- Email: [email protected]
A new search method for combinatorial optimization problem inspired by the spin glass system
International Congress Series 1291 (2006) 201 – 204
www.ics-elsevier.com
A new search method for combinatorial optimization problem inspired by the spin glass system Hiroshi Wakuya * Faculty of Science and Engineering, Saga University, Honjo-machi, Saga 840-8502, Japan
Abstract. A Hopfield network is a good tool for solving combinatorial optimization problems. Since its proposal, research activity around this area has been accelerated with the help of related ideas. But some of them are just techniques to get better solutions, so no meaningful phenomena might be observed in the field of the actual spin glass. In this paper, therefore, another search method inspired from an analogy between the Hopfield network and the spin glass is proposed. It is quite a simple idea to control a threshold of the Hopfield network corresponding to a magnetic field in the spin glass. In order to confirm its effectiveness, an N queens problem is adopted. As a result, it is found experimentally that the proposed method shows better score, e.g. a ratio of correct answer and an averaged final energy, than the conventional one. D 2006 Elsevier B.V. All rights reserved. Keywords: Virtual magnetic diminuendo method; Combinatorial optimization problem; Hopfield network; Spin glass; N queens problem
1. Introduction A combinatorial optimization problem is a task for searching an optimal solution from the plural combinations. In order to solve it in a short time, one of the neural network models is proposed by Hopfield and Tank about 20 years ago [1]. Because of its good performance, numerous studies have been carried out with the help of related ideas, such as a simulated annealing and a hill-climbing term, since its proposal. But some of them are just techniques to get better solutions, so there seems to be no meaningful phenomena in the actual spin glass system. Then, from the viewpoint of reconsideration referring to the
* Tel.: +81 952 28 8636; fax: +81 952 28 8865. E-mail address: [email protected]. 0531-5131/ D 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ics.2006.01.036
202
H. Wakuya / International Congress Series 1291 (2006) 201–204
original concept of the Hopfield network, a novel search method based on an analogy between the Hopfield network and the spin glass is proposed in this paper. 2. Basic idea of virtual magnetic diminuendo method Fig. 1 shows the correspondence between the Hopfield network and the spin glass. If neuron outputs (active/quiescent) V and weights W are assigned to spin states (up/down) S and interactions J, respectively, an energy E of the Hopfield network is defined by the Hamiltonian H of the spin glass system. If a solution of the combinatorial optimization problem is assigned to an energy minimum point, the Hopfield network will change the state toward reducing its energy following a theory constructed in the actual spin glass system, and finally it is expected that the state reaches the desired solution automatically. Also, as can be seen in Fig. 1, it is clear that the latter region Y is projected into the former region X. This fact implies that an idea A0 a Y inspired from some phenomenon in the spin glass system must be projected to A a X. One of the famous methods called a simulated annealing, which controls a virtual temperature from high to low gradually, belongs to this category. In contrast, a technique B a X adopted to improve the score might be projected to B0 g Y. A hill-climbing term, which changes some neuron output from 0 to 1 for escaping from a local minimum such as all neurons in certain row/column are 0, belongs to this category. The essence of this operation is to update the state of only one neuron in the network without any influence over the rest of all other neurons. From these considerations mentioned above, we summarize this underlying law as follows: An idea A0 a Y is also applicable in X, but a technique B might not show a corresponding phenomenon in Y. By the way, if a magnetic field is applied from outside the spin glass system, all spins must be aligned with their orientation. Then, as the magnetic field is reduced gradually, it makes the spin free depending on its interactions with other spins. Hereafter, a new search method inspired from this thought experiment is referred to as a virtual magnetic diminuendo method [2]. According to Fig. 1, it is clear that the magnetic field h corresponds to the threshold h. It is noticeable here that the proposed method control a position (bias) of the sigmoidal function, while the conventional simulated annealing control its gradient. So, the principle of the basic idea is completely different.
Fig. 1. Correspondence between Hopfield network and spin glass.
