Quadratic unconstrained binary optimization problem preprocessing: Theory and empirical analysis

Abstract

Abstract. The Quadratic Unconstrained Binary Optimization problem (QUBO) has become a unifying model for representing a wide range of combinatorial optimization problems, and for linking a variety of disciplines that face these problems. A new class of quantum annealing computer that maps QUBO onto a physical qubit network structure with specific size and edge density restrictions is generating a growing interest in ways to transform the underlying QUBO structure into an equivalent graph having fewer nodes and edges. In this paper we present rules for reducing the size of the QUBO matrix by identifying variables whose value at optimality can be predetermined. We verify that the reductions improve both solution quality and time to solution and, in the case of metaheuristic methods where optimal solutions cannot be guaranteed, the quality of solutions obtained within reasonable time limits.We discuss the general QUBO structural characteristics that can take advantage of these reduction techniques and perform careful experimental design and analysis to identify and quantify the specific characteristics most affecting reduction. The rules make it possible to dramatically improve solution times on a new set of problems using both the exact Cplex solver and a tabu search metaheuristic.

Logical and inequality implications for reducing the size and difficulty of quadratic unconstrained binary optimization problems

“…The data points represent average percent reductions for 16 problems, hence for the 1% edge dense 10000 variable problems about 600 variables were determined in the first pass (6%) and 2500 on average were determined in the second pass. In comparison to our earlier work (Lewis & Glover, 2016) using only 3 basic rules (R1, R2 and R3.4) embedded in a procedure called QPro, we find here that the additional rules yield about a 22% improvement in reduction overall and positively affect every problem. Most importantly the additional rules yield ten 100% reductions in contrast to 0 reductions using only rules 1, 2 and 3.4.…”

Leveraging quantum annealing for large MIMO processing in centralized radio access networks

“…11 show that tens of microseconds suffice to achieve a low enough (below 10 −3 ) FER to support high throughput communication for 60-user BPSK, 18-user QPSK, or four-user 16-QAM suffices to serve four users with the idealized median performance of Opt. QuAMax (mean Fix) achieves a similar performance with slightly smaller numbers of 8 Outlier mitigation methods for QA may address such outliers in future work [40]. users.…”

A QUBO Model for the Traveling Salesman Problem with Time Windows

“…In terms of classical approach, Boros et al presented a set of local search heuristics for Quadratic Unconstrained Binary Optimization (QUBO) problems, providing indicative simulation results on various benchmark tests [30]. The reduction of the large matrix size in QUBO was the main topic in [25] by Lewis and Glover. Glover et al showed in a step by step procedure how one can translate a problem with particular characteristics into a QUBO instance [29].…”

“…At this point, it is important to pause and confirm that the constraints in Equations (25) and (26) conform to the QUBO formulation requirements, in the sense that, after the expansion of the square, we get a sum of terms, where each term is the product of input data such as c u,v or l v and at most two binary decision variables.…”

“…A major category of optimization problems, particularly amenable to D-Wave's quantum annealing, are those that can be expressed as quadratic unconstrained binary optimization (QUBO) problems. QUBO refers to a pattern matching technique that, among other applications, can be used in machine learning and optimization, and which involves minimizing a quadratic polynomial over binary variables [21,[24][25][26][27][28][29][30]. We emphasize that QUBO is NP-hard [30].…”