Information processing device, information processing method, and program

Abstract

This information processing device includes: an input unit that receives an input of a first objective function that refers to a quadratic- or higher-order polynomial expression including a plurality of first binary variables; a conversion unit that converts the first objective function into a second objective function in a QUBO format comprising a plurality of second binary variables by replacing the plurality of first binary variables included in the first objective function with a plurality of second binary variables; an identification unit that identifies the values of the plurality of second binary variables that minimize the second objective function, and identifies the values of the plurality of first binary variables corresponding to the identified values of the plurality of second binary variables; and an output unit that outputs the identified values of the plurality of first binary variables.

Description

Information processing device, information processing method, and program CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application is based on Japanese Patent Application No. 2024-126654, filed on August 2, 2024, the contents of which are incorporated herein by reference.

[0002] The present invention relates to an information processing device, an information processing method, and a program.

[0003] In recent years, algorithms and hardware for solving quadratic unconstrained binary optimization (QUBO) problems, such as annealing processors, have made significant advances. This advancement has led to interest in formulating combinatorial optimization problems as quadratic polynomials. For example, Patent Literature 1 (PTL 1) discloses technology related to quantum hardware for implementing quantum annealing processes.

[0004] Japanese Patent Application Laid-Open No. 2022-58462

[0005] By expressing a combinatorial optimization problem in the QUBO format, it is possible to solve the problem using an annealing processor. However, even if a combinatorial optimization problem can be expressed in the QUBO format, it is known that the greater the number of binary variables, the more difficult it becomes to solve the problem.

[0006] The present invention has been made in view of the above, and aims to provide a technology that enables QUBO format expression with fewer binary variables.

[0007] An information processing device according to one aspect of the present invention includes an input unit that accepts input of a first objective function that is a polynomial of degree two or higher that includes a plurality of first binary variables; a conversion unit that converts the first objective function into a second objective function in QUBO format that is made up of the plurality of second binary variables by replacing the plurality of first binary variables included in the first objective function with a plurality of second binary variables; an identification unit that identifies values ​​of the plurality of second binary variables that minimize the second objective function and identifies values ​​of the plurality of first binary variables that correspond to the identified values ​​of the plurality of second binary variables; and an output unit that outputs the identified values ​​of the plurality of first binary variables.

[0008] According to the present invention, it is possible to provide a technique that enables QUBO format expression using fewer binary variables.

[0009] 1 is a diagram illustrating an example of the configuration of an information processing system according to the present embodiment; 2 is a diagram illustrating an example of the hardware configuration of an information processing device; 3 is a diagram illustrating an example of the functional block configuration of an information processing device; 4 is a flowchart illustrating an example of a processing procedure performed by an information processing device; 5 is a diagram illustrating an outline of the present embodiment; 6 is a diagram illustrating an outline of the present embodiment; 7 is a diagram illustrating an outline of the present embodiment; 8 is a diagram illustrating an outline of the present embodiment; 9 is a diagram illustrating an outline of the present embodiment; 10 is a diagram illustrating an outline of the present embodiment; 11 is a diagram illustrating an outline of the present embodiment; 12 is a diagram illustrating an outline of the present embodiment; 13 is a diagram illustrating an outline of the present embodiment; 14 is a diagram illustrating an outline of the present embodiment; 15 is a diagram illustrating an outline of the present embodiment; 16 is a diagram illustrating an outline of the present embodiment; 17 is a diagram illustrating an outline of the present embodiment; 18 is a diagram illustrating an outline of the present embodiment; 19 is a diagram illustrating an outline of the present embodiment; FIG. 1 is a diagram for explaining details of the present embodiment. FIG. 2 is a diagram for explaining details of the present embodiment. FIG. 3 is a diagram for explaining details of the present embodiment. FIG. 4 is a diagram for explaining details of the present embodiment. FIG. 5 is a diagram for explaining details of the present embodiment. FIG. 6 is a diagram for explaining details of the present embodiment. FIG. 7 is a diagram for explaining details of the present embodiment. FIG. 8 is a diagram for explaining details of the present embodiment. FIG. 9 is a diagram for explaining details of the present embodiment.

[0010] DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS The present invention will be described with reference to the accompanying drawings, in which the same reference numerals denote the same or similar components.

[0011] 1 is a diagram showing an example of the configuration of an information processing system according to this embodiment. The information processing system 1 includes an information processing device 10 and a terminal 20. The information processing device 10 and the terminal 20 are connected via a wireless or wired communication network N and can communicate with each other.

[0012] The information processing device 10 is a device that, upon receiving an input of an objective function for solving a combinatorial optimization problem, converts the input objective function into a QUBO format and solves the converted QUBO format objective function using a QUBO solver, thereby outputting a solution to the combinatorial optimization problem.

[0013] The terminal 20 is a computer, a tablet terminal, or the like used by a user of the information processing system. The terminal 20 is connected to the information processing device 10, and displays various screens for using the information processing device 10, accepts data input, and the like.

[0014] Combinatorial optimization problems involve finding the most optimal combination from among the many combinations (combinations of variable values) that exist under certain constraints.Combinatorial optimization problems generally involve a huge number of combinations, and solving them by brute force often takes an enormous amount of time.

