Compact enumeration for scheduling one machine

TL;DR

This paper introduces variable parameter analysis for scheduling on a single machine, providing algorithms with complexity depending only on specific problem parameters, and demonstrates the practical efficiency of these methods.

Contribution

It presents two VP-algorithms for a strongly NP-hard scheduling problem, including an implicit enumeration and a polynomial-time approximation scheme, with complexity depending on problem-specific parameters.

Findings

Partial solutions are polynomial-time computable without special jobs.

Exponential complexity depends only on a small set of variable parameters.

Experimental results suggest variable parameters are significantly fewer than total jobs.

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.