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arXiv:2104.01709

Constraint Programming to Discover One-Flip Local Optima of Quadratic Unconstrained Binary Optimization Problems

Amit Verma, Mark W. Lewis

Published Apr 4, 2021• 2 citations
arXivHTMLPDF

Abstract

The broad applicability of Quadratic Unconstrained Binary Optimization (QUBO) constitutes a general-purpose modeling framework for combinatorial optimization problems and are a required format for gate array and quantum annealing computers. QUBO annealers as well as other solution approaches benefit from starting with a diverse set of solutions with local optimality an additional benefit. This paper presents a new method for generating a set of one-flip local optima leveraging constraint programming. Further, as demonstrated in experimental testing, analysis of the solution set allows the generation of soft constraints to help guide the optimization process.

DOI
10.48550/arxiv.2104.01709

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The broad applicability of Quadratic Unconstrained Binary Optimization (QUBO) constitutes a general-purpose modeling framework for combinatorial optimization problems and are a required format for gate array and quantum annealing computers. QUBO annealers as well as other solution approaches benefit from starting with a diverse set of solutions with local optimality an additional benefit. This paper presents a new method for generating a set of one-flip local optima leveraging constraint programming. Further, as demonstrated in experimental testing, analysis of the solution set allows the generation of soft constraints to help guide the optimization process.
The Quadratic Unconstrained Binary Optimization (QUBO) modeling and solution framework is a requirement for quantum and digital annealers. However optimality for QUBO problems of any practical size is extremely difficult to achieve. In order to incorporate the problem-specific insights, a diverse set of solutions meeting an acceptable target metric or goal is the preference in high level decision making. In this paper, we present two alternatives for goal-seeking QUBO for minimizing the deviation from a given target as well as a range of values around a target. Experimental results illustrate the efficacy of the proposed approach over Constraint Programming for quickly finding a satisficing set of solutions.
Quadratic unconstrained binary optimization (QUBO) is a classic NP-hard problem with an enormous number of applications. Local search strategy (LSS) is one of the most fundamental algorithmic concepts and has been successfully applied to a wide range of hard combinatorial optimization problems. One LSS that has gained the attention of researchers is the r-flip (also known as r-Opt) strategy. Given a binary solution with n variables, the r-flip strategy “flips” r binary variables to obtain a new solution if the changes improve the objective function. The main purpose of this paper is to develop several results for the implementation of r-flip moves in QUBO, including a necessary and sufficient condition that when a 1-flip search reaches local optimality, the number of candidates for implementation of the r-flip moves can be reduced significantly. The results of the substantial computational experiments are reported to compare an r-flip strategy-embedded algorithm and a multiple start tabu search algorithm on a set of benchmark instances and three very-large-scale QUBO instances. The r-flip strategy implemented within the algorithm makes the algorithm very efficient, leading to very high-quality solutions within a short CPU time.
Combinatorial optimization problems are of great significance in many fields, and this paper explores the application of quantum annealing-based QUBO (Quadratic Unconstrained Binary Optimization) model to combinatorial optimization problems, including the 0/1 knapsack problem, the traveler's problem, the maximum cut problem, and the graph coloring problem. The quantum annealing algorithm can effectively avoid falling into local optimal solutions by virtue of its unique quantum tunneling advantage, and shows significant advantages in solving complex combinatorial optimization problems, especially in dealing with large-scale data and multiple constraints, quantum annealing provides an efficient solution.
With the advances in customized hardware for quantum annealing and digital/CMOS Annealing, Quadratic Unconstrained Binary Optimization (QUBO) models have received growing attention in the optimization literature. Motivated by an existing general-purpose approach that derives QUBO models from binary linear programs (BLP), we propose a novel Multilevel Constraint Transformation Scheme (MLCTS) that derives QUBO models with fewer ancillary binary variables. We formulate sufficient conditions for the existence of a compact QUBO formulation (i.e., in the original BLP decision space) in terms of constraint levelness and demonstrate the flexibility and applicability of MLCTS on synthetic examples and several well-known combinatorial optimization problems, i.e., the Maximum 2-Satisfiability Problem, the Linear Ordering Problem, the Community Detection Problem, and the Maximum Independence Set Problem. For a proof-of-concept, we compare the performance of two QUBO models for the latter problem on both a general-purpose software-based solver and a hardware-based QUBO solver. The MLCTS-derived models demonstrate significantly better performance for both solvers, in particular, solving up to seven times more instances with the hardware-based approach.
Quantum annealing computers are designed to produce high-quality solutions to optimization problems that can be formulated as quadratic unconstrained binary optimization (QUBO) problems. While most of the well known NP-hard problems can easily be represented as quadratic binary problems, such formulations usually contain constraints that have to be added as penalties to the objective function in order to obtain QUBOs. In this paper, we propose a method based on finite automaton representation of the constraints for generating penalty implementations for them, which uses fewer qubits than the alternatives and is general enough to be applied to a whole class of constraints.
Quantum Annealing is an alternative type of computation in which problems are encoded in quantum Hamiltonians (energy functions) and quantum dynamics is used to find solutions (ground states of minimal energy). Quantum computers such as the D-Wave systems are indeed implementing those ideas in hardware, as well as "quantum-inspired" devices based on classical electronics such as Fujitsu’s Digital Annealing Unit. All those systems use the same modeling language: Quadratic Unconstrained Binary Optimization (QUBO). However, QUBO is a low-level language and for modeling combinatorial problems such as constraint satisfaction and constrained optimization problems, we need to introduce higher-level abstractions in order to define complex constraints.