Isomorphic coupled-task scheduling problem with compatibility constraints on a single processor

TL;DR

This paper studies a generalized coupled-task scheduling problem with compatibility constraints on a single processor, proving its complexity, relating it to graph problems, and providing approximation algorithms with ratios as low as 1.37.

Contribution

It introduces a new generalized scheduling problem, proves its NP-completeness, relates it to the min-DCP graph problem, and develops approximation algorithms with improved ratios.

Findings

The problem is NP-complete.

A (3a+2b)/(2a+2b)-approximation algorithm is proposed.

A polynomial-time algorithm reduces the ratio to 1.37.

Abstract

The problem presented in this paper is a generalization of the usual coupled-tasks scheduling problem in presence of compatibility constraints. The reason behind this study is the data acquisition problem for a submarine torpedo. We investigate a particular configuration for coupled tasks (any task is divided into two sub-tasks separated by an idle time), in which the idle time of a coupled task is equal to the sum of durations of its two sub-tasks. We prove -completeness of the minimization of the schedule length, we show that finding a solution to our problem amounts to solving a graph problem, which in itself is close to the minimum-disjoint-path cover (min-DCP) problem. We design a (3a+2b)/(2a+2b)-approximation, where a and b (the processing time of the two sub-tasks) are two input data such as a>b>0, and that leads to a ratio between 3/2 and 5/4. Using a polynomial-time algorithm developed for some class of graph of min-DCP, we show that the ratio decreases to 1.37 .