This article is an excerpt from the Shortform book guide to "Algorithms to Live By" by Brian Christian and Tom Griffiths. Shortform has the world's best summaries and analyses of books you should be reading.

How do you approach problem-solving in day-to-day life? What do you do when faced with a complex problem—one that has no obvious solutions?

Some daily problems seem impossible to solve—for example, complex optimization problems in which you need to find the optimal arrangement of a set of variables to achieve the best outcome under specific constraints. If you’re trying to purchase the cheapest airline tickets available that get you the most vacation days with good weather or trying to minimize risk in your investment portfolio, you’re solving an optimization problem.

Here is how to solve complex problems with no obvious solutions.

## How to Solve Impossible Problems

Christian and Griffiths explain that when optimization problems reach a certain level of complexity, they become impossible to solve in the sense that they would take too long for even the fastest computer on earth to calculate the perfect solution. Since we don’t have access to this level of computing power in our lives, perfect solutions are often far out of our reach, even for relatively simple optimization problems.

Christian and Griffiths argue that the best way to solve complex problems like these is to strategically embrace imperfection.

Sometimes, simply lowering your standard for success is necessary to keep moving forward. In some cases where it’s impossible to find the perfect solution, getting close is just as good. Cristian and Griffiths argue that, at a certain point, the additional benefits earned by the “perfect solution” aren’t worth the cost of the time it would take to find it. Even when experts use computers to solve optimization problems, they often trade away accuracy to save time.

This strategy is far more effective than you would imagine. Christian and Griffiths explain that when computers are dealing with extremely complex optimization problems, they might calculate an answer half as good as the perfect solution in one quadrillionth of the time. Finding a solution that’s merely workable and moving on is often the best use of your time.

Rule #1: Temporarily change the rules to gain information about the true solution.

Even after accepting an imperfect solution, you may find it still isn’t good enough. In this case, Christian and Griffiths describe other ways of embracing imperfection to get closer to perfection.

First, by adjusting the rules of impossible problems, you can create solvable analogous problems that give you valuable information about the original problem. This strategy is known in mathematics as “relaxation.”

Christian and Griffiths explain that the most common form of relaxation is “constraint relaxation,” and it involves temporarily reducing or removing limitations on the problem. People using this strategy imagine the ideal form of the problem they have, solve it, and learn from that solution.

For example, imagine you’re struggling to compose a poem for a contest, and entries can only be a maximum of fifty words long. In a sense, this is an impossible optimization problem. You want to find the right combination of words that will earn you the win, but it’s obviously impossible to calculate the perfect poem.

If you’re feeling creatively stumped, with several drafts of “solutions” that aren’t good enough, you could try removing some constraints. You might remove the word count constraint, freewriting a thousand-word poem and see if that gives you any ideas. Alternatively, you could try removing some constraints on the language you’re choosing to use—dip into made-up words, or start using profanity.

Christian and Griffiths argue that the insights you gain from solving easier iterations of the problem will help you get closer to the original solution. The poem you wrote full of profanity might be your best one yet after you’ve reapplied the constraints and taken out the expletives.

Rule #2: If the rules are bendable, try bending them.

The final relaxation technique, and perhaps the most easily applicable one, is called “Lagrangian relaxation.” With this strategy, you remove some constraints on the problem as a part of your ultimate solution, penalizing broken rules instead of forbidding them entirely. By allowing the rules to bend, you’re not just relaxing constraints as a thought exercise anymore—you’re doing it in real life. Bending the rules and suffering the consequences is sometimes the only way to solve a complex problem.

Imagine a pregnant woman going into labor in the passenger seat of a car as her husband desperately looks for a place to park at the hospital. The closest lot is full, but there is a handicapped parking spot open nearby. Normally, the driver would never consider parking in that spot, but, using Lagrangian relaxation to solve the problem, he decides that it’s worth it to take the spot and pay whatever fine is necessary.

How to Solve Complex Problems: Embrace Imperfection

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• How to schedule your to-do list like a computer
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#### Darya Sinusoid

Darya’s love for reading started with fantasy novels (The LOTR trilogy is still her all-time-favorite). Growing up, however, she found herself transitioning to non-fiction, psychological, and self-help books. She has a degree in Psychology and a deep passion for the subject. She likes reading research-informed books that distill the workings of the human brain/mind/consciousness and thinking of ways to apply the insights to her own life. Some of her favorites include Thinking, Fast and Slow, How We Decide, and The Wisdom of the Enneagram.