[0015] Here, converting to the QUBO format (also referred to as QUBO conversion) refers to setting an objective function g expressed as a quadratic polynomial for solving a combinatorial optimization problem, such that the solution is p, which minimizes f(p). In other words, QUBO conversion refers to converting the objective function f into an objective function g whose degree is at most quadratic. By expressing the objective function in the QUBO format, it is possible to model various combinatorial optimization problems, such as image synthesis in the field of computer vision and machine learning tasks using neural networks and decision trees. QUBO can also be applied to structural learning of score-based Bayesian networks. Recent developments in annealing machines have attracted significant interest in QUBO formulation.

[0016] A QUBO solver is special hardware and software designed to heuristically solve QUBO problems. Examples of QUBO solvers include, but are not limited to, QUBO solvers that use an annealing processor and QUBO solvers that use a GPU (Graphics Processing Unit).

[0017] However, even if a combinatorial optimization problem can be modeled in the QUBO format, it is known that the greater the number of binary variables (two-valued variables) included in the objective function, the more difficult it becomes to solve. Furthermore, because there are limitations on the hardware circuitry of annealing machines, if the number of binary variables is too large, the problem cannot be solved by an annealing machine. Therefore, when modeling in the QUBO format, it is desirable to model using a quadratic polynomial with fewer binary variables.

[0018] When converting an objective function for solving a combinatorial optimization problem into the QUBO format, the information processing device 10 according to this embodiment converts the function so that the number of binary variables is minimized, thereby improving the performance of solving the combinatorial optimization problem and enabling the problem to be solved more efficiently.

[0019] <Hardware Configuration> Fig. 2 is a diagram showing an example of the hardware configuration of the information processing device 10. The information processing device 10 has a processor 11 such as a CPU (Central Processing Unit) or a GPU, a memory (for example, a RAM (Random Access Memory) or a ROM (Read Only Memory)), a storage device 12 such as an HDD (Hard Disk Drive) and / or an SSD (Solid State Drive), a network IF (Network Interface) 13 for wired or wireless communication, an input device 14 for accepting input operations, and an output device 15 for outputting information. The input device 14 is, for example, a keyboard, a touch panel, a mouse, and / or a microphone. The output device 15 is, for example, a display, a touch panel, and / or a speaker.

[0020] Furthermore, the information processing device 10 may include an annealing machine, or may be capable of communicating with an external annealing machine.

[0021] Furthermore, the information processing device 10 may be configured using one or more physical servers, or may be configured using a virtual server running on a hypervisor, or may be configured using a cloud server.

[0022] <Functional Block Configuration> FIG. 3 is a diagram showing an example of the functional block configuration of the information processing device 10. The information processing device 10 includes a storage unit 100, an input unit 101, a conversion unit 102, an identification unit 103, and an output unit 104. The storage unit 100 can be realized using the storage device 12 provided in the information processing device 10. The input unit 101, the conversion unit 102, the identification unit 103, and the output unit 104 can be realized by the processor 11 of the information processing device 10 executing a program stored in the storage device 12. The program can be stored in a storage medium. The storage medium storing the program may be a non-transitory computer-readable medium. The non-transitory storage medium is not particularly limited, and may be, for example, a storage medium such as a universal serial bus (USB) memory or a compact disc read-only memory (CD-ROM).

[0023] The storage unit 100 stores a program that performs processing according to this embodiment, various data used by the program, and values ​​of variables used when performing calculation processing.

[0024] The input unit 101 accepts input data representing an objective function f for solving a combinatorial optimization problem, which is a polynomial of degree two or higher that includes multiple variables. The input unit 101 may accept input of the objective function from the screen of the terminal 20. Here, the variables included in the objective function f are referred to as "main variables." The objective function f is an example of a "first objective function." The main variables are also an example of a "first binary variable." The input data representing the objective function f may be, for example, data representing a combination of the coefficients and main variables for each term that constitutes the objective function f.

[0025] The conversion unit 102 converts the objective function f, which is a polynomial of degree two or higher, into a QUBO-format objective function g consisting of the multiple variables by replacing multiple main variables included in the objective function f with multiple variables. Here, the variables included in the QUBO-format objective function g are called "auxiliary variables." The objective function g is an example of a "second objective function." Furthermore, the auxiliary variables x and x are examples of "second binary variables."

[0026] The identification unit 103 identifies values ​​of a plurality of auxiliary variables that minimize the objective function g in the QUBO format. Note that the values ​​of the plurality of auxiliary variables identified by the identification unit 103 may be any combination of values ​​of the auxiliary variables that minimizes the objective function g within the searched range, and it is not necessary to identify values ​​that minimize the objective function g for all combinations of values ​​that the auxiliary variables can take.

[0027] The output unit 104 outputs the values ​​of the main variables corresponding to the values ​​of the identified auxiliary variables. The output unit 104 may display the values ​​of the main variables on the screen of the terminal 20.

[0028] <Processing Procedure> Fig. 4 is a flowchart showing an example of a processing procedure performed by the information processing device 10. The processing performed by the information processing device 10 will be described with reference to Fig. 4 .

[0029] In step S10, the input unit 101 receives an input of an objective function f (first objective function) expressed by the following formula (1), which is a polynomial of degree two or higher including multiple main variables, from the terminal 20. Here, the objective function f is a function that represents a combinatorial optimization problem, and may be any function as long as it includes a polynomial of degree two or higher. Here, for a polynomial consisting of "I+1" terms, "F" indicates a set whose elements are all terms except for the constant term. ０ " and "π(V)" are real numbers representing the coefficients of the polynomial, and "p = (p １ ,…,p Ｉ ) is a set of main variables corresponding to each term "V" belonging to "F".

