Abstract
Combinatorial optimization to search for the best solution across a vast number of legal candidates requires the development of a domainspecific computing architecture that can exploit the computational power of physical processes, as conventional generalpurpose computers are not powerful enough. Recently, Ising machines that execute quantum annealing or related mechanisms for rapid search have attracted attention. These machines, however, are hard to map application problems into their architecture, and often converge even at an illegal candidate. Here, we demonstrate an analogue electronic computing system for solving the travelling salesman problem, which mimics efficient foraging behaviour of an amoeboid organism by the spontaneous dynamics of an electric current in its core and enables a high problemmapping flexibility and resilience using a resistance crossbar circuit. The system has high application potential, as it can determine a highquality legal solution in a time that grows proportionally to the problem size without suffering from the weaknesses of Ising machines.
Introduction
Combinatorial optimization problems are computationally demanding tasks appearing in various practical applications, such as optimization of traffic flow, path planning, nurse scheduling and advertisement allocation^{1,2,3,4}. Often these problems become intractable for conventional von Neumanntype computers, such as generalpurpose CPUs; they need to evaluate an enormous number of candidate solutions in a serial manner, where the number of the candidates grows exponentially with the problem size, leading to “combinatorial explosion”.
The travelling salesman problem (TSP) is one of the most widely investigated combinatorial optimization problems; given a map of \(N\) cities, the TSP is stated as a problem of finding the shortest route for visiting each city exactly once and returning to the starting city^{5,6}, where the number of all legal solutions (possible routes) grows factorially as \(\left(N1\right)!/2\). The TSP is a nondeterministic polynomial time (NP)hard problem, that is, any exact algorithm to find the optimal solution (exactly the shortest route) for a general instance in polynomial time is not known so far. On the other hand, various natureinspired approximation algorithms have been proposed to promptly derive a highquality legal solution (a satisfactory short route), such as the kopt algorithm with simulated annealing, genetic algorithm, particle swarm optimization algorithm and ant colony optimization algorithm^{7,8,9,10,11}.
Many of the natureinspired algorithms are formulated to update multiple variables in parallel to achieve rapid search, whereas the serial process of the CPU that manipulates a single bit at a time can only simulate the parallelism in a limited way. Therefore, it is required to develop a novel domainspecific computing architecture to implement these algorithms to maximize their parallel search capabilities by exploiting physical processes of specific hardware, expecting to cultivate new potentials and markets of combinatorial optimization. The first physical computing system for solving the TSP comprises the Hopfield’s recurrent neural network implemented by an electronic circuit^{12,13}. This system, however, was not so useful because it frequently converges at a local minimum state (a lowquality solution) and sometimes cannot reach even a candidate solution for some problem instances^{14,15,16}. In fact, for some instances of 10city TSP, it was reported that the rate of finding a legal solution was at most 20%^{15}.
In recent years, many proposals on rapid combinatorial optimization by physically implementing the “Ising machine” or “annealing machine” with several exploration mechanisms, such as quantum annealing (QA) and related methods, have attracted a lot of public attention^{17,18,19,20,21}. Each of these machines explores an optimal solution by mapping the problem to a process of finding a minimumenergy spin assignment in the Ising model that abstracts a ferromagnetic material^{17,22,23}. However, the problem mapping and parameter tuning of the Ising model are complex and costly. For the \(N\)city TSP, the regular layout of spin variables with sparse connectivity requires redundant variables to be introduced in the order of \({N}^{4}\) to handle irregularly distributed cities, leading to a rapid increase in the circuit area^{24,25,26,27,28}. Figure 1a shows a graph structure of the Ising model, which is referred to as a chimera graph^{17}. In such a structure, the consistency between the redundant variables can be broken potentially. When the parameter tuning cannot be made appropriately, the Ising model sometimes converges at an illegal state in which constraints of the TSP to exclude the revisiting of a oncevisited city and to exclude simultaneous visits to multiple cities are violated (see Supplementary Information [SI]). Actually, in the physical Ising machines solving the Maxcut problem, the probability of reaching a legal state varies depending on the connectivity among graph nodes, and the sparsely connected graph often results in even worse performance owing to the variable overhead^{29}.
