The triangle scheduling problem

TL;DR

This paper introduces a new triangular scheduling problem motivated by criticality levels, analyzes its complexity, and provides optimal solutions under certain conditions along with approximation schemes.

Contribution

It defines the binary tree ratio as a key measure, proves optimality of greedy algorithms for ratios up to 2, and establishes NP-hardness and approximation schemes for larger ratios.

Findings

Greedy algorithm is optimal when binary tree ratio ≤ 2.

The problem is NP-hard for ratios > 2.

A quasi polynomial time approximation scheme is provided.

Abstract

This paper introduces a novel scheduling problem, where jobs occupy a triangular shape on the time line. This problem is motivated by scheduling jobs with different criticality levels. A measure is introduced, namely the binary tree ratio. It is shown that the greedy algorithm solves the problem to optimality when the binary tree ratio of the input instance is at most 2. We also show that the problem is unary NP-hard for instances with binary tree ratio strictly larger than 2, and provide a quasi polynomial time approximation scheme (QPTAS). The approximation ratio of Greedy on general instances is shown to be between 1.5 and 1.05.