[0030] Here, we will give a specific example of the objective function f. For example, suppose there is problem No. 1, which asks who should be hired to maximize profits when the following conditions 1 and 2 exist. Condition 1: There is a job that can be done by worker A with a profit of 100 yen, and a job that can be completed by worker B and worker C working together with a profit of 300 yen. Condition 2: If three or more workers are hired, an additional 200 yen in room rent will be incurred. When problem No. 1 is expressed in terms of the objective function f, the objective function f is expressed as "f = -100p １ -200p ２ p ３ +300p １ p ２ p ３ Therefore, in formula (1), F is p １ , p ２ ×p ３ and p １ ×p ２ ×p ３ Also, V is a set of p １ , p ２ ×p ３ and p １ ×p ２ ×p ３ It is a set of any one of the above.

[0031] In step S20, the conversion unit 102 converts the objective function f into a QUBO-format objective function g by replacing multiple main variables included in the objective function f with multiple auxiliary variables. Specifically, the conversion unit 102 identifies multiple substitution patterns between multiple main variables and multiple auxiliary variables, identifies values ​​of multiple search binary variables that minimize an objective function including multiple search binary variables corresponding to each of the multiple substitution patterns so as to satisfy a constraint equation determined based on the multiple substitution patterns, and converts the objective function f into the objective function g based on the substitution patterns corresponding to the identified values ​​of the multiple search binary variables. Below, the processing performed by the conversion unit 102 in step S20 will be described separately in steps S21 to S23.

[0032] In step S21, the conversion unit 102 identifies multiple substitution patterns between multiple main variables and multiple auxiliary variables, and limits the substitution patterns by excluding combinations determined to be used for QUBO conversion from the identified substitution patterns. Here, the substitution patterns may be referred to as a "search space." Furthermore, limiting the substitution patterns may be referred to as "limiting the search space." Note that when replacing a main variable with an auxiliary variable, one or more main variables are replaced with one auxiliary variable. In other words, one main variable is not replaced with multiple auxiliary variables.

[0033] Here, we propose equation (2) as a specific procedure for limiting the search space.

[0034] First, if there is a first-order term in the objective function f, the conversion unit 102 replaces the main variable p of the first-order term with an auxiliary variable. In other words, it is determined that the combination of the main variable p and the auxiliary variable will be used in the QUBO conversion. The set of main variables p and auxiliary variables that are replaced in this way is called "V ｃｌｏｓｅ For example, in the case of the objective function f corresponding to the above problem No. 1, the term whose degree is first order is defined as "-100p １ Therefore, the conversion unit 102 converts "p １ It is decided to replace "V ｃｌｏｓｅ " is "p １ It becomes a set whose elements are substitution patterns in which "replace" with auxiliary variables.

[0035] In addition, when there are multiple terms with first order in the objective function f, for example, p １ , p ２ When two linear terms exist, the conversion unit 102 converts "p １ "Replace " with auxiliary variables", "p ２ Here, the different auxiliary variables that replace each main variable are determined as follows: １ , q ２、 These auxiliary variables are expressed by adding subscripts, such as .... These auxiliary variables are specifically determined in steps S21 and S22.

[0036] Next, the conversion unit 102 selects "V ｃｌｏｓｅ The set excluding the substitution patterns included in "V ｏｐｅｎ For example, in the case of the objective function f for the above problem No. 1, all the substitution patterns of the main variable p to the auxiliary variables are defined as "p １ "Replace " with auxiliary variables", "p ２ "Replace " with auxiliary variables", "p ３ "Replace " with auxiliary variables", "p １ p ２ "Replace " with auxiliary variables", "p ２ p ３ "Replace " with auxiliary variables", "p １ p ３ "Replace " with auxiliary variables", "p １ p ２ p ３ Among these substitution patterns, "p １ "Replace with auxiliary variables" is "V ｃｌｏｓｅ " is included in "V ｏｐｅｎ " is "p ２ "Replace " with auxiliary variables", "p ３ "Replace " with auxiliary variables", "p １ p ２ "Replace " with auxiliary variables", "p ２ p ３ "Replace " with auxiliary variables", "p １ p ３ "Replace " with auxiliary variables", "p １ p ２ p ３ The result is a set with six patterns as elements: "replace with auxiliary variables".

[0037] "V ｏｐｅｎ " is a substitution pattern "V" that is determined to be used for the QUBO conversion from all substitution patterns of the main variable p to the auxiliary variables. ｃｌｏｓｅ " is excluded from all substitution patterns. ｃｌｏｓｅ " and "V ｏｐｅｎ " corresponds to restricting the substitution pattern.

[0038] Next, the conversion unit 102 converts the term "Vｃｌｏｓｅ " is a set of combinations of multiple main variables that constitute terms that are not transformed by the substitution patterns included in " (i.e., terms of second or higher order). ｏｐｅｎ For example, the objective function f is defined as "f = -100p １ -200p ２ p ３ +300p １ ×p ２ ×p ３ ", "W ｏｐｅｎ " is p ２ ×p ３ and p １ ×p ２ ×p ３ It becomes a set whose elements are terms of.