H. Wakuya / International Congress Series 1291 (2006) 201–204
203
3. Computer simulations There are many kinds of combinatorial optimization problems. Among them, an N queens problem is adopted to confirm an advantage of the proposed method. It is a task to put N queens, which do not collide with each other, on the N N chessboard. First of all, an energy function E is constructed to satisfy the above-mentioned conditions. Next, through a comparison to the Hamiltonian H, a weight W xy,ab between the neurons (x, y) and (a, b), a threshold H xy of the neuron (x, y) are obtained. With these parameters, an output of the neuron (x, y) in the Hopfield network is defined as follows: ( ) N X N X Wxy;ab Vab ðt Þ Hxy ðt Þ ; ð1Þ Vxy ðt þ 1Þ ¼ f a¼1 b¼1
Hxy ðt Þ ¼ hxy þ he ðt Þ; f ð zÞ ¼
ð2Þ
1; zz0; 0; zb0;
ð3Þ
where h e is a virtual magnetic term newly introduced in this study, and defined its scheduling function changing with time t linearly as, he ðt Þ ¼ 500 þ 0:1t:
ð4Þ
Table 1 is a summary of the computer simulation results starting from 1000 different initial states. In general, a ratio of correct answer decreases with the number of queens N, but the proposed method shows somewhat better performance than the conventional one. Table 1 Results of computer simulations [I] — ratio of correct answer and averaged final energy Number of queens
Proposed method
Conventional method
Ratio of correct answer (%)
Averaged final energy
Ratio of correct answer (%)
Averaged final energy
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
0.0 77.4 81.7 24.1 49.1 27.7 6.4 1.7 1.6 0.6 0.6 1.6 0.4 0.2 0.0 0.0 0.0
2048 452 378 1572 1352 1938 2522 3102 3610 3698 3944 3566 3824 4540 4544 4586 3988
0.0 77.4 80.9 23.8 44.9 17.9 5.7 0.4 0.9 0.1 0.2 0.9 0.0 0.0 0.0 0.0 0.0
2192 724 432 1826 1892 2718 3576 4434 5222 5028 5484 6074 6120 5948 6538 5582 6248
204
H. Wakuya / International Congress Series 1291 (2006) 201–204
Fig. 2. Results of computer simulations [II]: (a) correct answer; (b) matching degree of final energy and their difference.
This property is clearly confirmed following the symbols . (proposed) and o (conventional) shown in Fig. 2(a), a graphical display version of Table 1. Also, as seen from Fig. 2(b), when the number of queens N is small, most trials starting from the same initial state reach the same final energy level (w), and few energy difference is observed between the proposed and conventional method (.). But the number of queens N becomes bigger, the matching degree of the final energy becomes smaller (w), and it makes their difference bigger and bigger (.). This fact strongly suggests as follows: Although the acquired answer is not correct, its berror levelQ (final energy) is lower because of an improvement achieved by the proposed method. Later, the same tendency has been confirmed through some computer simulations of other tasks with this proposed method. 4. Discussion The correspondence between the Hopfield network and the spin glass, shown in Fig. 1, is the origin of this study. The virtual magnetic diminuendo method proposed here is one of the examples ended in success, and it tells us another possibility to discover new search methods for combinatorial optimization problems. Even if all attempts might not reach the goal, this kind of examinations must be important from the viewpoint of reconsideration referring to the original concept of the Hopfield network. Of course, a proposal in this paper just shows a possibility to develop a new technique, therefore further considerations are required to refine it. 5. Conclusions In this paper, a new search method inspired from the analogy between the Hopfield network and the spin glass is investigated. As a result of computer simulations, it is found that the proposed method has a good advantage over the conventional one. References [1] J.J. Hopfield, D.W. Tank, Neural computation of decisions in optimization problems, Biol. Cybern. 52 (1985) 141 – 152. [2] H. Wakuya, E. Iwaki, A solution of combinatorial optimization problem with virtual magnetic diminuendo method. In: Proc. 2005 Ann. Conf. of Japanese Neural Network Society, 2005. P3–15, 186–187 (in Japanese).
