A Variable Parameter (VP) analysis aims to give precise time complexity expressions of algorithms with exponents appearing solely in terms of variable parameters. A variable parameter is the number of objects with specific properties. Here we describe two VP-algorithms, an implicit enumeration and a polynomial-time approximation scheme for a strongly $NP$-hard problem of scheduling $n$ independent jobs with release and due times on one machine to minimize the maximum job completion time $C_{\max}$. Thus variable parameters are amounts of some specially defined types of jobs. A partial solution without these jobs is constructed in a low degree polynomial time, and an exponential time (in the number of variable parameters) procedure is carried out to augment this solution to a complete optimal solution. We also give alternative time complexity expressions, where the exponential dependence is solely on some job parameters. Applying the fixed parameter analysis to these estimations, unexpectedly, we obtain a polynomial-time dependence. Both, intuitive probabilistic estimations and our extensive experimental study support our conjecture that the total number of the variable parameters is far less than $n$ and its ratio to $n$ asymptotically converges to 0.

翻译：变参数（VP）分析旨在给出算法精确的时间复杂度表达式，其中指数项仅出现于可变参数之中。可变参数是指具有特定属性的对象数量。本文针对强$NP$-难的调度问题（在单台机器上调度$n$个具有释放时间和截止时间的独立作业以最小化最大完工时间$C_{\max}$），提出了两种变参数算法：隐式枚举算法和多项式时间近似方案。其中变参数是某些特殊定义作业类型的数量。不含这些作业的部分解可在低阶多项式时间内构建，而通过指数时间（以变参数个数为指数）的增广过程可将其扩展为完整最优解。本文还给出了替代的时间复杂度表达式，其中指数依赖仅涉及部分作业参数。将这些估计应用于固定参数分析时，意外得到了多项式时间依赖性。直观的概率估计与广泛的实验研究均支持我们的猜想：变参数总数远小于$n$，且其与$n$的比值渐近收敛于0。