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

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.

Multi-spin-flip engineering in an Ising machine

“…They are NP-hard problems [17], [18]. It has been reported that existing Isingmachine hardware is unable to provide even a near-optimal solution of them [32], [33]. Section 5.2 shows the assignment process of {m i } and {s i } for each combinatorial optimization problem.…”

“…where n item is the number of items. The matrix {p i,j } is referred to as the profit matrix [32]. The overall weight of items within the knapsack is given by…”

“…( 19) is 0 only when the overall weight is smaller than C but larger than C − m + 1. We set m to the largest weight of items according to [32].…”

A Hybrid Integer Encoding Method for Obtaining High-Quality Solutions of Quadratic Knapsack Problems on Solid-State Annealers

“…This section first shows the experimental results of SA simulation for QUBO problems constructed using BE, UE, and HE for QKPs of different sizes and feature values. After that, utilizing well-known benchmark QKPs [22], we estimate the performance of HE on a solid-state annealer and compare it with the state-of-the-art results reported in [16].…”

“…Therefore, we can utilize the formulation with QUBO as well as the Ising model. Various methods for formulating COPs as Ising models or QUBOs have been reported so far [14]- [16], and standard techniques have been established for some representative COPs [17].…”