www.ics-elsevier.com
A new search method for combinatorial optimization problem inspired by the spin glass system Hiroshi Wakuya * Faculty of Science and Engineering, Saga University, Honjo-machi, Saga 840-8502, Japan
Abstract. A Hopfield network is a good tool for solving combinatorial optimization problems. Since its proposal, research activity around this area has been accelerated with the help of related ideas. But some of them are just techniques to get better solutions, so no meaningful phenomena might be observed in the field of the actual spin glass. In this paper, therefore, another search method inspired from an analogy between the Hopfield network and the spin glass is proposed. It is quite a simple idea to control a threshold of the Hopfield network corresponding to a magnetic field in the spin glass. In order to confirm its effectiveness, an N queens problem is adopted. As a result, it is found experimentally that the proposed method shows better score, e.g. a ratio of correct answer and an averaged final energy, than the conventional one. D 2006 Elsevier B.V. All rights reserved. Keywords: Virtual magnetic diminuendo method; Combinatorial optimization problem; Hopfield network; Spin glass; N queens problem
1. Introduction A combinatorial optimization problem is a task for searching an optimal solution from the plural combinations. In order to solve it in a short time, one of the neural network models is proposed by Hopfield and Tank about 20 years ago [1]. Because of its good performance, numerous studies have been carried out with the help of related ideas, such as a simulated annealing and a hill-climbing term, since its proposal. But some of them are just techniques to get better solutions, so there seems to be no meaningful phenomena in the actual spin glass system. Then, from the viewpoint of reconsideration referring to the
* Tel.: +81 952 28 8636; fax: +81 952 28 8865. E-mail address: [email protected]. 0531-5131/ D 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ics.2006.01.036
202
H. Wakuya / International Congress Series 1291 (2006) 201–204
original concept of the Hopfield network, a novel search method based on an analogy between the Hopfield network and the spin glass is proposed in this paper. 2. Basic idea of virtual magnetic diminuendo method Fig. 1 shows the correspondence between the Hopfield network and the spin glass. If neuron outputs (active/quiescent) V and weights W are assigned to spin states (up/down) S and interactions J, respectively, an energy E of the Hopfield network is defined by the Hamiltonian H of the spin glass system. If a solution of the combinatorial optimization problem is assigned to an energy minimum point, the Hopfield network will change the state toward reducing its energy following a theory constructed in the actual spin glass system, and finally it is expected that the state reaches the desired solution automatically. Also, as can be seen in Fig. 1, it is clear that the latter region Y is projected into the former region X. This fact implies that an idea A0 a Y inspired from some phenomenon in the spin glass system must be projected to A a X. One of the famous methods called a simulated annealing, which controls a virtual temperature from high to low gradually, belongs to this category. In contrast, a technique B a X adopted to improve the score might be projected to B0 g Y. A hill-climbing term, which changes some neuron output from 0 to 1 for escaping from a local minimum such as all neurons in certain row/column are 0, belongs to this category. The essence of this operation is to update the state of only one neuron in the network without any influence over the rest of all other neurons. From these considerations mentioned above, we summarize this underlying law as follows: An idea A0 a Y is also applicable in X, but a technique B might not show a corresponding phenomenon in Y. By the way, if a magnetic field is applied from outside the spin glass system, all spins must be aligned with their orientation. Then, as the magnetic field is reduced gradually, it makes the spin free depending on its interactions with other spins. Hereafter, a new search method inspired from this thought experiment is referred to as a virtual magnetic diminuendo method [2]. According to Fig. 1, it is clear that the magnetic field h corresponds to the threshold h. It is noticeable here that the proposed method control a position (bias) of the sigmoidal function, while the conventional simulated annealing control its gradient. So, the principle of the basic idea is completely different.
Fig. 1. Correspondence between Hopfield network and spin glass.
