Quantum Hypergraph Partitioning

TL;DR

This paper explores quantum optimization methods for balanced hypergraph partitioning, formalizing the problem and evaluating quantum algorithms on small instances to assess effectiveness and trade-offs.

Contribution

It introduces binary optimization formulations for quantum hypergraph partitioning, highlighting which cut functions are suitable for QUBO encodings and testing on small hypergraphs.

Findings

QUBO formulations are effective for small hypergraphs

Balance-penalty weight influences cut quality and balance trade-offs

Simulated QAOA and SA perform well on small instances

Abstract

Hypergraph partitioning is a fundamental optimization problem with applications in data management and other domains involving higher-order relations. In this paper, we study balanced hypergraph partitioning from the perspective of quantum optimization. We formalize balanced $k$-way hypergraph partitioning with general hyperedge cut functions, and derive corresponding binary optimization formulations targeted at quantum optimization methods in both the two-way and multi-way settings. Our discussion highlights which cut functions admit Quadratic Unconstrained Binary Optimization (QUBO) encodings and which instead lead to higher-order binary objectives or rational forms. As a preliminary empirical validation, we focus on balanced two-way partitioning with the all-or-nothing cut on 3-uniform hypergraphs, where a direct QUBO is available, and evaluate simulated Quantum Approximate Optimization Algorithm (QAOA) and Simulated Annealing (SA) on small instances against exact solutions. The results show that the formulation is effective on small hypergraphs and that the balance-penalty weight plays a critical role in trading off cut quality and balance.