[0039] In step S22, the conversion unit 102 converts "V ｏｐｅｎ " is selected from the multiple substitution patterns included in " ｏｐｅｎ Specifically, the conversion unit 102 determines an optimal substitution pattern when substituting the terms included in "V ｏｐｅｎ " and "W ｏｐｅｎ ", the optimal replacement pattern is identified by specifying values ​​of a plurality of binary variables (hereinafter referred to as "search binary variables" and "third binary variables") that minimize an objective function for identifying a replacement pattern (hereinafter referred to as an "objective function for identifying a replacement pattern" and a "third objective function") so as to satisfy a constraint equation determined based on the

[0040] Here, the objective function for identifying a substitution pattern is expressed by the following equation (3). For example, "V ｏｐｅｎ " is "p ２ p ３ "Replace " with auxiliary variables", "p １ p ２ p ３ is replaced with an auxiliary variable, the conversion unit 102 ２ p ３ "Replace with auxiliary variables" １ , "p １ p ２ p ３ "Replace with auxiliary variables" ２In addition, when the value of the search binary variable is 1, it corresponds to using the substitution pattern corresponding to the search binary variable in the QUBO conversion, and when the value of the search binary variable is 0, it corresponds to not using the substitution pattern corresponding to the search binary variable in the QUBO conversion.

[0041] The constraint equations are expressed by the following equations (4) and (5). ｃｌｏｓｅ " are terms of F that cannot be expressed only by the substitution patterns (elements) contained in " (i.e., "W ｏｐｅｎ " element) to "V ｏｐｅｎ " is a constraint expression to be expressed using substitution patterns (elements).

[0042] The first term of the left equation of equation (4) is "V ｃｌｏｓｅ " and / or "V ｏｐｅｎ " in the substitution pattern "W ｏｐｅｎ ", and the second term of the left equation of formula (4) means "V ｏｐｅｎ " in any two substitution patterns "W ｏｐｅｎ " More specifically, the formula (4) means "W ｏｐｅｎ " means that the sum of A and B below is 1 or more for each term included in ". A (the first term of the left equation of equation (4)): "V ｏｐｅｎ ", or one substitution pattern and "V ｃｌｏｓｅ " is the sum of the values ​​of the binary variables for search corresponding to one substitution pattern when the substitution is possible by multiplying it with a substitution pattern included in ". B (the second term of the left expression of the expression (4)): "V ｏｐｅｎ ", the product of two substitution patterns or the product of two substitution patterns and "V ｃｌｏｓｅ ", the sum of the values ​​of the binary variables for search corresponding to the product of the two replacement patterns when the replacement is possible by the product of the two replacement patterns included in ".

[0043]

[0044] Equation (5) is "W ｏｐｅｎ" can be replaced by the product of three or more elements, and if so, the term is replaced by "V ｏｐｅｎ The right side is constrained to be expressed as a combination of two elements contained in ". ｏｐｅｎ " Among the items included in "V ｃｌｏｓｅ " and "V ｏｐｅｎ If there is a term (here, "term X") that can be expressed as a product of any three or more elements in "V ｏｐｅｎ " represents the product of binary variables for search corresponding to the substitution pattern of ". ｃｌｏｓｅ " and "V ｏｐｅｎ Identify all combinations in which the product of any two elements in "V" becomes the term X. ｃｌｏｓｅ "V" in combinations containing elements included in ｏｐｅｎ " and one binary variable for searching corresponding to the elements of "V ｃｌｏｓｅ "V" in combinations that do not include elements included in ｏｐｅｎ " and add up all the products of the binary variables for search corresponding to each of the two elements of ".

[0045] In the case of the objective function f for the above problem No. 1, "V ｏｐｅｎ " is "p ２ "Replace " with auxiliary variables", "p ３ "Replace " with auxiliary variables", "p １ p ２ "Replace " with auxiliary variables", "p ２ p ３ "Replace " with auxiliary variables", "p １ p ３ "Replace " with auxiliary variables", "p １ p ２ p ３ "p" is replaced with an auxiliary variable. ２ "Replace " with auxiliary variables", "p ３ "Replace " with auxiliary variables", "p １ p ２ "Replace " with auxiliary variables", "p ２ p ３ "Replace " with auxiliary variables", "p １ p ３ "Replace " with auxiliary variables", "p １ p ２ p３ The binary variables for search corresponding to the six patterns of "replace with auxiliary variables" are respectively １ , v ２ , v ３ , v ４ , v ５ , v ６ In this case, equation (3) is expressed as equation (6) below.

[0046] Also, "W ｏｐｅｎ " is p ２ ×p ３ and p １ ×p ２ ×p ３ Since the two terms are included, the constraint equation corresponding to equation (4) is expressed by the following equation (7). ２ ×p ３ The lower row corresponds to the term p １ ×p ２ ×p ３ This corresponds to the section. Regarding the upper part of equation (7), "W ｏｐｅｎ "p ２ ×p ３ The term is calculated by using the first term of the left equation of equation (4) as "V ｏｐｅｎ "V ４ "p" corresponding to ２ p ３ can be expressed by substituting "W ｏｐｅｎ "p ２ ×p ３ The term is expressed as "V ｏｐｅｎ "V １ "Replace p2 with auxiliary variables" and v ２ Therefore, the upper part of the formula (7) "v ４ +v １ ×v ２ ≧1" is derived.