Our approach is not based on the Ising model. We focus on a living amoeboid organism that performs trialanderror behaviour to survive efficiently and resiliently in a harsh environment, deforming its gellike body^{30,31}. Here, we demonstrate, as a proof of concept, an analogue electronic computing system called an “electronic amoeba”^{32,33}, inspired by the food search and risk avoidance behaviour of a singlecelled amoeboid organism, Physarum polycephalum^{30,31,34,35,36,37,38,39}. In the electronic amoeba, an arbitrary TSP instance can be mapped on the resistor network of a crossbar structure shown in Fig. 1b, which we call the “instancemapping circuit (IMC)”. The architecture of the IMC is similar to that of the Hopfield’s recurrent neural network^{12,13}. However, as shown in Fig. 2a, it is connected with the “amoeba core”, which contributes to avoiding the convergence at an illegal state. In the author’s previous works^{35,38,39}, we formulated an algorithm representing a primitive idea of the electronic amoeba and predicted its potential performance without any physical implementation. In this paper, with results obtained from numerical simulations using a conventional computer and laboratory experiments using a physically fabricated circuit (Fig. 2b), for the first time, we show that the electronic amoeba finds a highquality TSP solution in a time that is proportional to \(N\). The electronic amoeba is highly scalable and energyefficient as it comprises existing complementary metaloxide semiconductor (CMOS) devices and is expected to be useful for wide range of applications.
Results
Figure 2a shows a schematic of the electronic amoeba composed of the amoeba core and IMC, which electronically mimics the solutionsearching dynamics of a socalled “amoebabased computer” that employs a living amoeba to search for a solution to the TSP^{38,39} (see Supplementary Information). The state of each unit in the amoeba core represents the decision on where and when to visit. The IMC implements a type of feedback control, called the “bounceback control”, which refers to the TSP constraints and intercity distances from the given map and sends a “bounceback signal” to each unit in the amoeba core^{36,37,38,39}; the signal is determined in accordance with the recurrent neural network dynamics defined by Eq. (1) in “Methods”.
We first conduct numerical simulations using a circuit simulator run on a conventional computer (see “Methods”) to determine if the electronic amoeba can solve the 4city TSP instance shown in Fig. 3a. Figure 3b demonstrates an example of output waveforms obtained from the circuit simulation. The subscripts of the state of each unit \({X}_{V,k}\), \(V\) and \(k\), mean that city \(V\) is visited at the \(k\)th order. Initially, every unit takes a state of 1 because the capacitor charge is set to zero. The IMC then sends the bounceback signals to all units to flip their states from 1 to 0, as the allone states are violating constraints of the TSP. The state of each unit gradually approaches to 0 when charging the capacitor by injecting a current from the current source. After several flips have been induced by the bounceback control, the dynamics of all units became stable as shown in the hatching area in Fig. 3b. At this moment, the electronic amoeba finds an optimal solution, D → A → B → C → D, which corresponds to the shortest route.
Using a circuit simulator, we have investigated the solutionsearching performance of the electronic amoeba for \(N\) ranging from 10 to 30. We performed 50 trials for each instance for \(N\) not greater than 20 but only once for each \(N\) more than 20 because the simulation time increased rapidly; it took 5 h for the 20city instances but took 6 days for the 30city cases. In each trial, resistances in the units were randomly assigned from 1 Ω to 10 kΩ to lead the electronic amoeba to explore a wider state space. To evaluate the rate of finding a legal solution in the electronic amoeba, we performed 560 trials for solving the TSP of 10–30 cities. The rate was found to be 100%; the system certainly converged to one of the legal solutions for every try. This is because the amoeba core always stabilizes at a steady state in which no variable violates the TSP constraints^{39}; in such a state, no further change in all units in the amoeba core is induced by the bounceback signals. Figure 4a shows the length of the route obtained by the circuit simulator. In Fig. 4a, the vertical axis is normalized by the average route length obtained from random sampling of 10,000 trials; if a value on the vertical axis is less than 1.0, it implies that the quality of the legal solution found is higher than that found by random sampling. The results indicate that the electronic amoeba finds higherquality solutions than random sampling. Moreover, the average of the route length was on a declining trend; the quality did not degrade even when the problem size \(N\) became larger. By introducing random variations in the resistances of the units, each unit varies the velocity of the transition from state 1 to 0, and a wide variety of legal solutions were found as shown in Fig. 4b; it reached different solutions even for an identical instance, although it did not guarantee to reach the optimal one.
