This letter presents a method for solving several linear equations in max-plus algebra. The essential part of these equations is reduced to constraint satisfaction problems compatible with mixed integer programming. This method is flexible, compared with optimization methods, and suitable for scheduling of certain discrete event systems.

A method for flexibly allocating and reallocating optical access network (OAN) resources, including fibers and equipment, using the constraint satisfaction problem (CSP) is described. OAN resource allocation during service delivery provisioning involves various input conditions and allocation sequences, so an OAN resource allocation method has to support various workflow patterns. Furthermore, exception processing, such as reallocating OAN resources once they are allocated, is inevitable, especially during the spread of service using optical fiber and during the deployment of an optical access network. However, it is almost impossible to describe all workflow patterns including exception processes. Improving the efficiency of these exception processes, as well as that of the typical processes, is important for reducing the service delivery time. Describing all these patterns and process flows increases development cost. The CSP can be used to search for solutions without having to fix the process sequence and input conditions beforehand. We have formulated the conditions for OAN resource allocation and reallocation as a CSP. Use of this method makes it possible to handle various allocation workflow patterns including exception processes. Evaluation of the solution search time demonstrated its feasibility.

We have proposed a neural network called the Lagrange programming neural network with polarized high-order connections (LPPH) for solving the satisfiability problem (SAT) of propositional calculus. The LPPH has gradient descent dynamics for variables and gradient ascent dynamics for Lagrange multipliers, which represent the weights of the clauses of the SAT. Each weight wr increases according to the degree of unsatisfaction of clause Cr. This causes changes in the energy landscape of the Lagrangian function, on which the values of the variables change in the gradient descent direction. It was proved that the LPPH is not trapped by any point that is not a solution of the SAT. Experimental results showed that the LPPH can find solutions faster than existing methods. In the LPPH dynamics, a function hr(x) calculates the degree of unsatisfaction of clause Cr via multiplication. However, this definition of hr(x) has a disadvantage when the number of literals in a clause is large. In the present paper, we propose a new definition of hr(x) in order to overcome this disadvantage using the "min" operator. In addition, we extend the LPPH to solve the constraint satisfaction problem (CSP). Our neural network can update all neurons simultaneously to solve the CSP. In contrast, conventional discrete methods for solving the CSP must update variables sequentially. This is advantageous for VLSI implementation.

We propose hysteresis neural network solving combinatorial optimization problems, Box Puzzling Problem. Hysteresis neural network searches solutions of the problem with nonlinear dynamics. The output vector becomes stable only when it corresponds with a solution. This system does never become stable without satisfying constraints of the problem. After estimating hardware calculating time, we obtain that numerical calculating time increases extremely comparing with hardware time as problem's scale increases. However the system has possibility of limit cycle. Though it is very hard to remove limit cycle completely, we propose some methods to remove this phenomenon.