[0047] Similarly, for the lower part of equation (7), "W ｏｐｅｎ "p １ ×p ２ ×p ３ The term is calculated by using the first term of the left equation of equation (4) as "Vｃｌｏｓｅ " in " １ "Replace with auxiliary variables" and "V ｏｐｅｎ "V ４ "p" corresponding to ２ p ３ can be expressed as a product of "substituting auxiliary variables" (i.e., v ４ Similarly, "W ｏｐｅｎ "p １ ×p ２ ×p ３ The term is calculated by using the first term of the left equation of equation (4) as "V ｏｐｅｎ "V ６ "p" corresponding to １ p ２ p ３ can be expressed by substituting auxiliary variables (i.e., v ６ Similarly, "W ｏｐｅｎ "p １ ×p ２ ×p ３ The term is expressed as "V ｏｐｅｎ "V １ "p" corresponding to ２ "Replace with auxiliary variables" and v ５ "p" corresponding to ２ p ３ can be expressed as a product of "substituting v １ v ５ Similarly, "W ｏｐｅｎ "p １ ×p ２ ×p ３ The term is expressed as "V ｏｐｅｎ "V ２ "p" corresponding to ３ "Replace with auxiliary variables" and v ３ "p" corresponding to １ p ２ can be expressed as a product of "substituting v ２ v ３ is derived).

[0048] Also, "W ｏｐｅｎ "p １ ×p ２ ×p ３ The term p １ ×p １ ×p２ ×p ３ It can be transformed into (P １ If is 1, then p ２ ×p ３ and P １ (When W is 0, it becomes 0, so the value is the same before and after deformation.) Therefore, "W ｏｐｅｎ "p １ ×p ２ ×p ３ The term is expressed as "V ｏｐｅｎ "V ３ "p" corresponding to １ p ２ "Replace with auxiliary variables" and v ５ "p" corresponding to １ p ３ can be expressed as a product of "substituting v ３ v ５ is derived). Also, ｏｐｅｎ "p １ ×p ２ ×p ３ The term p １ ×p ２ ×p ２ ×p ３ Therefore, using the second term of the left equation of equation (4), "V ｏｐｅｎ "V ３ "p" corresponding to １ p ２ "Replace with auxiliary variables" and v ４ "p" corresponding to ２ p ３ can be expressed as a product of "substituting v ３ v ４ is derived). Also, ｏｐｅｎ "p １ ×p ２ ×p ３ The term p １ ×p ２ ×p ３ ×p ３ Therefore, using the second term of the left equation of equation (4), "V ｏｐｅｎ "V ４ "p" corresponding to ２ p ３ "Replace with auxiliary variables" and v ５ "p" corresponding to １ p３ can be expressed as a product of "substituting v ４ v ５ Therefore, the lower part of equation (7) "v ４ +v ６ +v １ v ５ +v ２ v ３ +v ３ v ５ +v ３ v ４ +v ４ v ５ ≧1" is derived. ４ +v ６ +v １ v ５ +v ２ v ３ +v ３ v ５ +v ３ v ４ +v ４ v ５ ≧1” is “v ４ +v ６ +v １ v ５ +v ２ v ３ +v ３ v ５ ≧1”.

[0049] The constraint equation corresponding to equation (5) is expressed by the following equation (8). Regarding formula (8), "W ｏｐｅｎ "p １ ×p ２ ×p ３ The section "V ｃｌｏｓｅ " and "V ｏｐｅｎ Among the substitution patterns in "V ｃｌｏｓｅ " in " １ "Replace with auxiliary variables" and "V ｏｐｅｎ "V １ "p" corresponding to ２ "Replace with auxiliary variables" and v ２ "p" corresponding to ３ It can be expressed as a product of three substitution patterns: "P1 × p2 × p3" and "P2 × p3" ｏｐｅｎ"The substitution pattern is v ６ Therefore, v, which is the left side of equation (8), ６ v １ v ２ As mentioned above, considering that p1×p2×p3 can be transformed into p1×p1×p2×p3, p1×p2×p2×p3, or p1×p2×p3×p3, "V ｏｐｅｎ " The product of two substitution patterns is "W ｏｐｅｎ The combination of substitution patterns that replaces the p1×p2×p3 term of "V ｃｌｏｓｅ " in " １ "Replace with auxiliary variables" and "V ｏｐｅｎ "V ４ "p" corresponding to ２ p ３ "Replace with auxiliary variables" and "V ｏｐｅｎ "V １ "p" corresponding to ２ "Replace with auxiliary variables" and v ５ "p" corresponding to １ p ３ "Replace with auxiliary variables" and "V ｏｐｅｎ "V ２ "p" corresponding to ３ "Replace with auxiliary variables" and v ３ "p" corresponding to １ p ２ "Replace with auxiliary variables" and "V ｏｐｅｎ "V ３ "p" corresponding to １ p ２ "Replace with auxiliary variables" and v ５ "p" corresponding to １ p ３ "Replace with auxiliary variables" and "V ｏｐｅｎ "V ３ "p" corresponding to １ p ２ "Replace with auxiliary variables" and v ４ "p" corresponding to ２ p ３ "Replace with auxiliary variables" and "V ｏｐｅｎ "V ４ "p" corresponding to ２ p ３"Replace with auxiliary variables" and v ５ "p" corresponding to １ p ３ is replaced with an auxiliary variable, and by taking the sum of these, we get the right side of equation (8), "v ４＋ v １ v ５ +v ２ v ３ +v ３ v ５ +v ３ v ４ +v ４ v ５ " is derived. The right side of equation (8) is "v ４ +v １ v ５ +v ２ v ３ +v ３ v ５ " can also be expressed as ".

