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Simple, polynomial-time, heuristic algorithms for finding approximate solutions to various polynomial complete optimization problems are analyzed with respect to their worst case behavior, measured by the ratio of the worst solution value that can be chosen by the algorithm to the optimal value. For certain problems, such as a simple form of the knapsack problem and an optimization problem based on satisfiability testing, there are algorithms for which this ratio is bounded by a constant, independent of the problem size.For a number of set covering problems, simple algorithms yield worst case ratios which can grow with the log of the problem size. And for the problem of finding the maximum clique in a graph, no algorithm has been found for which the ratio does not grow at least as fast as 0(ne), where n is the problem size and e > 0 depends on the algorithm.

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.

In this and two companion papers, we report on an extended empirical study of the simulated annealing approach to combinatorial optimization proposed by S. Kirkpatrick et al. That study investigated how best to adapt simulated annealing to particular problems and compared its performance to that of more traditional algorithms. This paper (Part I) discusses annealing and our parameterized generic implementation of it, describes how we adapted this generic algorithm to the graph partitioning problem, and reports how well it compared to standard algorithms like the Kernighan-Lin algorithm. (For sparse random graphs, it tended to outperform Kernighan-Lin as the number of vertices become large, even when its much greater running time was taken into account. It did not perform nearly so well, however, on graphs generated with a built-in geometric structure.) We also discuss how we went about optimizing our implementation, and describe the effects of changing the various annealing parameters or varying the basic annealing algorithm itself.

Using questions expressly added to the Consumer Expenditure Survey, we estimate the change in consumption expenditures caused by the 2001 federal income tax rebates and test the permanent income hypothesis. We exploit the unique, randomized timing of rebate receipt across households. Households spent 20 to 40 percent of their rebates on nondurable goods during the three-month period in which their rebates arrived, and roughly two-thirds of their rebates cumulatively during this period and the subsequent three-month period. The implied effects on aggregate consumption demand are substantial. Consistent with liquidity constraints, responses are larger for households with low liquid wealth or low income. (JEL D12, D91, E21, E62, H24, H31)

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We measure the response of household spending to the economic stimulus payments (ESPs) disbursed in mid-2008, using special questions added to the Consumer Expenditure Survey and variation arising from the randomized timing of when the payments were disbursed. We find that, on average, households spent about 12-30% (depending on the specification) of their stimulus payments on nondurable expenditures during the three-month period in which the payments were received. Further, there was also a substantial and significant increase in spending on durable goods, in particular vehicles, bringing the average total spending response to about 50-90% of the payments. Relative to research on the 2001 tax rebates, these spending responses are estimated with greater precision using the randomized timing variation. The estimated responses are substantial and significant for older, lower-income, and home-owning households. We find little evidence that the propensity to spend varies with the method of disbursement (paper check versus electronic transfer).