Figure 4c shows that the average time required for the electronic amoeba to find a legal solution increases almost only linearly as a function of \(N\). This result reproduces the observation confirmed in the amoebabased computer well; the living amoeba reached an approximate solution of the TSP in linear time^{38,39}. A hypothetical mechanism of the lineartime solution has been proposed with an abstract mathematical model of the solutionsearching dynamics of the amoebabased computer; the numerical simulation of the model, named “AmoebaTSP”, suggested that the lineartime solution can be achieved if the hub of the singlecelled organism could supply the intracellular resources to grow its branches with a constant rate, even while responding to the bounceback signals^{38,39}. The lineartime operation is attributed to the design of the bounceback control together with parallel operations of all units in the amoeba core. The units try to choose a path between two cities having a shorter distance, according to information on shortness of the distances of possible paths through accumulating and comparing experiences of being inhibited by the bounceback signals; the two units representing a shorterdistance path are less frequently inhibited compared to the others and are allowed to take relatively larger state values. The bounceback rule is designed so that once a path and their visiting orders are decided, the rule restricts the amoeba core from changing the decision after that. Thereby the system decides each path one by one, avoiding illegal paths.
The electronic amoeba will reach a highquality solution in linear time even for largersize instances if it follows a similar mechanism as AmoebaTSP. Moreover, the solution search time of the electronic amoeba can be reduced further as the flipping of the state of each unit can be accelerated by increasing the current and/or decreasing the capacitance (see Supplementary Information).
We compared the solutionsearching performance of the circuit simulatorbased electronic amoeba with that of a representative stochastic local search algorithm, the 2opt (see “Methods”), which is a simple and fast method requiring no parameter optimization^{7}. The quality of the solution obtained by the 2opt gets higher (and saturates eventually) as its main operation iterates for a larger number of times, but we terminated the iteration when the quality become equal to that obtained by the electronic amoeba and measured the time required by the termination. As shown in Fig. 4d, the 2opt required an amount of time that grows as a quadratic function of the problem size, whereas the electronic amoeba needed only linear time to present the solution with the same level of quality (Fig. 4c). Accordingly, the electronic amoeba, once physically implemented, will be more useful for finding a highquality solution in a shorter search time than the 2opt when run on a conventional computer and when the number of cities exceeds 50.
By fabricating the electronic amoeba physically (Fig. 2b) using CMOS devices, we have verified that the fabricated system can solve various 4city TSP instances as shown in Fig. 5a–e, where the optimal and worst solutions are summarized in Table 1. The amoeba core comprises 16 branches, and we can map an arbitrary 4city instance by changing the resistance values in the IMC. The time evolution of the state variables in the fabricated system is shown in Fig. 3c. At an initial stage, the variables behaved similarly to those in the circuit simulation (Fig. 3b), and they became stable after reaching a solution. The bottom of Fig. 5a–e shows that the system found the shortest route for instances A–C and E where we performed 50 trials for each instance without changing the resistance values.
The fabricated system could not find the shortest route for instance D as shown in Fig. 5d, although the shortest route length of D equals that of C. This would be attributed to a number of device variations in the fabricated circuit, such as the threshold voltage variation in the CMOS inverter, offset voltage variation in the operational amplifier and difference in the wiring length in the IMC, which might create a preference when making a decision. However, for the cases where the route lengths are widely distributed such as instance E, the system reached the optimal solution, overcoming the preference. When solving largersized instances and introducing more noise, the system performance is expected to become reliable and robust (Supplementary Fig. S6), because the differences in qualities (route lengths) of legal solutions of those instances become relatively smaller than that of smallersized instances, depending on noise amplitude.