[0050] Next, the conversion unit 102 specifies values ​​of the search binary variables that minimize the objective function for identifying a replacement pattern so as to satisfy the constraint condition equation. Any method can be used to specify the values ​​of the search binary variables, but for example, the conversion unit 102 may specify values ​​of the search binary variables that minimize the objective function for identifying a replacement pattern by searching for values ​​of the search binary variables using integer programming. For example, if v is found to be a solution as a result of the search, １ = 0, v ２ = 0, v ３ = 0, v ４ = 1, v ５ = 0, v ６ If V = 0 is found, the conversion unit 102 determines that the optimal substitution pattern is "V ｃｌｏｓｅ " is a substitution pattern included in １ with the auxiliary variable, and a binary variable v for search having the value 1. ４ "p" corresponding to ２ p ３ "Replace with auxiliary variables" and "Replace with auxiliary variables".

[0051] In step S23, the conversion unit 102 converts the objective function f into an objective function g by replacing the main variables with auxiliary variables using the substitution pattern identified in the processing procedure of step S22. In the case of the objective function f for the above problem No. 1, in the processing procedures of steps S21 and S22, the optimal substitution pattern is "p １ "Replace with auxiliary variables" and "p ２ p ３ Therefore, the former auxiliary variable is replaced by q １ The latter auxiliary variable is q ２ Then, the objective function g is "g = -100q １ -200q ２ +300q １ q ２ " is expressed as:

[0052] In step S30, the identification unit 103 uses the QUBO solver to identify values ​​of a plurality of auxiliary variables that minimize the objective function g. In the case of the objective function g for the above problem No. 1, q １ and q ２ will be specified as 0 and 1, respectively.

[0053] In step S40, the identification unit 103 identifies the value of the main variable from the value of the auxiliary variable identified in step S30. In this embodiment, the auxiliary variable corresponds to the product of one or more main variables. In other words, when the value of the auxiliary variable is 1, it can be identified that the values ​​of one or more main variables replaced by the auxiliary variable are all 1. Also, when the value of the auxiliary variable is 0, it can be identified that the values ​​of one or more main variables replaced by the auxiliary variable are such that any one of the main variables is 0, and the values ​​of the other main variables are arbitrary (either 0 or 1). For example, in the case of the above problem No. 1, the auxiliary variable q １ and q ２ The values ​​of p are specified to be 0 and 1, respectively. １ , p ２ and p ３ The values ​​of will be specified as 0, 1 and 1, respectively.

[0054] In addition, when there are multiple values ​​of the main variable corresponding to the value of the auxiliary variable, the specifying unit 103 may specify the value of the main variable by selecting one of the multiple values ​​of the auxiliary variable. For example, in the case of the above problem No. 1, the auxiliary variable q ２ Assume that the value of is specified to be 0. In this case, the main variable p ２ and p ３ The combination of the values ​​of the main variable p is one of three combinations: 0 and 0, 1 and 0, and 0 and 1. ２ and p ３ The value of may be selected from 0 and 0, 1 and 0, or 0 and 1.

[0055] In step S50, the output unit 104 outputs the value of the main variable identified in the processing procedure of step S40 to the screen of the terminal 20.

[0056] <Other Specific Examples> Although the processing procedure performed by the information processing device 10 has been explained above using Problem No. 1, specific examples of the processing procedure performed by the information processing device 10 will be explained using other problems. For example, suppose there is Problem No. 2, which asks who should be hired to maximize profits when the following conditions 1' and 2' exist. Condition 1': There is a job that can be completed by workers A and B in cooperation, with a profit of 200 yen, and a job that can be completed by workers B and C in cooperation, with a profit of 400 yen. Condition 2': If two or more workers are hired, an additional room fee of 300 yen will be incurred. When Problem No. 2 is expressed by an objective function f, the objective function f is expressed as "f = -200p １ p ２ -400p ２ p ３ +300 (p １ p ２ +p ２ p ３ +p １ p ３ -2p １ p ２ p ３ )"

[0057] In step S10, the input unit 101 receives input data representing the objective function f from the terminal 20.

[0058] In step S20, the conversion unit 102 converts the objective function f into an objective function g in the QUBO format by replacing a plurality of main variables included in the objective function f with a plurality of auxiliary variables.

[0059] In step S21, the conversion unit 102 identifies multiple substitution patterns between multiple main variables and multiple auxiliary variables, and limits the substitution patterns by excluding combinations that have been determined to be used for QUBO conversion from the identified substitution patterns.

[0060] First, in the case of the objective function f corresponding to problem No. 2, there is no term whose degree is first order. ｃｌｏｓｅ " becomes an empty set. Then, in the objective function f, all substitution patterns of the main variable p to the auxiliary variables are １ "Replace " with auxiliary variables", "p ２ "Replace " with auxiliary variables", "p ３ "Replace " with auxiliary variables", "p １ p ２ "Replace " with auxiliary variables", "p ２ p ３ "Replace " with auxiliary variables", "p １ p ３ "Replace " with auxiliary variables", "p １ p ２ p ３ There are seven patterns of "V ｃｌｏｓｅ " is an empty set, so "V ｏｐｅｎ " is a collection of these seven patterns.

