Multi-constraint Graph Partitioning Problems Via Recursive Bipartition Algorithm Based on Subspace Minimization Conjugate Gradient Method

TL;DR

The paper tackles multi-constraint graph/hypergraph partitioning, formulating a 0-1 quadratic program for graphs with vertex-weight and fixed-vertex constraints and solving it via a recursive bipartition framework. It relaxes the constrained problem to an unconstrained one using an equilibrium term and a trigonometrically parameterized representation, solved with an accelerated subspace minimization conjugate gradient method (ASMCG_BB), followed by hyperplane rounding and local refinements. The approach yields competitive cut-sizes on knapsack-constrained graphs and industrial VLSI hypergraph partitions while delivering substantial time savings over existing methods. This work offers a scalable and practical tool for constrained partitioning in large-scale graph/hypergraph settings such as VLSI design and related engineering problems.

Abstract

The graph partitioning problem is a well-known NP-hard problem. In this paper, we formulate a 0-1 quadratic integer programming model for the graph partitioning problem with vertex weight constraints and fixed vertex constraints, and propose a recursive bipartition algorithm based on the subspace minimization conjugate gradient method. To alleviate the difficulty of solving the model, the constrained problem is transformed into an unconstrained optimization problem using equilibrium terms, elimination methods, and trigonometric properties, and solved via an accelerated subspace minimization conjugate gradient algorithm. Initial feasible partitions are generated using a hyperplane rounding algorithm, followed by heuristic refinement strategies, including one-neighborhood and two-interchange adjustments, to iteratively improve the results. Numerical experiments on knapsack-constrained graph partitioning and industrial examples demonstrate the effectiveness and feasibility of the proposed algorithm.