Tackling Hard Computational Problems

The notion that some computational problems in math and computer science can be hard should come as no surprise. There is, in fact, an entire class of problems deemed impossible to solve algorithmically. Just below this class lie slightly “easier” problems that are less well-understood — and may be impossible, too. David Gamarnik, professor of operations research at the MIT Sloan School of Management and the Institute for Data, Systems, and Society, is focusing his attention on the latter, less-studied category of problems, which are more relevant to the everyday world because they involve randomness — an integral feature of natural systems. He and his colleagues have developed a potent tool for analyzing these problems called the overlap gap property (or OGP). Gamarnik described the new methodology in a recent paper in the Proceedings of the National Academy of Sciences.

Fifty years ago, the most famous problem in theoretical computer science was formulated. Labeled “P ≠ NP,” it asks if problems involving vast datasets exist for which an answer can be verified relatively quickly, but whose solution — even if worked out on the fastest available computers — would take an absurdly long time. The P ≠ NP conjecture is still unproven, yet most computer scientists believe that many familiar problems — including, for instance, the traveling salesman problem — fall into this impossibly hard category. The challenge in the salesman example is to find the shortest route, in terms of distance or time, through N different cities. The task is easily managed when N=4, because there are only six possible routes to consider. But for 30 cities, there are more than 1030 possible routes, and the numbers rise dramatically from there. The biggest difficulty comes in designing an algorithm that quickly solves the problem in all cases, for all integer values of N. Computer scientists are confident, based on algorithmic complexity theory, that no such algorithm exists, thus affirming that P ≠ NP. Computer scientists have long recognized that you can’t create a fast algorithm that can, in all cases, efficiently solve problems like the saga of the traveling salesman. “Such a thing is likely impossible for reasons that are well-understood,” Gamarnik notes. “But in real life, nature doesn’t generate problems from an adversarial perspective. It’s not trying to thwart you with the most challenging, hand-picked problem conceivable.” In fact, people normally encounter problems under more random, less contrived circumstances, and those are the problems the OGP is intended to address.

To understand what the OGP is all about, it might first be instructive to see how the idea arose. Since the 1970s, physicists have been studying spin glasses — materials with properties of both liquids and solids that have unusual magnetic behaviors. Research into spin glasses has given rise to a general theory of complex systems that’s relevant to problems in physics, math, computer science, materials science, and other fields. (This work earned Giorgio Parisi a 2021 Nobel Prize in Physics.) One vexing issue physicists have wrestled with is trying to predict the energy states, and particularly the lowest energy configurations, of different spin glass structures. The situation is sometimes depicted by a “landscape” of countless mountain peaks separated by valleys, where the goal is to identify the highest peak. In this case, the highest peak actually represents the lowest energy state (though one could flip the picture around and instead look for the deepest hole). This turns out to be an optimization problem similar in form to the traveling salesman’s dilemma, Gamarnik explains: “You’ve got this huge collection of mountains, and the only way to find the highest appears to be by climbing up each one” — a Sisyphean chore comparable to finding a needle in a haystack. Physicists have shown that you can simplify this picture, and take a step toward a solution, by slicing the mountains at a certain, predetermined elevation and ignoring everything below that cutoff level. You’d then be left with a collection of peaks protruding above a uniform layer of clouds, with each point on those peaks representing a potential solution to the original problem.

In a 2014 paper, Gamarnik and his coauthors noticed something that had previously been overlooked. In some cases, they realized, the diameter of each peak will be much smaller than the distances between different peaks. Consequently, if one were to pick any two points on this sprawling landscape — any two possible “solutions” — they would either be very close (if they came from the same peak) or very far apart (if drawn from different peaks). In other words, there would be a telltale “gap” in these distances — either small or large, but nothing in-between. A system in this state, Gamarnik and colleagues proposed, is characterized by the OGP.

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