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SIAM. J. on Algebraic and Discrete Methods 7, pp. 247-257 (11 pages)
A Packing Problem You Can Almost Solve by Sitting on Your Suitcase
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Add to FacebookIn this paper, we present a novel approach for approximating solutions to the bin-packing and machine scheduling problems. In obtaining our results, we exploit a certain dual relationship that exists between these two problems.
We introduce the notion of a dual approximation algorithm, where for the bin-packing problem, the aim is to find approximate packings where at most the optimal number of bins are used, but the bins are allowed to be filled beyond their capacity. For this approach, the objective is to minimize the tardiness of the machine that finishes last. For bin-packing instances where the size of each piece is at least $( 1 /3 - \varepsilon )$ times the capacity of the bin, we give an approximation algorithm $A_\varepsilon $ that is guaranteed to produce a solution where no bin contains more than $( 1 + 3\varepsilon /2 )$ times the bin capacity. Thus we have a family of dual approximation algorithms, dependent on the problem instance, where the “closer” the instance is to belonging to a class that can be solved in polynomial-time, the better performance is guaranteed.
Using this result, we construct an approximation algorithm for the minimum makespan scheduling problem, that always finds a schedule where all jobs are completed by $\frac{5} {4}$ times the best completion time.
© 1986 Society for Industrial and Applied Mathematics
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