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Edited by: Víctor M. Eguíluz, Institute of Interdisciplinary Physics and Complex Systems (IFISC), Spain

Reviewed by: Jonas Maziero, Universidade Federal de Santa Maria, Brazil; Hsi-Sheng Goan, National Taiwan University, Taiwan

This article was submitted to Quantum Computing, a section of the journal Frontiers in Computer Science

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

For e-commerce websites, deciding the manner in which items are listed on webpages is an important issue because it can dramatically affect item sales. One of the simplest strategies for listing items to improve the overall sales is to do so in a descending order of popularity representing sales or sales numbers aggregated over a recent period. However, in lists generated using this strategy, items with high similarity are often placed consecutively. In other words, the generated item list might be biased toward a specific preference. Therefore, this study employs penalties for items with high similarity being placed next to each other in the list and transforms the item listing problem to a quadratic assignment problem (QAP). The QAP is well-known as an NP-hard problem that cannot be solved in polynomial time. To solve the QAP, we employ quantum annealing, which exploits the quantum tunneling effect to efficiently solve an optimization problem. In addition, we propose a problem decomposition method based on the structure of the item listing problem because the quantum annealer we use (i.e., D-Wave 2000Q) has a limited number of quantum bits. Our experimental results indicate that we can create an item list that considers both popularity and diversity. In addition, we observe that using the problem decomposition method based on a problem structure can provide to a better solution with the quantum annealer in comparison with the existing problem decomposition method.

Several companies have recently started operating e-commerce websites to sell their items and services to the public considering the widespread use of the internet. For these companies, deciding on the order in which items are listed on their website's pages is important because this ordering has the potential to dramatically affect the sales of their items or services.

An example of an item list on a hotel reservation website.

To improve sales on e-commerce websites, placing items in the descending order of popularity representing sales or sales numbers aggregated over a recent period is a simple strategy for determining the list order of items (Long and Chang, _{ij} is the estimated popularity of an item

where _{ij} is a binary variable that indicates whether or not to assign item

is referred to as the popularity term for _{11}, _{12}, ⋯). This problem can be interpreted as a network flow problem, and an efficient technique to solve such a problem in polynomial time exists. Furthermore, the solution obtained by solving this network flow problem with _{ij} ∈ [0, 1] coincides with the solution of the abovementioned integer programming problem (Vazirani,

However, in the case of the list of items generated using such a strategy, the relationship between the different objects is ignored because the popularity of each item _{ij} is considered independently. For example, let us assume that customers visit an e-commerce website and browse the page of a particular item group. If the relationships among different items are not considered while placing items in an item list, several items with high similarities can possibly be placed close to each other, thereby reducing the value of the item list for customers in terms of item diversity. Considering this, several attempts have been made to include item diversity in item recommendation lists for users to ensure that they find the recommendation lists useful (Adomavicius and Kwon,

In this study, we introduce diversity into the item list for the entire user base, as well as methods for improving the usefulness of recommendation lists. An item list is generated by solving an optimization problem that imposes a penalty when items with high similarity are placed in adjacent to each other. Considering both popularity and diversity, the item list generation problem can be formulated as a quadratic assignment problem (QAP) as detailed below.

We employ quantum annealing (QA) herein to solve the QAP (Kadowaki and Nishimori,

In particular, our problem can be solved using the quantum annealer by formulating our QAP as a QUBO. However, a QUBO for such a large number of items cannot be directly solved in one instance with the current state-of-the-art quantum annealer, namely D-Wave 2000Q because it employs the chimera graph. The physical qubits available on D-Wave 2000Q are less than 2048 because the qubits might have defects. In addition, the connection between the physical qubits is sparse and limited on the chimera graph. Thus, several embedding techniques have been proposed; however, the number of logical qubits available to represent the optimization problems to be solved is drastically reduced (Boothby et al.,

The primary contributions of our study are summarized as follows:

We propose a method of creating item lists on an e-commerce website as a QAP considering the popularity and diversity of the items.

We convert the QAP to the QUBO to solve the abovementioned problem with D-wave 2000Q.

We propose a decomposition technique exploiting the structure of the item list.

We introduce the diversity term in our proposed model to add diversity in the item list. In particular, we calculate the similarity

where

Two methods for calculating the similarity

The advantage of the first approach is that the interpretation of the result is straightforward. Also, the semantic features of each item are available when the item is added to the database, that is, so-called cold start problems are avoided. Nevertheless, it suffers from a disadvantage in that appropriate semantic features must be created and quantified. In contrast, the advantage of the second approach is that it involves easy calculations and can consider various information reflecting customer behavior; however, its disadvantage is that semantic interpretation might be difficult. Section 3.1 describes the two methods for the calculation of the similarity

As previously specified, the optimization problem (2) is a QAP. The QAP is well-known as an NP-hard problem that cannot be solved in polynomial time (Anstreicher,

We utilize QA to solve our optimization problem. The details of QA pertaining to D-Wave 2000Q are outlined in

where ^{N×N}. Thus, our optimization problem can be transformed into a QUBO by employing a penalty function for violating constraints and adding this penalty function to the objective function:

where

Therefore, we focus herein on the structure of the assignment problem and propose a method to extract problems with feasible solutions. Particularly in the case of an assignment problem, one condition involves each item being necessarily assigned to one position and another condition, in which each position is necessarily assigned to one item. Therefore, while dividing the problem, we have to select variables with candidate combinations of items and positions that are already assigned.

Example of problem decomposition. The red frames represent the variables of the subproblem to be selected. The left figure is a selection of a subproblem without considering the problem structure. A subproblem has no feasible solution. The right figure is a selection of a subproblem based on the logical structure of the problem. A subproblem has feasible solutions.

