A (2+ɛ)-approximation for scheduling parallel jobs in platforms

We consider the problem of \textsc{Scheduling parallel Jobs in heterogeneous Platforms}: We are given a set $\mathcal{J}=\{1,\ldots,n\}$ of $n$ jobs, where a job $j\in\mathcal{J}$ is described by a pair $(p_j,q_j)$ of a processing time $p_j\in\mathbb{Q}_{>0}$ and the number of processors $q_j\in\mathbb{N}$ that are required to execute $j$. We are also given a set $\mathcal{B}$ of $N$ heterogeneous platforms $P_1,\ldots,P_N$, where each $P_i$ contains $m_{i}$ processors for $i\in\{1,\ldots, N\}.$ The objective is to find a schedule for the jobs in the platforms minimizing the makespan, i.e. the latest finishing time of a job. Unless $\mathcal{P}=\mathcal{NP}$ there is no approximation algorithm with absolute ratio strictly better than $2$ for the problem. We give a $(2+\epsilon)$-approximation for the problem improving the previously best known absolute approximation ratio $3$