We discuss several mappings from well-known NP-hard problems to Quadratic Unconstrained Binary Optimisation problems which are treated incorrectly by Lucas. We provide counterexamples and correct the mappings. We also extend the body of QUBO formulations of NP-complete and NP-hard optimisation problems by discussing additional problems.
With the recent advancements in the field of combinatorial optimization, QUBO (quadratic unconstrained binary optimization) has become a topic of intense interest, as a problem format acceptable to several hardware heuristics such as quantum and digital annealers. Accordingly, transforming hard mathematical problems into QUBO problems is of particular importance. In this paper, we restate three problems in NP-complete and NP-hard classes as QUBO problems with proofs of equivalence. Our problems of interest are the numerical three-dimensional matching problem, the monochromatic triangle problem and the optimum facility location problem. Furthermore, we comment on the feasibility of these formulations on sophisticated QUBO solving platforms.
Quadratic Unconstrained Binary Optimization (QUBO) models are defined by quadratic expressions composed of binary variables, each taking a value of either 0 or 1. The objective of the QUBO problem is to find an optimal assignment of these binary variables that minimizes the value of the QUBO model. The QUBO problem has attracted significant research interest due to its versatility in reformulating various NP-complete problems and its potential for efficiently finding heuristic solutions through quantum annealing. This paper focuses on converting Cubic Unconstrained Binary Optimization (CUBO) problems, defined by cubic expressions, into equivalent QUBO models. Through this conversion, CUBO problems can be solved using quantum annealing. Since the important NP-hard problem 3-SAT can be straightforwardly represented as a CUBO problem, investigating the conversion from CUBO to QUBO models is a meaningful endeavor.
Quadratic Unconstrained Binary Optimization (QUBO) is a combinatorial optimization to find an optimal binary solution vector that minimizes the energy value defined by a quadratic formula of binary variables in the vector. The graph isomorphism problem is one of the applications of combinatorial optimization, and is not known to be solvable in polynomial time. The problem is also not known to be NP-complete. We propose four QUBO formulations for graph isomorphism problem instances. We also propose four QUBO formulations for induced subgraph isomorphism problem instances and two QUBO formulations for subgraph isomorphism problem instances, which are NP-complete problems. We solve QUBO instances defined by our QUBO formulations, using known QUBO solvers: Gurobi optimizer, Fixstars Amplify AE and OpenJij with SA.
The Quadratic Unconstrained Binary Optimization problem (QUBO) is the optimization of a quadratic function where the variables are binary; it is frequently used to model combinatorial problems in operational research and graph theory.QUBO is an NP-Complete problem and a challenge to exact methods.Therefore, the problem has been approached with heuristics.Here, we propose new ideas to solve QUBO with exact methods.We develop a new quadratic formulation that maps the binary problem into an equivalent continuous optimization problem with a single quadratic constraint.A further development delves into Lagrangian inspired concepts to obtain an equivalent unconstrained formulation.The case studies illustrate the benefit and limitations of both formulations.Also, the computational experiments show that problems with up to 2500 variables were solved with processing times under 500 seconds to achieve an average gap under 1%; these results also show that the approach allowed to solve, with an exact method, instances of 1500 more binary variables than the instances considered in previous papers.
We present a classical algorithm to find approximate solutions to instances of quadratic unconstrained binary optimisation. The algorithm can be seen as an analogue of quantum annealing under the restriction of a product state space, where the dynamical evolution in quantum annealing is replaced with a gradient-descent based method. This formulation is able to quickly find high-quality solutions to large-scale problem instances, and can naturally be accelerated by dedicated hardware such as graphics processing units. We benchmark our approach for large scale problem instances with tuneable hardness and planted solutions. We find that our algorithm offers a similar performance to current state of the art approaches within a comparably simple gradient-based and non-stochastic setting.