LP rounding and combinatorial algorithms for minimizing active and busy time

We consider fundamental scheduling problems motivated by energy issues. In this framework, we are given a set of jobs, each with release time, deadline and required processing length. The jobs need to be scheduled so that at most g jobs can be running on a machine at any given time. The duration for which a machine is active (i.e., "on") is referred to as its active time. The goal is to find a feasible schedule for all jobs, minimizing the total active time. When preemption is allowed at integer time points, we show that a minimal feasible schedule already yields a 3-approximation (and this bound is tight) and we further improve this to a 2-approximation via LP rounding. Our second contribution is for the non-preemptive version of this problem. However, since even asking if a feasible schedule on one machine exists is NP-hard, we allow for an unbounded number of virtual machines, each having capacity of g. This problem is known as the busy time problem in the literature and a 4-approximation is known for this problem. We develop a new combinatorial algorithm that is a 3-approximation. Furthermore, we consider the preemptive busy time problem, giving a simple and exact greedy algorithm when unbounded parallelism is allowed, that is, where g is unbounded. For arbitrary g, this yields an algorithm that is 2-approximate.