Compact enumeration for scheduling one machine

A Variable Parameter (VP) analysis, that we introduce here, aims to give a precise algorithm time complexity expression in which an exponent appears solely in terms of a variable parameter. A variable parameter is the number of objects with specific problem-dependent properties. Here we describe two VP-algorithms, an implicit enumeration algorithm 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 lateness. For the problem considered, a variable parameter is the number of a special kind of the so-called ``emerging'' 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 that we derive, the exponential dependence is solely on some job parameters. Applying the fixed parameter analysis to these estimations, a purely polynomial-time dependence is obtained. Both, the intuitive probabilistic estimation and an extensive experimental study support an intuitively evident conjecture that the total number of the variable parameters is far less than $n$. In particular, its ratio to $n$ asymptotically converges to 0.