Mapping NP-hard and NP-complete optimisation problems to Quadratic Unconstrained Binary Optimisation problems

Solving Quadratic Unconstrained Binary Optimization with divide-and-conquer and quantum algorithms

Quadratic Unconstrained Binary Optimization (QUBO) is a broad class of optimization problems with many practical applications. To solve its hard instances in an exact way, known classical algorithms require exponential time and several approximate methods have been devised to reduce such cost. With the growing maturity of quantum computing, quantum algorithms have been proposed to speed up the solution by using either quantum annealers or universal quantum computers. Here we apply the divide-and-conquer approach to reduce the original problem to a collection of smaller problems whose solutions…

A Polynomial-Time Algorithm for Unconstrained Binary Quadratic Optimization

In this paper, an exact algorithm in polynomial time is developed to solve unrestricted binary quadratic programs. The computational complexity is $O\left( n^{\frac{15}{2}}\right) $, although very conservative, it is sufficient to prove that this minimization problem is in the complexity class $P$. The implementation aspects are also described in detail with a special emphasis on the transformation of the quadratic program into a linear program that can be solved in polynomial time. The algorithm was implemented in MATLAB and checked by generating five million matrices of arbitrary dimensions …

Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation

We propose a solution approach for the problem (P) of minimizing an unconstrained binary polynomial optimization problem. We call this method PQCR (Polynomial Quadratic Convex Reformulation). The resolution is based on a 3-phase method. The first phase consists in reformulating (P) into a quadratic program (QP). For this, we recursively reduce the degree of (P) to two, by use of the standard substitution of the product of two variables by a new one. We then obtain a linearly constrained binary program. In the second phase, we rewrite the quadratic objective function into an equivalent and para…

Symmetry Detection for Quadratically Constrained Quadratic Programs Using Binary Layered Graphs

Symmetry in mathematical programming may lead to a multiplicity of solutions. In nonconvex optimisation, it can negatively affect the performance of the branch-and-bound algorithm. Symmetry may induce large search trees with multiple equivalent solutions, i.e. with the same optimal value. Dealing with symmetry requires detecting and classifying it first. This work develops methods for detecting groups of symmetry in the formulation of quadratically constrained quadratic optimisation problems via adjacency matrices. Using graph theory, we transform these matrices into Binary Layered Graphs (BLG…

量子启发张量网络用于QUBO、QUDO和张量QUDO问题的k-邻居

Quantum-inspired Tensor Network for QUBO, QUDO and Tensor QUDO Problems with k-neighbors

本文提出一种新的张量网络算法，用于解决QUBO、QUDO和张量QUDO问题，基于MeLoCoToN方法，结合叠加、虚时间演化和投影测量，通过4阶张量收缩和矩阵向量乘法两种方法解决k-邻居交互问题，并与二次优化求解器比较展示方法优势。

扩展伊辛机在二次背包问题中的性能

Performance of the Extended Ising Machine for the Quadratic Knapsack Problem

本文提出利用扩展伊辛机（EIM）处理二次无约束二元优化（QUBO）难以求解的二次背包问题（QKP），通过与多种传统和先进求解器对比，证明了EIM的优越性。