The original problem can be decomposed as follows if the number of items in the original problem is _{org} and the number of items solved by a partial problem is _{sub}:

Let _{sub} items extracted from _{org} items.

Let

Let

This procedure involves _{Norg}C_{Nsub} exploiting the structure of the item list.

In practice, it is most important to determine the order in which items are listed in the upper positions of the item list because they are the items that are browsed most often. Therefore, it is effective to solve the entire list as an integer programming problem as in Problem (1) first, then only resolve the particularly important upper positions of the list using the QAP (2).

For our experiments, we used the actual access log data of the online hotel reservation site Jalan^{1}_{ij} and similarity _{ij} and _{ij} and

We conducted two experiments in this study:

evaluating the effect of the diversity term by comparison of solutions when the diversity control parameter is changed for the QAP (2), and

evaluating the performance of problem decomposition by comparison of the objective values when the structure of the item list is considered for the

As previously specified, we used D-Wave 2000Q (DW_2000Q_VFYC_2) for our experiments. Coupler strengths mapping logical to physical couplers with two physical couplers connecting each pair of logical qubits were set as 3.0.

Parameters used for solving the problem in our experiments.

1,000 | |

20 [μs] | |

True | |

Optimization | |

4 | |

20 [s] | |

5 | |

64 |

D-Wave 2000Q has less than 2048 qubits because the qubits typically have defects. In addition, as previously specified, the connection between the physical qubits is sparse and limited on the chimera graph, in which D-Wave 2000Q has been based on. Thus, we can consider the problem with eight items per subproblem because complete graph embedding can be applied to arbitrary problem graphs with less than 64 logical variables. Therefore, we first compare the popularity term

In _{ij} and

Changes in the popularity term when the diversity control parameter is changed.

Changes in the diversity term when the diversity control parameter is changed.

Note that the behavior of

Locations and type of a particular area's hotels.

A particular area's item lists when the diversity control parameter

So far, we have used the co-browsing similarity as

Item lists using co-browsing similarity and semantic similarity as similarity measures in a particular area. The diversity control parameter

We compared the method of extracting partial problems by

Comparison of the objective values for problem decomposition.

12 | −160.038 | −160.337 | −0.299 |

16 | −268.777 | −270.176 | −1.399 |

20 | −391.428 | −393.051 | −1.623 |

24 | −505.634 | −509.266 | −3.632 |

In terms of application, our proposed method is not only limited to the item list optimization problem, but can also be widely applied to other problems involving similar constraints, such as the assignment problem (1). For example, our method can be applied to the traveling salesman problem, which typically includes two constants

This study proposed a method of creating item lists on an e-commerce website as a QAP considering item popularity and diversity. We converted the QAP to a QUBO such that it can be directly solved by the quantum annealer, D-Wave 2000Q. Direct manipulation to solve the resulting QUBO was not possible in the case with a large number of items because of the limited number of qubits available in the current version of the quantum annealer and the restriction on specifying connections between the qubits. Therefore, we proposed a decomposition technique exploiting the structure of the problem. The original large problem was divided into several subproblems, which can eventually be solved by D-Wave 2000Q individually. Our experiments using actual real-world data demonstrated the efficiency of our proposed approach. A remarkable observation made from the experimental results was that the output item list changed based on the diversity control parameter. Our formulation led to the antiferromagnetic Ising model with a random field. The resulting lists were “aligned” along the random field when the diversity control parameter was small. In contrast, increasing the diversity control parameter eliminated the order in the item list and introduced diversity.

However, our research has some limitations. The item list created by our method will not work well when the data needed to calculate the popularity _{ij} and similarity _{ij} and

As the number of qubits available in the quantum annealer increases in the future, our method for division of a large-sized problem into smaller subproblems will become more useful. The experiments in this study clearly showed that our method performed better than

Thus, our method has a wide range of applications involving optimization problems that can be solved via QUBO formulation, such as those using the QA, D-Wave 2000Q, and other types of QUBO solvers. Our results indicate that not only the evolution of hardware devices, but also the development of better software based on the structure of problems are essential for future QA applications.

The datasets generated for this study can be found in the GitHub repository of the Recruit Communications Co., Ltd^{2}

NN contributed to the conception and design of the study, performed all the experiments, and wrote the first draft of the manuscript. KT implemented the program of our problem decomposition method. MO, MM, and KS contributed to the manuscript revision. All authors discussed this study, then reviewed and approved the final version of the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors would like to thank Recruit Lifestyle Co., Ltd. and Recruit Communications Co., Ltd. for their support in this exploratory research project.

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QA belongs to a class of meta-heuristic algorithms, which exploit the quantum tunneling effect to efficiently solve an optimization problem (Kadowaki and Nishimori,

where _{i} is the on-site energy of qubit _{ij} denotes the interaction energies of two qubits _{i} ∈ {−1, +1} are called spins, and are fixed in a lattice graph _{0}, with an easy-to-implement ground state, and a problem Hamiltonian _{P}, whose minimal configuration needs to be explored. Then, we change the Hamiltonian slowly such that it is the spin glass Hamiltonian at time

If _{P}.

For computation on D-Wave 2000Q, the problem is first mapped to the Ising binary and quadratic structures. Then, it is embedded in the available qubit lattice. The qubits are arranged according to a chimera graph on D-Wave 2000Q. Each qubit couples to five or six others, except when the qubit has defects. If the problem does not embed directly, auxiliary qubits can be introduced to augment the available couplings. However, introducing auxiliary qubits is a significant cost in qubits. Both mapping and embedding imply restrictions on the types of problems that can effectively solved with the D-Wave 2000Q. For more details on QA in the D-Wave 2000Q, see D-Wave Systems Inc. (