FPTAS

AcronymDefinition
FPTASfully polynomial time approximation scheme
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One of the standard approaches to generate an FPTAS is the technique of structuring the execution of an algorithm.
[v.sup.([alpha])#.sub.k] is a set of states generated by FPTAS for the first jobs, except for job [alpha].
[v.sup.#.sub.k] is the minimum penalty generated by FPTAS for problem 1 [absolute value of [d.sub.i]] = d] BTP with a fixed straddling job [alpha].
[Z.sup.#] is the minimum penalty generated by FPTAS for problem 1 [absolute value of [d.sub.i]] = d] BTP.
The FPTAS algorithm works on the reduced state space [v.sup.([alpha])#.sub.k] instead of [v.sup.([alpha]).sub.k] and can be described as in Algorithm 2.
Therefore, the FPTAS algorithm generates the state [[t'.sup.#] + [p.sub.k], [f'.sup.#]] that may be eliminated when cleaning up the state subset.
Therefore, the FPTAS algorithm generates the state [[t'.sup.#], [f'.sup.#] + [u.sub.k] + [w.sub.k] max{0, [P.sub.sum] - [[summation].sup.k-1.sub.i=1] [p.sub.i] + [t'.sup.#] - d}] at iteration k that may be eliminated during the cleaning up procedure.
Given an arbitrary [epsilon] > 0, the FPTAS algorithm outputs a sequence with [Z.sup.#] penalty such that [Z.sup.*] [less than or equal to] (1 + [epsilon]) [Z.sup.*].
Since the FPTAS algorithm checks all jobs as straddling, then obviously job [[alpha].sup.*] will be selected in one of its iterations.
According to Lemma 9, the FPTAS algorithm generates a state [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and
It is clear that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], denote the tardiness of job [[alpha].sup.*] in the optimal and FPTAS solutions, respectively.
Finally, step 2 requires O(n) time, and the final complexity of the FPTAS algorithm is computed as O([n.sup.2]/[epsilon]).