Relaxation: Why the Next Best Option is the Best Option

When somebody tells you to relax, it’s probably because you make a bigger deal out of your problems than you should. Here’s some counterintuitive advice: you don’t have to relax yourself. Relax your problems instead.

Let’s take a problem. A man named Lincoln Abraham works as a lawyer in a city. He has to ride hundreds of miles over many weeks and move through towns in numerous states to try cases. It’s a hard job and wastes a lot of time. One fine day Mr. Abraham decides to optimise his route. He asks himself, how I can visit all the necessary towns while covering as few miles as possible and without going to any town twice? Mr. Abraham soon realises that this is gonna be harder than he initially thought. What he is trying to solve is a popular (intractable) problem known as the travelling salesman problem.

The question isn’t whether we can find the shortest route. One can simply crank out a list of all the possibilities and measure each one. The real issue is that as the number of towns grows, the list of possible routes connecting them explodes. As the scale increases this problem is beyond the reach by even the most powerful computers.

In simple terms, a route is just an ordering of the towns. Trying to find all of them by brute force is the equivalent of sorting a deck of cards by throwing them in the air until they land in order. This works only in theory.

The travelling salesman problem is an example of an intractable problem. One that is practically impossible to solve! But this isn’t the end of the story. This is an opportunity for us to learn something: how to best approach problems whose optimal answers are unreachable. How to relax problems.

The simplest way to relax any problem is to relax one or more of its constraints. In other words, instead of solving the real problem we try solving the problem we wish we had.

For instance, you can relax the travelling salesman problem by letting the salesman visit the same town more than once, thereby letting Mr. Abraham retrace his steps for free. You’ll soon see that solving this looser problem takes no time at all.

While this loose solution isn’t the answer to the real problem, it is quite useful all the same. It’s the next best thing you have to an ideal solution. Trying to solve it any further may have diminishing returns—the effort for accuracy far outweighs the benefit of the difference. In practical problems, the next best solution is the best solution.

We often relax problems in real life without knowing it. If you have ever asked yourself, “If money wasn’t a problem would I be doing what I’m about to do?” you’ve engaged in relaxation. What you are doing here is making the intractable real life problem tractable by removing some of its constraints. These questions help you make progress in a loose form of the problem before porting it back to reality. If not a practical solution, this loose version gives you direction.

We may not have to face Mr. Abraham’s routing problem but there are multiple cases in real life where relaxation in necessary. For example, when you have to deliver a five-day-work in two days, or when you have to squeeze a one-hour presentation in 10 minutes.

You can either relax the quality of the output, for example deliver a less-than-ideal solution within time. Or, you can take more time and face the consequences. Both are strategies of relaxation and chances are you already use them in real life.

Relaxation is the best way to mitigate analysis paralysis. An answer at least half as good as the perfect solution in less time is a more practical solution. Always! Businesses that move fast and deal with a lot of ambiguity follow this method. Rather than wasting time searching for a perfect answer, they simply ask, “How close can we get to the perfect solution with a relaxed problem?” Turns out, pretty close!

Unless you’re willing to waste aeons striving for perfection every time you encounter a hitch, hard problems demand that you imagine easier versions and tackle those first. When applied correctly, it’s one of the best ways of making progress.

The message is simple but profound: if you’re willing to accept solutions that are close enough, then even some of the hairiest problems can be tamed with the right techniques. We all practise it in some form. This essays helps us create a system that codifies the decision-making process.