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 $C_{\max}$. Our variable parameters are the 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 procedure (in the number of variable parameters) is carried out to augment it to a complete optimal solution. In the alternative time complexity expressions, the exponential dependence is solely on the some job parameters. Applying the fixed parameter analysis to these estimations, a polynomial-time dependence is achieved. Both, the 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 $n$ converges to 0 asymptotically.

翻译：变参数（VP）分析旨在给出算法时间复杂度的精确表达式，其中指数项仅由变参数决定。变参数指具有特定属性的对象数量。本文针对一个强$NP$难问题——在单机上调度$n$个具有释放时间和截止时间的独立作业以最小化最大作业完成时间$C_{\max}$，提出了两种变参数算法：一种隐式枚举算法和一种多项式时间近似方案。我们的变参数定义为若干特殊类型作业的数量。算法首先在低阶多项式时间内构造不含这些作业的部分解，随后通过指数时间过程（依赖于变参数数量）将其扩展为完整最优解。在另一种时间复杂度表达式中，指数依赖性仅与某些作业参数相关。对这些估计应用固定参数分析，可获得多项式时间依赖关系。直观的概率估计和我们大量的实验研究均支持以下猜想：变参数总数远小于$n$，且其与$n$的比值渐近收敛于0。