Note that it is necessary for the Ising machines to expend considerable efforts on problemmapping and parametertuning processes prior to solving the problem. If this preprocessing could be made properly, the Ising model combined with simulated annealing exhibits a better performance in finding a higherquality TSP solution than the electronic amoeba, otherwise it fails to reach even a legal solution (see Supplementary Information). In contrast, there is no need for the electronic amoeba to execute complex and costly preprocessing because the IMC offers a high problemmapping flexibility with its unrestricted connectivity between an arbitrary pair of variables and requires only simple numerical operations to determine the parameters (see “Methods”). Moreover, when the resistors in the IMC are replaced with memristors or atomic switches, enabling dynamic resistance rewriting^{40,41,42,43,44}, the TSP instance to be solved can be updated promptly even in the middle of the solutionsearching process, and the bounceback control will enable the amoeba core to find a new solution of the updated instance effortlessly by slightly revising the previous one. Such a dynamically rewritable IMC will enable the electronic amoeba to respond resiliently to sudden changes in the problem constraints caused by unexpected failures occurring in everchanging practical situations.
The scalability is an important issue for the implementation of the electronic amoeba. For solving \(N\)city TSP, the amoeba core needs \({N}^{2}\) units, and the crossbar IMC requires \(2{N}^{2}\) wires and \({N}^{4}\) resistors at cross points of the wires. The circuit area of the electronic amoeba, therefore, grows in the order of \({N}^{2}\), which is less costly than the Ising machines that require the area in the order of \({N}^{4}\). This order is apparently comparable to that of the Ising modelbased digital machines with full connectivity in logical level. However, the physical crossbar in the electronic amoeba produces advantages in terms of execution time and energy consumption; the crossbar allows to execute the bounceback control for all variables in a fully parallel manner, whereas the FPGAbased digital Ising machine requires a lot of memory accesses to achieve the logical fullconnectivity, consuming higher time and energy costs. Owing to the modern digital LSI technology, each capacitance in the amoeba core will be downsized to the minimum level where its charging time is distinguishable from that of the parasitic elements. It is expected that stateoftheart nanotechnology to fabricate a nanoscale crossbar structure equipped with memristors to represent analogue resistance values^{42,43} will suppress the increase in the physical size of the IMC. The electronic amoeba can be built and maintained with a significantly cheaper cost than the Quantum annealing machine that requires a lot of elaborate equipment for refrigeration and maintenance of quantum coherence. Furthermore, while the computing speed of other Ising machines are unscalable as they suffer from the "Neumann bottleneck" that limits the data transfer rate in a data bus between memory and operation sections, the electronic amoeba does not encounter such a limitation.
In this paper, we demonstrated that, given an arbitrary TSP instance, the electronic amoeba enables to start the computing readily after simple resistance determination operations in the IMC and to find a highquality solution certainly in only linear time, exploiting the spontaneous dynamics of the electric current in the amoeba core. This reliable and swift solutionsearching capability could be beneficial for particular applications that prioritize the search time over the quality of a solution found. For example, in a situation at a disaster site where presenting reliable evacuation routes for residents is necessary, making a swift announcement should be prioritized than deriving the exactly optimal routes. The electronic amoeba would be more useful than using conventional computers to run the stochastic local search algorithm when the number of cities exceeds a hundred or more. Moreover, the compactness of the IMC suggests that the systemonchip approach supported by semiconductor LSI technologies will further enhance the scalability and energyefficiency, making it useful for wider cloud and edgecomputing applications. One of our future subjects is to improve the quality of the solution found by the electronic amoeba. Possible approaches are to assign appropriate initial states to the amoeba core units using the genetic algorithm, to impose stochastic fluctuations using a hardware random number generator to forcibly escape local minimum solutions (see SI), and to introduce delays in the bounceback control to induce the oscillation of state variables^{45}.
