Fast Approximation for Scheduling One Machine

Abstract

We propose an approximation algorithm for scheduling jobs with release and delivery times on a single machine with the objective to minimize the makespan. The algorithm is based on an implicit enumeration of the set of complete solutions in a search tree. By analyzing specific structural properties of the solutions created in each branch of the solution tree, a certain approximation factor of each solution from that branch is calculated. Our algorithm guarantees approximation factor 1+1/κ for a generated solution if there are κ jobs with a specified property in that solution (typically, the longer the length of the path from the root to the node representing that solution in the solution tree, the larger the κ value). We have carried out an extensive computational study to verify the practical performance of our algorithm and the effectiveness of the approximation factor provided by us. While our problem instances are generated randomly, we have discarded a considerable number of the instances, ones which were already solved optimally by the earlier known dominance rules. For the vast majority of the tested problem instances, within a short running time of our algorithm parameter κ becomes sufficiently large so that the approximation factor which the algorithm guarantees becomes better than that provided by the earlier known approximation algorithms.

Compact enumeration for scheduling one machine

“…Based on the computational experiments from [1], the asymptotic behavior of the parameter ν implied certain "practical relationship" between this parameters and the input length O(n). In particular, as we have mentioned in the introduction, ν turns out to be a fraction of n and asymptotically, this fraction becomes less.…”

“…For small problem instances, in average, ν was about n/4. However, for larger instances up to 10000 jobs it became less than n/10 (Table 1 in [1]).…”

“…Note that the worst-case exponential dependence on ν and κ in the proposed algorithms is expressed explicitly without using the big O-notation, i.e., no constant is hidden in the exponent (such "clean" computational complexity with respect to parameters ν and κ is not only beneficial in the complexity analysis but also it is helpful in the analysis of the inherent properties of the problem and in finding approximation solutions in polynomial time). Although an accurate theoretical estimation of the values of parameters ν and κ is difficult, a recently conducted experimental study for about thousand randomly generated instances of problem 1|r j , q j |C max in [1] has revealed, in particular, the values that parameter ν takes in practice. As it can be seen from the experimental study reported in [1], for some amount of the tested instances, independently of their size, ν turned out to be a relatively small integer number.…”

“…Although an accurate theoretical estimation of the values of parameters ν and κ is difficult, a recently conducted experimental study for about thousand randomly generated instances of problem 1|r j , q j |C max in [1] has revealed, in particular, the values that parameter ν takes in practice. As it can be seen from the experimental study reported in [1], for some amount of the tested instances, independently of their size, ν turned out to be a relatively small integer number. For small problem instances, in average, ν was about n/4.…”

“…They described two PTASs with time complexities O(16 k (nk) 3+4k ) and O(n log n + n(4k) 8k 2 +8k+2 ). Mastrolilli [10] has proposed another approximation scheme for a more general identical processor setting and showed that his algorithm has an aggregated time complexity O(nk O (1)…”