On the Complexity of Minimizing Energy Consumption of Partitioning DAG Tasks

TL;DR

This work analyzes the energy-aware DAG partitioning problem (ESP-DAG), where each vertex (task) is assigned to one of $k$ heterogeneous machines to minimize the total energy of computation plus inter-machine data transfer. The authors establish a clear complexity landscape: ESP-DAG is NP-hard for all $k\ge3$, but solvable in polynomial time for $k=2$; it remains polynomial on directed-path DAGs via dynamic programming. They further study the size-bounded two-machine variant SB-ESBP-DAG and prove it is W[1]-hard with respect to the bound $\ell$, and they provide reductions (notably from Multiway Cut) to substantiate the hardness results. Collectively, the results delineate when efficient energy-aware partitioning is feasible and when it becomes intractable, offering a foundation for further research into restricted or practical instances of energy-constrained DAG scheduling.

Abstract

We study a graph partition problem where we are given a directed acyclic graph (DAG) whose vertices and arcs can be respectively regarded as tasks and dependencies among tasks. The objective of the problem is to minimize the total energy consumed for completing these tasks by assigning the tasks to k heterogeneous machines. We first show that the problem is NP-hard. Then, we present polynomial-time algorithms for two special cases where there are only two machines and where the input DAG is a directed path. Finally, we study a natural variant where there are only two machines with one of them being capable of executing a limited number of tasks. We show that this special case remains computationally hard.

Figures (4)

• Figure 1: An illustration of the ESP-DAG problem. (a) A DAG $G$. The number next to each task (vertex) represents the energy consumption required to transfer the task's output. (b) Energy consumption for executing the tasks on three machines: M1, M2, and M3. (c) and (d) are two different assignments of the tasks to the machines. The corresponding energy consumption are respectively $(1+8+2+2+4)+(2+3)=22$ (c) and $(1+1+2+2+4)+(2+3)=15$ (d).
• Figure 2: An illustration of the reduction in the proof of Theorem \ref{['thm-DAG-k-NP-hard']}. The digraph $\overrightarrow{G}$ is constructed based on the linear ordering $(t_1,x,y,z,t_2,t_3)$. Dark vertices in $G'$ are newly introduced vertices in the reduction.
• Figure 3: Construction of the digraph $G'$ from the DAG $G$ as described in the proof of Theorem \ref{['thm-p-k=2']}. Red arcs have positive infinite weights.
• Figure 4: An illustration of the reduction in the proof of Theorem \ref{['thm-bounded-wah']}. The DAG $\overrightarrow{G}$ is constructed based on the ordering $(s,x,y,z,v,t)$ on $V(G)$.

Theorems & Definitions (26)

• Theorem 1
• proof
• Corollary 1
• Theorem 2
• proof
• Claim 1
• proof
• Claim 2
• proof
• Claim 3