H. Wakuya / International Congress Series 1291 (2006) 201–204
203
3. Computer simulations There are many kinds of combinatorial optimization problems. Among them, an N queens problem is adopted to confirm an advantage of the proposed method. It is a task to put N queens, which do not collide with each other, on the N N chessboard. First of all, an energy function E is constructed to satisfy the above-mentioned conditions. Next, through a comparison to the Hamiltonian H, a weight W xy,ab between the neurons (x, y) and (a, b), a threshold H xy of the neuron (x, y) are obtained. With these parameters, an output of the neuron (x, y) in the Hopfield network is defined as follows: ( ) N X N X Wxy;ab Vab ðt Þ Hxy ðt Þ ; ð1Þ Vxy ðt þ 1Þ ¼ f a¼1 b¼1
Hxy ðt Þ ¼ hxy þ he ðt Þ; f ð zÞ ¼
ð2Þ
1; zz0; 0; zb0;
ð3Þ
where h e is a virtual magnetic term newly introduced in this study, and defined its scheduling function changing with time t linearly as, he ðt Þ ¼ 500 þ 0:1t:
ð4Þ
Table 1 is a summary of the computer simulation results starting from 1000 different initial states. In general, a ratio of correct answer decreases with the number of queens N, but the proposed method shows somewhat better performance than the conventional one. Table 1 Results of computer simulations [I] — ratio of correct answer and averaged final energy Number of queens
Proposed method
Conventional method
Ratio of correct answer (%)
Averaged final energy
Ratio of correct answer (%)
Averaged final energy
3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
0.0 77.4 81.7 24.1 49.1 27.7 6.4 1.7 1.6 0.6 0.6 1.6 0.4 0.2 0.0 0.0 0.0
2048 452 378 1572 1352 1938 2522 3102 3610 3698 3944 3566 3824 4540 4544 4586 3988
0.0 77.4 80.9 23.8 44.9 17.9 5.7 0.4 0.9 0.1 0.2 0.9 0.0 0.0 0.0 0.0 0.0
2192 724 432 1826 1892 2718 3576 4434 5222 5028 5484 6074 6120 5948 6538 5582 6248
204
H. Wakuya / International Congress Series 1291 (2006) 201–204
Fig. 2. Results of computer simulations [II]: (a) correct answer; (b) matching degree of final energy and their difference.
This property is clearly confirmed following the symbols . (proposed) and o (conventional) shown in Fig. 2(a), a graphical display version of Table 1. Also, as seen from Fig. 2(b), when the number of queens N is small, most trials starting from the same initial state reach the same final energy level (w), and few energy difference is observed between the proposed and conventional method (.). But the number of queens N becomes bigger, the matching degree of the final energy becomes smaller (w), and it makes their difference bigger and bigger (.). This fact strongly suggests as follows: Although the acquired answer is not correct, its berror levelQ (final energy) is lower because of an improvement achieved by the proposed method. Later, the same tendency has been confirmed through some computer simulations of other tasks with this proposed method. 4. Discussion The correspondence between the Hopfield network and the spin glass, shown in Fig. 1, is the origin of this study. The virtual magnetic diminuendo method proposed here is one of the examples ended in success, and it tells us another possibility to discover new search methods for combinatorial optimization problems. Even if all attempts might not reach the goal, this kind of examinations must be important from the viewpoint of reconsideration referring to the original concept of the Hopfield network. Of course, a proposal in this paper just shows a possibility to develop a new technique, therefore further considerations are required to refine it. 5. Conclusions In this paper, a new search method inspired from the analogy between the Hopfield network and the spin glass is investigated. As a result of computer simulations, it is found that the proposed method has a good advantage over the conventional one. References [1] J.J. Hopfield, D.W. Tank, Neural computation of decisions in optimization problems, Biol. Cybern. 52 (1985) 141 – 152. [2] H. Wakuya, E. Iwaki, A solution of combinatorial optimization problem with virtual magnetic diminuendo method. In: Proc. 2005 Ann. Conf. of Japanese Neural Network Society, 2005. P3–15, 186–187 (in Japanese).