pheromone trail was the key. As more and more ants picked the shorter branch, it accumulated more and more of their pheromone, increasing the likelihood that other ants would choose it. Here’s how it works: Let’s say two ants set out across the bridge at the same time. The first ant takes the shorter branch, and the second the longer one. By the time the first ant reaches the food, the second is only halfway across the bridge. By the time the first ant returns all the way to the colony, the second ant has just arrived at the food. To a third ant standing at the split in the bridge at this point, the pheromone trail left by the first ant would be twice as strong as that left by the second (since the first ant went out and returned), making it more likely to take the shorter branch. The more this happens, the stronger the pheromone trail grows, and the more ants follow it.
Ant colonies, in other words, have evolved an ingenious way to determine the shortest path between two points. Not that any of the ants are doing so on their own. None of them attempts to compare the length of the two branches independently. Instead, the colony builds the best solution as a group, one individual after another, using pheromones to “amplify” early successes in an impressive display of self-organization.
Taking this idea one step further, Deneubourg and his colleagues proposed a relatively simple mathematical model to describe this behavior. If you know how many ants have taken the shorter branch at any particular time, Deneubourg said, you can reliably calculate the probability of the next ant choosing that branch. To demonstrate this, he plugged his team’s equations into a computer simulation of the double-bridge experiment and ran it for a thousand ants. The results mirrored those of real ants. When the branches were the same length, the odds of an ant picking either one were fifty-fifty. But when one branch was twice as long as the other, the odds of picking the shorter one shot up dramatically.
The key to the colony’s system, in short, lay in the simple rules that each ant applied to local information. If you changed these rules, you would change the behavior pattern of the whole colony.
The implications of this discovery were not lost on Dorigo: If real ant colonies could find the shortest path between two points, then why couldn’t researchers do the same thing with “virtual ants”? Dorigo knew how to design software “agents” that could follow simple rules just like real ants do. Why couldn’t these software agents find the shortest path, too? Only, what if the path wasn’t the distance between an ant nest and a pile of food? What if it was the shortest route for a message across the Internet between two computers? Or the shortest distance for a package going from a factory warehouse in California to a customer in Florida? Or the shortest path between multiple steps in an industrial process? Then there was the concept of the “shortest path” itself. What if you redefined “shortest” as “most efficient” or “least costly”? Wouldn’t that be a handy tool?
“When I went back to Milan to discuss these ideas with Professor Alberto Colorni, who was supervising my work, he asked me to write a simple program as a proof of principle, to show it wasn’t just a crazy idea,” Dorigo says. At the time, Dorigo was working on a class of mathematical puzzles known as combinatorial optimization problems, which are relatively easy to describe but deceptively difficult to solve. One of the best-known examples is the traveling salesman problem, which involves the following scenario: A salesman needs to visit customers in a number of cities. What’s the shortest path he can take to visit each city once before returning back home?
When the problem involves just a few cities—let’s say, Moscow, Hong Kong, and Paris—you can figure out the answer on the back of an envelope. Leaving from the airport near his home in Cleveland, the salesman has three options for his first stop: Moscow, Hong Kong, or Paris. Let’s say he chooses Hong Kong. From there he has two choices: Moscow or Paris. Let’s say he flies to Paris. That leaves only Moscow before he can fly home. If you made a list of all the other possible sequences (such as Paris to Moscow to Hong Kong, or Moscow to Hong Kong to Paris, and so on), you would have a total of six to consider. Compare the mileage for each sequence and you have your answer.
But here’s the tricky part. If you add a fourth city to the salesman’s journey, you make the problem significantly more difficult. Now you have four times as many possible routes to consider—twenty-four instead of six. Add a fifth city and you get 120 possible routes. Jump ahead to ten cities and you’re talking about more than 3.6 million possible routes. The number of solutions, in other words, increases exponentially with each new city. When you get to thirty cities, there aren’t enough years in a lifetime to list all the possible routes.
Dorigo thought it might be interesting instead to let virtual ants give it a try. Rather than trying to identify every possible outcome of the salesman’s decision making, the ant system used trial-and-error shortcuts to find a handful of good ones. Instead of being straightforward and linear, it was decentralized and distributed. Instead of calling for complicated calculations, it relied on simple rules of thumb. Instead of getting swamped by the exponential nature of the math, it took advantage of that same snowballing effect to rapidly turn small differences into big advantages. It was different, in other words, because it harnessed self-organization in a smart way.
So Dorigo and Vittorio Maniezzo, a fellow graduate student, created a set of virtual ants capable of cooperating with one another to find the shortest route for the traveling salesman. Their secret weapon: “virtual pheromones” that the ants would leave along the way. Imagine a map with fifteen cities that a salesman needed to visit. At the beginning of the first cycle, ants were placed randomly on all of the cities. Then each ant used a formula based on probability to decide which town to visit next. This formula considered two factors: which city was closest, and which city had the strongest pheromone trail leading to it. At the start, there were no pheromone trails, so the closest cities tended to be selected. As soon as each ant completed a tour of all fifteen cities, it retraced its path, depositing virtual pheromone on its route. The shortest routes discovered by the ants tended to receive the most pheromone, while the longest ones were allowed to “evaporate” more rapidly. This enabled the ants, as a group, to remember the best routes. So when the second cycle was run, and the ants worked their way from city to city again, each ant built upon the successes of the first cycle by favoring the strongest pheromone trails. After repeating this time and again, the ants kept reducing their travel times, until the pheromone trails on the shortest segments were so strong that none of them could resist choosing them.
The results were quite encouraging.
“We discovered that the ants could find nearly optimal solutions for as many as thirty, fifty, or even a hundred cities,” Dorigo says.
Not that the ants didn’t sometimes make mistakes. If a particular ant, while hopping from city to city, happened to get caught in a loop along the way—like a twig in a river eddy—other ants occasionally followed, resulting in a nonsolution to the problem. To prevent this, Dorigo and Maniezzo instructed the ants to forget such loops when depositing pheromone on completed trails. Other problems required similar fixes. But none of the ants’ bad habits were so serious that they couldn’t be tweaked in one way or another, or made more effective by pairing them with more specialized algorithms.
“The important thing was that the ant colony optimization worked because of the implicit cooperation among many agents,” Dorigo says. “On their own, each artificial ant built a solution which was usually not very good. But together, by exchanging information—not talking to each other, but simply by exchanging information through virtual pheromones—the ants ended up finding some very good solutions.” In this way, cooperation became cumulative. Instead of representing simply the sum of the ants’ individual efforts, the search got smarter as it went forward, powered by the mechanisms of self-organization—decentralized control, distributed problem-solving, and multiple interaction between agents.
It wasn’t long before other computer scientists were adapting the ant-colony approach to solve a variety of difficult problems. A few researchers even experimented with
