MIT’s New Tool for Tackling Hard Computational Problems

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Some difficult computational problems drawn by finding the highest peaks in the “landscape” of the myriad mountain peaks separated by valleys can take advantage of the overlap gap property. At a high enough “altitude”, any two points will be near or far. It’s far away, but there’s nothing in between.

David Gamarnik has developed Overlap Gap Property, a new tool for understanding computational problems that seem out of control.

The idea that some computational problems in mathematics and computer science are difficult is not surprising. In fact, there is an entire class of problems that seems impossible to solve with an algorithm. Immediately below this class is a slightly “simple” problem that is not well understood. This may not be possible either.

David Gamalnick, Professor of Operations Research MIT The Sloan School of Business and the Institute for Data Systems and Social Studies are focusing on issues in the latter, less-studied category, which are more relevant to the everyday world because of the randomness that is an integral feature of natural systems. increase. He and his colleagues have developed a powerful tool for analyzing these problems called overlap gap properties (or OGP). Gamarnik described a new methodology in a recent paper. Minutes of the National Academy of Sciences..

P ≠ NP

Fifty years ago, the most famous problem in theoretical computer science was formulated. “P ≠ NP, “It asks if there is a problem associated with a large dataset that can validate the answer relatively quickly, but the solution, if solved by the fastest computer available, is tremendous. It will take a long time.

The P ≠ NP conjecture has not yet been proven, but most computer scientists believe that many well-known problems, such as the traveling salesman problem, fall into this impossiblely difficult category. The challenge in the salesman example is to find the shortest route in terms of distance or time. N Various cities.. Tasks can be easily managed if: N= 4, because there are only 6 routes to consider.But in 30 cities, there are more than 10 cities30 Possible routes, and their numbers, increase dramatically from there. The biggest difficulty is designing an algorithm that solves the problem quickly in all cases for all integer values. N. Computer scientists are convinced that such an algorithm does not exist, based on the theory of algorithm complexity, and therefore assert that P ≠ NP.

Addressing difficult computational problems

In some cases, the diameter of each peak is much smaller than the distance between the different peaks. Therefore, if you choose any two points (two possible “solutions”) of this vast landscape, they are either very close (if they come from the same peak) or very far apart (from different peaks). If it is withdrawn), it will be either. In other words, there is an obvious “gap” at these distances — either small or large, but nothing in between. Credits: Images are courtesy of researchers.

There are many other examples of such unmanageable problems. For example, suppose you have a huge numeric table with thousands of rows and thousands of columns. Can you find the exact placement of 10 rows and 10 columns so that 100 entries are the best total achievable of all possible combinations? “We call them optimization tasks,” says Gamarnik. “Because you are always trying to find the maximum or best number, the sum of the maximum numbers, the best route through the city, etc.”

Computer scientists have long recognized that it is not possible to create fast algorithms that can efficiently solve problems such as the traveling salesman’s story in all cases. “For well-understood reasons, that’s probably not possible,” Gamarnik said. “But in the real world, nature doesn’t cause problems from a hostile point of view. It’s not trying to interfere with you with the most difficult, carefully selected problems you can think of.” In fact, people are usually more than You will run into problems in random and unnatural situations. These are issues that OGP is intended to address.

Mountains and valleys

To understand what OGP is, it may be useful to first see how the idea came about. Physicists have been studying spin glass since the 1970s. Spin glass is a material that has both liquid and solid properties and behaves abnormally magnetically. Spinglass research has yielded general theories of complex systems related to problems in physics, mathematics, computer science, materials science, and other disciplines. (This work awarded Giorgio Parisi the 2021 Nobel Prize in Physics.)

One of the thorny problems that physicists have been tackling is trying to predict the energy state of various spin glass structures, especially the lowest energy composition. This situation can be represented by a myriad of “landscapes” of mountain peaks separated by valleys. The goal here is to identify the highest peak. In this case, the highest peak actually represents the lowest energy state (although you can flip the image over and look for the deepest hole instead). This turned out to be an optimization problem that resembled the traveling salesman dilemma. Gamarnik explains: A Sisyphus chore comparable to finding a needle in a haystack.

Physicists have shown that by slicing a mountain at a particular altitude and ignoring everything below its cutoff level, this figure can be simplified and a step towards a solution can be taken. Then there remains a collection of peaks protruding above a uniform layer of clouds. Each point of these peaks represents a potential solution to the original problem.

In a 2014 paper, Gamarnik and his co-authors noticed something that was previously overlooked. In some cases, I noticed that the diameter of each peak is much smaller than the distance between the different peaks. Therefore, if you choose any two points (two possible “solutions”) of this vast landscape, they are either very close (if they come from the same peak) or very far apart (from different peaks). If it is withdrawn), it will be either. In other words, there is an obvious “gap” at these distances — either small or large, but nothing in between. The system in this state, proposed by Gamarnik and colleagues, is characterized by OGP.

“We have found that all known problems of algorithmically difficult random properties have a version of this property.”-That is, the peak diameter of the schematic model is much smaller than the space between the peaks. Gamarnik insists. “This allows us to measure the hardness of the algorithm more accurately.”

Uncover the secrets of algorithmic complexity

The advent of OGP helps researchers assess the difficulty of creating fast algorithms to tackle specific problems.And it already made them “mathematically possible” [and] As a potential candidate, we strictly exclude large classes of algorithms, “says Gamarnik. “In particular, we learned that stable algorithms (algorithms that change little input but not much output) fail to solve this type of optimization problem.” This negative result is traditional. This applies not only to computers, but also to quantum computers, especially the so-called “quantum approximation optimization algorithm” (QAOA), which some researchers expected to solve these same optimization problems. Now, with the discovery of Gamarnik and his co-authors, the recognition that a successful QAOA-type algorithm requires many layers of manipulation and can be technically difficult eases these expectations. I did.

“Whether it’s good news or bad news depends on your point of view,” he says. “I think this is good news in the sense that it helps uncover the secrets of the complexity of the algorithm and increase our knowledge of what is and isn’t in the realm of potential. Nature creates problems and happens randomly. Even so, these issues are bad news in the sense that they are difficult. “The news isn’t all that surprising, he added. “Many of us have been expecting it for a long time, but now we have a stronger foundation for making this claim.”

Still, we cannot prove that there is no fast algorithm that can solve these optimization problems with random settings. With such a proof, we have a definitive answer to the P ≠ NP problem. “If you can show that you can’t use an algorithm that works in most cases, you can’t use an algorithm that works all the time,” he says.

Predicting the time it takes for the P ≠ NP problem to be resolved seems to be an unmanageable problem. There may be more mountains to climb and more valleys to cross before researchers get a clearer picture of the situation.

See also: Overlap Gap Properties: Topological Barriers for Optimizing Random Structures, David Gamarnik, October 12, 2021 Minutes of the National Academy of Sciences..
DOI: 10.1073 / pnas.2108492118

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