[0061] Next, the conversion unit 102 converts the "W ｏｐｅｎ " is sought. "V ｃｌｏｓｅ " is an empty set, so "W ｏｐｅｎ " is p １ ×p ２ and p ２ ×p ３ and p １ ×p ３ and p １ ×p ２ ×p ３ This becomes the term.

[0062] In step S22, the conversion unit 102 converts "V ｏｐｅｎ" is selected from the multiple substitution patterns included in " ｏｐｅｎ " to identify the optimal substitution pattern when substituting terms that need to be substituted for auxiliary variables contained in "V ｏｐｅｎ The seven substitution patterns included in " are １ , v ２ , v ３ , v ４ , v ５ , v ６ and v ７ Then, the objective function for specifying a substitution pattern corresponding to equation (4) is expressed by the following equation (9). Also, "W ｏｐｅｎ " is p １ ×p ２ and p ２ ×p ３ and p １ ×p ３ and p １ ×p ２ ×p ３ Term p １ ×p ２ and p ２ ×p ３ and p １ ×p ３ and p １ ×p ２ ×p ３ Since the four terms of (1) and (2) are included, the constraint equation corresponding to equation (5) is expressed by the following equation (10). The constraint equation corresponding to equation (6) is expressed by the following equation (11). Next, the conversion unit 102 identifies the value of the search binary variable that minimizes the objective function for identifying a replacement pattern so as to satisfy the constraint condition equation. １ = 0, v ２ = 0, v ３ = 0, v ４ = 1, v ５ = 1, v ６ = 1, v ７ In other words, the conversion unit 102 determines that the optimal substitution pattern is "V ｃｌｏｓｅ Among the substitution patterns included in "p １ p ２ "Replace " with auxiliary variables", "p ２ p３ "Replace with auxiliary variables" and "p １ p ３ "Replace with auxiliary variables."

[0063] In step S23, the conversion unit 102 converts the objective function f into the objective function g by replacing the main variables with auxiliary variables using the substitution pattern identified in the processing procedure of step S22. １ p ２ "Replace " with auxiliary variables", "p ２ p ３ "Replace with auxiliary variables" and "p １ p ３ "Replace p with auxiliary variables." １ p ２ q １ "Replace with ", "p ２ p ３ q ２ "Replace with" and "p １ p ３ q ３ Then, the objective function g is "g = -200q １ -400q ２ +300 (q １ +q ２ +q ３ -2q １ q ２ )) where p １ ×p ２ ×p ３ The term p １ , p ２ and p ３ Since is a binary variable, p １ ×p ２ ×p ２ ×p ３ Therefore, p １ ×p ２ ×p ３ The term is q １ q ２ can be replaced by

[0064] In step S30, the identification unit 103 uses the QUBO solver to identify values ​​of a plurality of auxiliary variables that minimize the objective function g. In the case of the objective function g for the above problem No. 2, q １ , q ２ and q ３ are respectively, q １ = 1, q ２ = 1 and q ３ = 1.

[0065] In step S40, the identification unit 103 identifies the value of the main variable from the value of the auxiliary variable identified in step S30. １ and q ２ The values ​​of q １ = 1, q ２ = 1 and q ３ = 1. Therefore, the main variable p １ , p ２ and p ３ The values ​​of p １ = 1, p ２ = 1 and p ３ =1.

[0066] In step S50, the output unit 104 outputs the value of the main variable identified in the processing procedure of step S40 to the screen of the terminal 20.

[0067] In the prior art, when performing QUBO transformation, secondary or higher order variables are replaced with primary variables, and a constraint term indicating the relationship between the secondary or higher order variables and the replaced primary variables is added to the objective function f. For example, in the case of the objective function f for the above problem No. 1, the objective function g after the replacement is expressed as follows: ２ p ３ The auxiliary variable q １ and the main variable p ２ p ３ The auxiliary variable q １ A constraint term was added to show the relationship between In the above formula (12), "1000(p ２ p ３ -2p ２ q １－ 2p ３q １ +3q １ )) is the main variable p ２ p ３ The auxiliary variable q １ corresponds to the constraint term that indicates the relationship between

[0068] On the other hand, in this embodiment, it has been confirmed that, as a result of identifying the values ​​of the search binary variables that minimize the objective function for identifying a replacement pattern in the processing procedure of step S22, a search result can be obtained in which the number of auxiliary variables included in the objective function g is at least equal to or less than the variables used in the conventional QUBO conversion (see Chapter 3 of FIGS. 31 and 32). Therefore, when the objective function g that has been QUBO converted using this embodiment is solved by a QUBO solver, it is possible to achieve QUBO solution performance that is similar to or better than that of the conventional technology.

[0069] In this embodiment, in the processing procedure of step S21, multiple substitution patterns (search spaces) between multiple main variables and multiple auxiliary variables are identified. These substitution patterns include the substitution patterns used in the QUBO conversion in the conventional technology described above. In other words, the number of auxiliary variables when QUBO conversion is performed according to this embodiment is the same as or smaller than the total number of main variables and auxiliary variables when QUBO conversion is performed using the conventional technology. In other words, when using this embodiment, the number of variables cannot increase compared to when QUBO conversion is performed using the conventional technology, and it is possible to discover an objective function g that is expressed with the same number of variables as or fewer than the conventional technology.

