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Utilizing Graph Sparsification for Pre-processing in Maxcut QUBO Solver

Vorapong Suppakitpaisarn, Jin-Kao Hao

TL;DR

The paper tackles the high communication cost of solving max-cut via QUBO on cloud-based or quantum-inspired platforms by preprocessing input graphs with effective-resistance graph sparsification. It proves a $(1+ε)$-approximation guarantee and demonstrates, across multiple datasets, that sparsification can deliver a near-optimal cut (≈$0.9$-approx) while reducing edges by up to ~90%, dramatically lowering data transmission and enabling faster classical solves. Empirical results show substantial speedups for classical solvers like Gurobi and consistent performance for quantum-inspired solvers, with preliminary quantum hardware experiments suggesting shallower circuits improve robustness to noise. Overall, the method offers a practical, scalable preprocessing step that enhances cloud- and quantum-assisted max-cut solving by reducing communication overhead without sacrificing solution quality, and it opens avenues for applying sparsification to larger problem instances and diverse solvers.

Abstract

We suggest employing graph sparsification as a pre-processing step for maxcut programs using the QUBO solver. Quantum(-inspired) algorithms are recognized for their potential efficiency in handling quadratic unconstrained binary optimization (QUBO). Given that maxcut is an NP-hard problem and can be readily expressed using QUBO, it stands out as an exemplary case to demonstrate the effectiveness of quantum(-inspired) QUBO approaches. Here, the non-zero count in the QUBO matrix corresponds to the graph's edge count. Given that many quantum(-inspired) solvers operate through cloud services, transmitting data for dense graphs can be costly. By introducing the graph sparsification method, we aim to mitigate these communication costs. Experimental results on classical, quantum-inspired, and quantum solvers indicate that this approach substantially reduces communication overheads and yields an objective value close to the optimal solution.

Utilizing Graph Sparsification for Pre-processing in Maxcut QUBO Solver

TL;DR

The paper tackles the high communication cost of solving max-cut via QUBO on cloud-based or quantum-inspired platforms by preprocessing input graphs with effective-resistance graph sparsification. It proves a -approximation guarantee and demonstrates, across multiple datasets, that sparsification can deliver a near-optimal cut (≈-approx) while reducing edges by up to ~90%, dramatically lowering data transmission and enabling faster classical solves. Empirical results show substantial speedups for classical solvers like Gurobi and consistent performance for quantum-inspired solvers, with preliminary quantum hardware experiments suggesting shallower circuits improve robustness to noise. Overall, the method offers a practical, scalable preprocessing step that enhances cloud- and quantum-assisted max-cut solving by reducing communication overhead without sacrificing solution quality, and it opens avenues for applying sparsification to larger problem instances and diverse solvers.

Abstract

We suggest employing graph sparsification as a pre-processing step for maxcut programs using the QUBO solver. Quantum(-inspired) algorithms are recognized for their potential efficiency in handling quadratic unconstrained binary optimization (QUBO). Given that maxcut is an NP-hard problem and can be readily expressed using QUBO, it stands out as an exemplary case to demonstrate the effectiveness of quantum(-inspired) QUBO approaches. Here, the non-zero count in the QUBO matrix corresponds to the graph's edge count. Given that many quantum(-inspired) solvers operate through cloud services, transmitting data for dense graphs can be costly. By introducing the graph sparsification method, we aim to mitigate these communication costs. Experimental results on classical, quantum-inspired, and quantum solvers indicate that this approach substantially reduces communication overheads and yields an objective value close to the optimal solution.
Paper Structure (14 sections, 2 theorems, 4 equations, 3 figures, 2 tables)

This paper contains 14 sections, 2 theorems, 4 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Works
  4. Preliminaries
  5. Max Cut Problem
  6. QUBO Formulation for the Max Cut Problem
  7. Graph Sparsification by Effective Resistances spielman2008graph
  8. Proposed Method
  9. Experimental Results
  10. Gap in Solutions Due to Graph Sparsification
  11. Computation Time in Classical Solver
  12. Experiments on Quantum-Inspired Solvers
  13. Discussions on Results on Classical and Quantum-Inspired Solvers
  14. Conclusion and Future Works

Key Result

theorem 1

If $q = 9 |V| \cdot \log |V| / \epsilon^2$, then, for all $S \subseteq V$,

Figures (3)

  • Figure 1: Outline of our proposal
  • Figure 2: Comparisons of the optimal values derived from the original max cut instances against the objective values from the sparsified graphs for (a) be120.3.1, (b) be250.1, and (c) mannino_k487.c
  • Figure 3: Comparisons of the computation time of Gurobi when the inputs are sparsified graphs for (a) be120.3.1, (b) be250.1, and (c) mannino_k487.c

Theorems & Definitions (2)

  • theorem 1
  • theorem 2