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A Simple and Elegant Mathematical Formulation for the Acyclic DAG Partitioning Problem

This work addresses the NP-Hard problem of acyclic directed acyclic graph (DAG) partitioning problem. The acyclic partitioning problem is defined as partitioning the vertex set of a given directed acyclic graph into disjoint and collectively exhaustive subsets (parts). Parts are to be assigned such that the total sum of the vertex weights within each part satisfies a common upper bound and the total sum of the edge costs that connect nodes across different parts is minimized. Additionally, the quotient graph, i.e., the induced graph where all nodes that are assigned to the same part are contracted to a single node and edges of those are replaced with cumulative edges towards other nodes, is also a directed acyclic graph. That is, the quotient graph itself is also a graph that contains no cycles. Many computational and real-life applications such as in computational task scheduling, RTL simulations, scheduling of rail-rail transshipment tasks and Very Large Scale Integration (VLSI) design make use of acyclic DAG partitioning. We address the need for a simple and elegant mathematical formulation for the acyclic DAG partitioning problem that enables easier understanding, communication, implementation, and experimentation on the problem.