for the path information in a directed graph. A directed graph G is said to be a transitive reduction of the directed graph G provided that (i) G has a directed path from vertex u to vertex v if and only if G has a directed path from vertex u to vertex v, and (ii) there is no graph with fewer arcs than G satisfying condition (i). Though directed graphs with cycles may have more than one such representation, we select a natural canonical representative as the transitive reduction for such graphs. It is shown that the time complexity of the best algorithm for finding the transitive reduction of a graph is the same as the time to compute the transitive closure of a graph or to perform Boolean matrix multiplication.

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The following abstract problem models several practical problems in computer science and operations research: given a list L of real numbers between 0 and 1, place the elements of L into a minimum number L* of "bins" so that no bin contains numbers whose sum exceeds 1. Motivated by the likelihood that an excessive amount of computation will be required by any algorithm which actually determines an optimal placement, we examine the performance of a number of simple algorithms which obtain "good" placements. The first-fit algarithm places each number, in succession, into the first bin in which it fits. The best-fit algorithm places each number, in succession, into the most nearly full bin in which it fits. We show that neither the first-fit nor the best-fit algorithm will ever use more than 17. a-6. + 2 bins. Furthermore, we outline a proof that, if L is in decreasing order, then neither algorithm will use more than L* + 4 bins. Examples are given to show that both upper bounds are essentially the best possible. Similar results are obtained when the list L contains no numbers larger than < 1.1. Introduction. Recent results in complexity theory 3], [10] indicate that many combinatorial optimization problems may be effectively impossible to solve, in the sense that a prohibitive amount of computation is required to construct optimal solutions for all but very small cases. In order to solve such problems in practice, one is forced to use approximate, heuristic algorithms which hopefully compute "good" solutions in an acceptable amount of computing time. Thus, instead of seeking the fastest algorithm from the set of exact optimization algorithms, one seeks the best approximation algorithm from the set of "sufficiently fast" algorithms. Unfortunately it is usually difficult to evaluate and compare the performance of heuristic algorithms, other than by running them on large problem

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