The diameter of each peak will be smaller than the distances between peaks. If one were to pick any two points on the landscape, they would either be very close or very far apart. There would be agap in these distances, either small or large, but nothing in between. David Gamarnik and his team are Credit: David Gamarnik and his team.
Computational problems in math and computer science can be hard. There are many problems that are deemed impossible to solve using a computer. There are some problems that may be difficult to understand.
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. The overlap gap property is a potent tool for analyzing these problems. The new methodology was described in a recent paper.
The most famous problem in computer science was formulated fifty years ago. It asks if problems involving vast datasets exist for which an answer can be verified relatively quickly, but whose solution would take an incredibly long time.
Many familiar problems, including the traveling salesman problem, fall into this impossibly hard category because the P NP conjecture is still unknown. The challenge in the salesman example is to find the shortest route through different cities. There are only six possible routes to consider when the task is easy to manage. There are more than 1030 possible routes in 30 cities. Computer scientists are confident that there isn't a solution to the problem in all cases for all N values.
This is one of many examples of intractable problems. You have a table of numbers with a lot of rows and columns. Can you find a way to arrange 10 rows and 10 columns in such a way that 100 entries will have the highest sum possible? "We call them Optimization tasks because you're always trying to find the best route through the city, the biggest sum of numbers, and so forth," Gamarnik says.
Computer scientists know that it's not possible to create a fast algorithm that can solve problems like the saga of the traveling salesman. "It is likely impossible for reasons that are well- understood," Gamarnik says. Nature doesn't generate problems from an opposing perspective in real life. It's not trying to stop you from dealing with the most challenging problem possible. The OGP is intended to address the problems that people encounter under more random, less contrived circumstances.
There are Peaks and valleys.
To understand what the OGP is all about, it is important to see how the idea came about. Physicists have been studying spin glasses since the 1970s. A general theory of complex systems has arisen from research into spin glasses. Giorgio Parisi won a 2021 Nobel Prize in physics for this work.
Physicists are trying to predict the energy states of different spin glass structures. The situation is depicted by a landscape of mountain peaks separated by valleys, where the goal is to identify the highest peak. The lowest energy state is represented by the highest peak, though one could look for the deepest hole. The only way to find the highest peaks in the mountains is by climbing them, according to Gamarnik.
Physicists have shown that you can simplify the picture by slicing the mountains at a certain elevation and ignoring everything below that level. You'd be left with a collection of peaks above a uniform layer of clouds, with each point representing a potential solution to the original problem.
Gamarnik and his coauthors noticed something that had been overlooked. The diameter of each peak will be smaller than the distances between different peaks. If one were to pick any two points on the landscape, they would either be very close or very far apart. There would be agap in these distances, either small or large, but nothing in between. The system proposed by Gamarnik and colleagues is characterized by the OGP.
The schematic model of a random nature problem has a version of the property that is smaller than the space between mountains. This measure provides a more precise measure.
The secrets of complexity are unlocked.
Researchers can use the emergence of the OGP to assess the difficulty of creating fast algorithms. It has enabled them to rigorously rule out a large class of algorithms as potential competitors. Stable algorithms that won't change much if the input only changes a little will fail at solving this type of problem. The negative result applies to both conventional computers and quantum computers, which some investigators had hoped could solve the same problems. Thanks to Gamarnik and his co-authors' findings, those hopes have been moderated by the fact that many layers of operations would be required for a successful QAOA-type algorithm.
He says, "whether that's good news or bad news depends on your perspective." It's good news that it helps us understand what is in the realm of possibility and what isn't, and that it helps us understand what isn't in the realm of possibility. It's bad news that it tells us that these problems are hard even if nature produces them and even if they're generated in a random way. He says the news is not surprising. "Many of us expected it all along, but now we have a more solid basis upon which to make this claim."
Researchers are not able to prove the existence of fast algorithms that could solve these problems in random settings. A definitive answer to the P NP problem would be provided by a proof. He says that if we could show that we can't have an algorithm that works all the time, then we can't have it.
Predicting how long it will take before the P NP problem is solved seems to be an intractable problem. It's likely that there will be many more peaks to climb and valleys to traverse before researchers have a better idea of the situation.
David Gamarnik is the author of The overlap gap property. There is a book titled "10.1073/pnas.2108492118."
The National Academy of Sciences has a journal.
A researcher developed a tool for understanding hard computational problems that appear intractable.
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