🤖 AI Summary
This paper addresses the single-machine scheduling problem with release times and deadlines, aiming to minimize maximum lateness (L_max)—a strongly NP-hard problem. To overcome its exponential complexity bottleneck, we introduce, for the first time in exact scheduling algorithms, the fixed-parameter tractability paradigm, proposing a Variable-Parameter (VP) analytical framework that strictly confines exponential dependence to the “number of emerging jobs”—a dynamic parameter significantly smaller than n. We prove the algorithm’s time complexity is O*(c^k), where k is the number of emerging jobs and c is a constant, with the dominant component being polynomial-time. Probabilistic analysis and empirical evaluation confirm that k/n asymptotically tends to zero. Our approach enables efficient exact resolution of this strongly NP-hard scheduling problem in practical scenarios. It constitutes the first fixed-parameter exact algorithm for scheduling that dynamically parameterizes based on intrinsic problem structure.
📝 Abstract
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.