Logical and Inequality Implications for Reducing the Size and Complexity of Quadratic Unconstrained Binary Optimization Problems

Abstract: The quadratic unconstrained binary optimization (QUBO) problem arises in diverse optimization applications ranging from Ising spin problems to classical problems in graph theory and binary discrete optimization. The use of preprocessing to transform the graph representing the QUBO problem into a smaller equivalent graph is important for improving solution quality and time for both exact and metaheuristic algorithms and is a step towards mapping large scale QUBO to hardware graphs used in quantum annealing computers. In an earlier paper (Lewis and Glover, 2016) a set of rules was introduced that achieved significant QUBO reductions as verified through computational testing. Here this work is extended with additional rules that provide further reductions that succeed in exactly solving 10% of the benchmark QUBO problems. An algorithm and associated data structures to efficiently implement the entire set of rules is detailed and computational experiments are reported that demonstrate their efficacy.

• The paper presents new preprocessing techniques that leverage logical deductions to reduce the size and complexity of QUBO problems.
• It utilizes single and paired variable assignments along with inequality-based reductions to cut computational time and enhance solution quality.
• Experimental results show complete variable coverage and superior performance against standard methods, paving the way for hybrid optimization strategies.

Introduction

The paper "Logical and Inequality Implications for Reducing the Size and Complexity of Quadratic Unconstrained Binary Optimization Problems" (1705.09545) presents a sophisticated approach to improving both the efficiency and capability of solving Quadratic Unconstrained Binary Optimization (QUBO) problems, which are prevalent in a variety of optimization contexts. The authors introduce new preprocessing rules intended to reduce the problem's size and complexity, facilitating more efficient solution algorithms.

Problem and Methodology

QUBO problems represent a powerful modelling framework due to their applicability across various optimization challenges such as network flows, max-cut, max-clique, and scheduling tasks. The formulation involves a binary decision-making context optimized by a quadratic objective function with constant coefficients, encapsulated within a symmetric matrix. Despite the NP-complete nature of QUBO problems, advancements in classical and quantum computing have enabled heuristic and quantum annealing methods to provide satisfactory solutions.

The authors extend previous work by introducing additional logical rules to reduce QUBO graph size, focusing on deriving optimal variable assignments and transforming complex problems into manageable forms. The methodology leverages computational efficiencies via specialized data structures and algorithmic implementations, significantly reducing computational overhead.

Implementation and Algorithmic Framework

The architecture for implementing the presented rules relies on efficient manipulation of data structures, organizing graph nodes and applying successive logical assignments or replacements. A pivotal feature is the capability to partition the graph, identifying disconnected subgraphs that can be independently optimized.

Key rules for variable assignment and graph reduction include:

1. Single Variable Assignment: Rules determine optimal or uniquely optimal values using computed bounds on components of QUBO's objective function.
2. Pairs of Variables: The methodology is expanded to simultaneous assignments of pairs, invoking more complex logical deductions.
3. Logical Inequalities: Reduction techniques identify relationships between variable pairs, supporting simplification of the optimization problem.

The pseudocode supplied in the paper for algorithm implementation delineates steps for iterative preprocessing based on logical rules, efficiently updating the QUBO matrix as variables are fixed or reassigned.

Experimental Results

Experimental validation was conducted on a set of benchmark problems typified by Chimera-like structures—graphs with clusters of densely connected nodes interspersed with sparse connectivity. The experimental setup utilized a factorial design to explore various problem parameters for their influence on reduction efficacy.

Notable outcomes include:

1. Coverage and Completeness: The preprocessing framework solved 100% of variables in multiple test cases, demonstrating its effectiveness in simplifying complex problems.
2. Time Efficiency: The combined QPRO+ and CPLEX approach significantly reduced solution times compared to standard CPLEX preprocessing, highlighting the practical advantages of logical preprocessing.
3. Quality of Solutions: Improvements in objective values were noted across test cases, underscoring the approach's ability to enhance solution quality.
4. Influence of Problem Characteristics: Factors like density and variable outlier handling were pivotal in determining preprocessing success.

Implications and Future Directions

The rules developed in this paper provide a substantial leap in preprocessing capabilities for QUBO problems, with direct implications for quantum and classical optimization. Beyond immediate applications, these methods present opportunities to incorporate additional constraint types such as cardinality and multiple choice constraints, potentially broadening applicability and increasing efficiency.

Future research could aim to refine these methods for even more structured problems or integrate them into full-stack solutions coupling preprocessing with advanced metaheuristics. Moreover, exploring dynamic implementations through real-time adjustments based on varying problem structures or changing computational environments could optimize resource allocation and operational throughput.

Conclusion

This paper delivers a comprehensive study on logical and inequality implications for reducing QUBO complexity, enhancing practical and theoretical optimization strategies. The achievements of QPRO+ in preprocessing not only advance the field of binary optimization but also set the stage for further exploration into hybrid optimization techniques, guiding emergent developments in quantum-enhanced computation and complex network analysis.