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Solving Subset Sum Problems using Binary Optimization with Applications in Auditing and Financial Data Analysis

David Biesner, Rafet Sifa, Christian Bauckhage

Published Jan 28, 2022• 0 citations
Publisher

Abstract

Many applications in automated auditing and the analysis and consistency check of financial documents can be formulated in part as the subset sum problem: Given a set of numbers and a target sum, find the subset of numbers that sums up to the target. The problem is NP-hard and classical solving algorithms are therefore not practical to use in real applications. We tackle the problem as a QUBO (quadratic unconstrained binary optimization) problem and show how gradient descent on Hopfield Networks reliably finds solutions for both artificial and real data. We give an outlook for the application of specialized hardware and quantum algorithms.

DOI
10.36227/techrxiv.18994160

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Many applications in automated auditing and the analysis and consistency check of financial documents can be formulated in part as the subset sum problem: Given a set of numbers and a target sum, find the subset of numbers that sums up to the target. The problem is NP-hard and classical solving algorithms are therefore not practical to use in real applications. We tackle the problem as a QUBO (quadratic unconstrained binary optimization) problem and show how gradient descent on Hopfield Networks reliably finds solutions for both artificial and real data. We give an outlook for the application of specialized hardware and quantum algorithms.
Many applications in automated auditing and the analysis and consistency check of financial documents can be formulated in part as the subset sum problem: Given a set of numbers and a target sum, find the subset of numbers that sums up to the target. The problem is NP-hard and classical solving algorithms are therefore not practical to use in real applications. We tackle the problem as a QUBO (quadratic unconstrained binary optimization) problem and show how gradient descent on Hopfield Networks reliably finds solutions for both artificial and real data. We give an outlook for the application of specialized hardware and quantum algorithms.
Many applications in automated auditing and the analysis and consistency check of financial documents can be formulated in part as the subset sum problem: Given a set of numbers and a target sum, find the subset of numbers that sums up to the target. The problem is NP-hard and classical solving algorithms are therefore not practical to use in many real applications.We tackle the problem as a QUBO (quadratic unconstrained binary optimization) problem and show how gradient descent on Hopfield Networks reliably finds solutions for both artificial and real data. We outline how this algorithm can be applied by adiabatic quantum computers (quantum annealers) and specialized hardware (field programmable gate arrays) for digital annealing and run experiments on quantum annealing hardware.
Quadratic unconstrained binary optimization (QUBO) formulas of quantum annealing (QA) algorithms are classified into four categories. QA algorithms using different QUBO formulas solve specific NP-hard problems as examples of the classification. The NP-hard problems solved are the subset sum, the vertex cover, the graph coloring, and the traveling salesperson problems. The QA algorithms are compared with their classical counterparts in terms of the quality of the solution and the time to the solution. Based on the comparison results, observations and suggestions are given for each QUBO formula category, so that proper actions can be adopted to improve the performance of QA algorithms. Compared with classical algorithms, QA algorithms are competitive in the current noisy intermediate-scale quantum (NISQ) era and beyond.
A key strategy to enhance specialized hardware accelerators, such as GPUs and FPGA, is to reduce the numerical precision in arithmetic operations, which increases processing speed and lowers latency-both crucial for real-time AI applications. In this work, we consider NP-hard Quadratic Unconstrained Binary Optimization (QUBO) problems, which arise in machine learning and beyond. We show that these problems often require high numerical precision, posing challenges for hardware solvers. We introduce a principled Branch-andBound algorithm for reducing the precision requirements of QUBO problems by utilizing the dynamic range as measure of complexity. Experiments demonstrate that our method reduces the dynamic range in subset sum, clustering, and vector quantization problems, thereby increasing their solvability on actual quantum annealers and FPGA-based digital annealers.
We introduce a variational quantum algorithm to solve unconstrained black box binary optimization problems, i.e., problems in which the objective function is given as black box. This is in contrast to the typical setting of quantum algorithms for optimization where a classical objective function is provided as a given Quadratic Unconstrained Binary Optimization problem and mapped to a sum of Pauli operators. Furthermore, we provide theoretical justification for our method based on convergence guarantees of quantum imaginary time evolution.To investigate the performance of our algorithm and its potential advantages, we tackle a challenging real-world optimization problem: feature selection . This refers to the problem of selecting a subset of relevant features to use for constructing a predictive model such as fraud detection. Optimal feature selection---when formulated in terms of a generic loss function---offers little structure on which to build classical heuristics, thus resulting primarily in ‘greedy methods’. This leaves room for (near-term) quantum algorithms to be competitive to classical state-of-the-art approaches. We apply our quantum-optimization-based feature selection algorithm, termed VarQFS, to build a predictive model for a credit risk data set with 20 and 59 input features (qubits) and train the model using quantum hardware and tensor-network-based numerical simulations, respectively. We show that the quantum method produces competitive and in certain aspects even better performance compared to traditional feature selection techniques used in today's industry.
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.