Methods
Bounceback control of the electronic amoeba
To implement the bounceback control for solving the TSP on the crossbar IMC, we followed the basic scheme of the amoebabased computing system (see SI)^{36,37,38}. For the \(N\)city TSP, \(N\times N\) state variables are needed where each variable \({X}_{Vk}\) with subscript \(Vk\) indicates that city \(V\) is visited at the \(k\)th order: when \({X}_{Vk}=1\), the city \(V\) is visited at the \(k\)th order. The bounceback signal in the electronic amoeba is generated by the crossbar IMC shown in Fig. 2a. The IMC evaluates all bounceback signals in a parallel and continuous fashion. The IMC circuit computes the bounceback signal is as follows,
where \({R}_{f}\) is a feedback resistor for the opamp and \({R}_{Vk,Ul}\) is a resistor at cross point of the output line \(Vk\) and the input line \(Ul\) in the crossbar corresponding to the intercity distance. \({V}_{T}\) is a threshold value of the threshold function \(F\) in the output portion of the crossbar IMC, which is implemented by a comparator. In this study we used \({V}_{T}=1.5\) V and \({R}_{f}=10\) kΩ. \({R}_{f}/{R}_{Vk,Ul}\) is given by
where \(\text{dist}(V,U)\) is an intercity distance between cities \(V\) and \(U\) and \(\lambda\) is a normalization factor. \({R}_{f}/{R}_{Vk,Ul}\) defines the interaction between the state variables so as to (1) prohibit the system from visiting the city once visited, (2) prohibit the system from visiting different cities at once, and (3) minimize the total length of a travel route. \(\lambda\) is a normalized coefficient and we use \(\lambda =\text{max}(\text{dist}({V}^{^{\prime}},{V}^{{^{\prime}}{^{\prime}}})+\text{dist}({V}^{{^{\prime}}{^{\prime}}},{V}^{{^{\prime}}{^{\prime}}{^{\prime}}}))/\theta\), where \(\theta\) is a threshold in the sigmoid function and \(\text{dist}({V}{^{\prime}},{V}{{^{\prime}}{^{\prime}}})+\text{dist}({V}{{^{\prime}}{^{\prime}}},{V}{{^{\prime}}{^{\prime}}{^{\prime}}})\) is the maximum value of the total distance between the cities \({V}^{^{\prime}}\) and \({V}^{{^{\prime}}{^{\prime}}{^{\prime}}}\) via \({V}^{{^{\prime}}{^{\prime}}}\).
A CMOS inverter with resistors shown as square boxes in Fig. 2a represents a sigmoid function. Here, the slope of the sigmoid function is defined by the resistance values \({R}_{s1}=390\) kΩ and \({R}_{s2}=2.2\) MΩ. The measured behaviour of a pseudopodlike branch in the fabricated amoeba core is shown in Supplementary Information.
The bounceback control ensures the system does not stabilize whenever there are constraints that remain unsatisfied as the bounceback signals flip the constraintviolating variables from 1 to 0. The IMC does not have any free parameters, although the degree of resistance variation in the amoeba core is adjustable. The electronic amoeba, therefore, does not require any complex and costly preprocessing for problem mapping and parameter tuning.
Circuit simulator
We used the electronic circuit simulator, LTspice XVII (Simulation Program for Integrated Circuit Emphasis simulator, SPICE, https://www.analog.com). The current source is \(5{N}^{2}\) µA for the number of cities \(N\), capacitance \(C\) is 500 pF and initial voltage of capacitors is set to 1.5 V in the circuit simulation. In the fabricated electronic amoeba integrating the commercial discrete devices, \(C\) is the sum of discrete components near 470 pF and the current source is set to 80 µA.
2opt
The stochastic local search algorithm, 2opt, updates the route by iterating the following main operation starting from a randomly chosen route (a legal solution); it chooses two cities randomly and inverts a path between the two locally in the current route if the inversion resulted in a reduction in route length. We performed the 2opt calculation using a conventional computer (Intel Xeon processor E51650 v2 @ 3.50 GHz).
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Acknowledgements
We thank Prof. Tadao Nakamura at Keio University and Stanford University for his significant comments. This work was partly supported by JSPS KAKENHI Grant number H8H01487, PRESTOJST Grant no. JPMJPR1321, and NEDO Grant no. 161010090.
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All authors conceived and designed the experiments; K.S. carried out the experiments, collected the overall data, analyzed the data, and wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Saito, K., Aono, M. & Kasai, S. Amoebainspired analog electronic computing system integrating resistance crossbar for solving the travelling salesman problem. Sci Rep 10, 20772 (2020). https://doi.org/10.1038/s41598020776177
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DOI: https://doi.org/10.1038/s41598020776177
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