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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Analysis and Acceleration of the Quadratic Knapsack Problem on an Ising Machine

Matthieu Parizy, Nozomu Togawa

Published Jul 7, 2021• 16 citations
Publisher

Abstract

The binary quadratic knapsack problem (QKP) aims at optimizing a quadratic cost function within a single knapsack. Its applications and difficulty make it appealing for various industrial fields. In this paper we present an efficient strategy to solve the problem by modeling it as an Ising spin model using an Ising machine to search for its ground state which translates to the optimal solution of the problem. Secondly, in order to facilitate the search, we propose a novel technique to visualize the landscape of the search and demonstrate how difficult it is to solve QKP on an Ising machine. Finally, we propose two software solution improvement algorithms to efficiently solve QKP on an Ising machine.

DOI
10.1587/transfun.2020kep0007

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IEEE Access•Jan 1, 2021•56 citations
The binary quadratic knapsack problem (QKP) aims at optimizing a quadratic cost function within a single knapsack. Its applications and difficulty make it appealing for various industrial fields. In this paper we present an efficient strategy to solve the problem by modeling it as an Ising spin model using an Ising machine to search for its ground state which translates to the optimal solution of the problem. Secondly, in order to facilitate the search, we propose a novel technique to visualize the landscape of the search and demonstrate how difficult it is to solve QKP on an Ising machine. Finally, we propose two software solution improvement algorithms to efficiently solve QKP on an Ising machine.
Combinatorial optimization has wide applications from industry to natural science. Ising machines bring an emerging computing paradigm for efficiently solving a combinatorial optimization problem by searching a ground state of a given Ising model. Current cutting-edge Ising machines achieve fast sampling of near-optimal solutions of the max-cut problem. However, for problems with additional constraint conditions, their advantages have been hardly shown due to difficulties in handling the constraints. In this work, we focus on benchmarks of Ising machines on the quadratic knapsack problem (QKP). To bring out their practical performance, we propose fast two-stage post-processing for Ising machines, which makes handling the constraint easier. Simulation based on simulated annealing shows that the proposed method substantially improves the solving performance of Ising machines and the improvement is robust to a choice of encoding of the constraint condition. Through evaluation using an Ising machine called Amplify Annealing Engine, the proposed method is shown to dramatically improve its solving performance on the QKP. These results are a crucial step toward showing advantages of Ising machines on practical problems involving various constraint conditions.
The extended Ising machine (EIM) enhances conventional Ising models, which handle only binary quadratic forms by allowing constraints through real-valued dependent variables. We address the quadratic knapsack problem (QKP), hard to solve using Ising machines when formulated as a quadratic unconstrained binary optimization (QUBO). We demonstrated the EIM's superiority by comparing it with the conventional Ising model-based approach, a commercial exact solver, and a state-of-the-art heuristic solver for QKP.
Ising machines (IM) are physics-inspired alternatives to von Neumann architectures for solving hard optimization tasks. By mapping binary variables to coupled Ising spins, IMs can naturally solve unconstrained combinatorial optimization problems such as finding maximum cuts in graphs. However, despite their importance in practical applications, constrained problems remain challenging to solve for IMs that require large quadratic energy penalties to ensure the correspondence between energy ground states and constrained optimal solutions. To relax this requirement, we propose a self-adaptive IM that iteratively shapes its energy landscape using a Lagrange relaxation of constraints and avoids prior tuning of penalties. Using a probabilistic-bit (p-bit) IM emulated in software, we benchmark our algorithm with multidimensional knapsack problems (MKP) and quadratic knapsack problems (QKP), the latter being an Ising problem with linear constraints. For QKP with 300 variables, the proposed algorithm finds better solutions than state-of-the-art IMs such as Fujitsu's Digital Annealer and requires 7,500x fewer samples. Our results show that adapting the energy landscape during the search can speed up IMs for constrained optimization.
The Quadratic Knapsack Problem (QKP) deals with maximizing a quadratic objective function subject to given constraints on the capacity of the Knapsack. We assume all coefficients to be non-negative and all variables to be binary. Solution to QKP generalizes the problem of finding whether a graph contains a clique of given size. We propose in this paper a Novel Quantum Evolutionary Algorithm (NQEA) for QKPs. These algorithms are general enough and can be used for similar subsection of problems. We report in this paper solutions which lie in less than 1% of the optimal solutions. We also show that our algorithm is scalable to much larger problem sizes and is capable of exploiting the search space to its maximum.
Solving combinatorial optimization problems is a promising application area for quantum algorithms in real-world scenarios. In this work, we extend the “Quantum Tree Generator” (QTG), previously proposed for the 0-1 Knapsack Problem, to the 0-1 Quadratic Knapsack Problem (QKP) and the Multidimensional Knapsack Problem (MDKP). The QTG constructs a superposition of all feasible solutions for a given instance and can therefore be utilized as a promising state preparation routine within amplitude amplification to produce high-quality solutions. Previously, QTG-based search was tested on the 0-1 Knapsack Problem, where it demonstrated the potential for practical quantum advantage, once quantum computers with a few hundred logical and fully connected qubits are available. Here, we evaluate the algorithm's performance on QKP and MDKP against the classical solver Gurobi. To facilitate large-scale evaluations, we employ an advanced benchmarking technique that enables runtime predictions for instances with up to 2000 variables for QKP and up to 1500 variables and 100 constraints for MDKP. Our results indicate that QTG-based search can produce high-quality solutions with competitive runtimes for QKP. However, its performance declines for MDKP, highlighting the challenges quantum algorithms face when tackling highly constrained optimization problems.
The differences in performance among binary-integer encodings in an Ising machine, which can solve combinatorial optimization problems, are investigated. Many combinatorial optimization problems can be mapped to find the lowest-energy (ground) state of an Ising model or its equivalent model, the Quadratic Unconstrained Binary Optimization (QUBO). Since the Ising model and QUBO consist of binary variables, they often express integers as binary when using Ising machines. A typical example is the combinatorial optimization problem under inequality constraints. Here, the quadratic knapsack problem is adopted as a prototypical problem with an inequality constraint. It is solved using typical binary-integer encodings: one-hot encoding, binary encoding, and unary encoding. Unary encoding shows the best performance for large-sized problems.