[0070] <Modification> In the case of structural learning, assumptions regarding "inclusion relationships" are not necessary, so as shown in equation (13), it is possible to "limit the search space" to a wider range than in general problems. The fourth line of equation (13) means to execute the above equation (2).

[0071] <Summary> According to the embodiment described above, the information processing device 10 replaces the main variables with auxiliary variables so that the number of auxiliary variables is at least equal to or less than the number of main variables, and then searches for a solution to the objective function using a QUBO solver. Furthermore, the solution for the main variables is obtained by replacing the auxiliary variable solution obtained by the search with the value of the main variable that is equivalent to the solution for the auxiliary variable. This makes it possible to express the QUBO format with fewer binary variables.

[0072] The above-described embodiments are intended to facilitate understanding of the present invention and are not intended to limit the present invention. The flowcharts, sequences, elements included in the embodiments, and their arrangements, materials, conditions, shapes, sizes, etc., described in the embodiments are not limited to those illustrated and can be modified as appropriate. Furthermore, configurations shown in different embodiments can be partially substituted or combined with each other.

[0073] <Supplementary Information> As shown in Figures 28 to 44, this embodiment proposes a QUBO formulation entitled "Decomposed Quadratization: An Efficient QUBO Formulation for Bayesian Network Structure Learning," which offers advantages over conventional quadratization methods in terms of bit capacity. Figures 5 to 27 provide an overview of the proposals described in Figures 28 to 44. A key application of this formulation is to significantly reduce the number of binary variables required for score-based Bayesian network structure learning. For example, experimental results for 16 cases involving 37 to 223 variables demonstrated that orders of magnitude fewer bits were required than in the paper [O'Gorman et al., 2014]. Furthermore, an annealing machine implementing this formulation can outperform existing algorithms in score maximization.

[0074] In Figures 22 and 30, the arrow from X2 to X1 on the right (directed graph) corresponds to d12 on the left (undirected graph). The arrow from X1 to X3 on the right (directed graph) corresponds to d31 on the left (undirected graph). The arrow from X2 to X3 on the right (directed graph) corresponds to d32 on the left (undirected graph).

[0075] 23 and 33, the arrow from X2 to X1 on the right (directed graph) corresponds to u11 on the left (undirected graph). The arrow from X1 to X3 on the right (directed graph) corresponds to u31 on the left (undirected graph). The arrow from X2 to X3 on the right (directed graph) corresponds to u31 on the left (undirected graph).

[0076] 1 Information processing system, 10 Information processing device, 11 Processor, 12 Storage device, 13 Network IF, 14 Input device, 15 Output device, 20 Terminal, 100 Storage unit, 101 Input unit, 102 Conversion unit, 103 Identification unit, 104 Output unit

Claims

an input unit that receives input data representing a first objective function that is a polynomial of degree two or higher that includes a plurality of first binary variables; a conversion unit that converts the first objective function into a second objective function in QUBO format that is made up of the second binary variables by replacing the first binary variables included in the first objective function with a plurality of second binary variables; an identification unit that identifies values ​​of the plurality of second binary variables that minimize the second objective function, and identifies values ​​of the plurality of first binary variables that correspond to the identified values ​​of the plurality of second binary variables; an output unit that outputs the identified values ​​of the plurality of first binary variables; An information processing device having the above. 　 The conversion unit identifying a plurality of permutation patterns between the plurality of first binary variables and the plurality of second binary variables; identifying values ​​of a plurality of third binary variables that minimize an objective function including a plurality of third binary variables corresponding to each of the plurality of permutation patterns so as to satisfy a constraint equation determined based on the plurality of permutation patterns and the first objective function; transforming the first objective function into the second objective function based on a substitution pattern corresponding to the identified values ​​of the plurality of third binary variables; The information processing device according to claim 1 . 　 The identification unit identifies the value of the second binary variable using a QUBO solver. The information processing device according to claim 1 . 　 when there are a plurality of values ​​of the first binary variable corresponding to the value of the second binary variable, the identification unit identifies the value of the first binary variable by selecting one of the plurality of values ​​of the second binary variable; The information processing device according to claim 1 . 　 An information processing method performed by an information processing device, receiving input data representing a first objective function that is a polynomial of degree two or higher that includes a plurality of first binary variables; a step of converting the first objective function into a second objective function in QUBO format that is made up of the second binary variables by replacing the first binary variables included in the first objective function with a plurality of second binary variables; identifying values ​​of the plurality of second binary variables that minimize the second objective function, and identifying values ​​of the plurality of first binary variables that correspond to the identified values ​​of the plurality of second binary variables; outputting the identified values ​​of the first plurality of binary variables; An information processing method including: 　 On the computer, receiving input data representing a first objective function that is a polynomial of degree two or higher that includes a plurality of first binary variables; a step of converting the first objective function into a second objective function in QUBO format that is made up of the second binary variables by replacing the first binary variables included in the first objective function with a plurality of second binary variables; identifying values ​​of the plurality of second binary variables that minimize the second objective function, and identifying values ​​of the plurality of first binary variables that correspond to the identified values ​​of the plurality of second binary variables; outputting the identified values ​​of the first plurality of binary variables; A program to execute.

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