In contrast to the fixed parameter analysis (FPA), in the variable parameter analysis (VPA) the value of the target problem parameter is not fixed, it rather depends on the structure of a given problem instance and tends to have a favorable asymptotic behavior when the size of the input increases. While applying the VPA to an intractable optimization problem with $n$ objects, the exponential-time dependence in enumeration of the feasible solution set is attributed solely to the variable parameter $\nu$, $\nu<<n$. As opposed to the FPA, the VPA does not imply any restriction on some problem parameters, it rather takes an advantage of a favorable nature of the problem, which permits to reduce the cost of enumeration of the solution space. Our main technical contribution is a variable parameter algorithm for a strongly $\mathsf{NP}$-hard single-machine scheduling problem to minimize maximum job lateness. The target variable parameter $\nu$ is the number of jobs with some specific characteristics, the ``emerging'' ones. The solution process is separated in two phases. At phase 1 a partial solution including $n-\nu$ non-emerging jobs is constructed in a low degree polynomial time. At phase 2 less than $\nu!$ permutations of the $\nu$ emerging jobs are considered. Each of them are incorporated into the partial schedule of phase 1. Doe to the results of an earlier conducted experimental study, $\nu/n$ varied from $1/4$ for small problem instances to $1/10$ for the largest tested problem instances, so that that the ratio becomes closer to 0 for large $n$s.
翻译:与固定参数分析(FPA)相比,在变量参数分析(VPA)中,目标问题参数值的值没有固定,而是取决于特定问题实例的结构,在输入量增加时,它往往具有一种有利的无症状行为。在将 VPA应用于一个用美元天体的棘手优化问题的同时,在列举可行解决方案集时的指数-时间依赖完全归因于变量参数$nnu$, $\nu ⁇ n。与FPA相比,目标问题参数的比值并不表示对一些问题参数有任何限制,相反,目标问题参数的比值最大的比值最大,目标比值最大值值的比值,目标比值最大的值参数值是 $n-nu\ 美元, 测试的比值最大的值, 测试值最大的值, 问题分为两个阶段。 在第一阶段, 包括 $n-n\ nu\ 美元 美元 美元 的计算出解决方案的计算成本 。