One summer afternoon in 2001, while visiting relatives in India, Subhash Khot drifted into his default mode — quietly contemplating the limits of computation. For hours, no one could tell whether the third-year Princeton University graduate student was working or merely sinking deeper into the living-room couch. That night, he woke up, scribbled something down and returned to bed. Over breakfast the next morning, he told his mother that he had come up with an interesting idea. She didn’t know what it was, but her reserved older son seemed unusually happy.Khot’s insight — now called the — helped him make progress on a problem he was working on at the time, but even Khot and his colleagues did not realize its potential. “It just sounded like an idea that would be nice if it was true,” recalled, now a 36-year-old computer science professor at New York University’s Courant Institute of Mathematical Sciences.When Khot returned to Princeton, he mentioned the idea to, his doctoral adviser, who advised him to hold off on publishing it. “I wasn’t sure it was going to be a good paper,” Arora said.
“I thought it was maybe a little premature, that it was just a month since he had the idea.”. In, the International Mathematical Union also credited Khot’s work for generating “new exciting interactions between computational complexity, analysis and geometry.”Khot, who tends to keep his thoughts close and acknowledgment of his achievements even closer, was as surprised as his colleagues by the success of the Unique Games Conjecture.
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“I definitely didn’t expect that this small proposal would go so far,” he said.Although Arora, like others studying the limits of approximation algorithms, was initially unconvinced by Khot’s “pie in the sky” idea, he now credits his former graduate student for sensing that his proposal could clear “a fundamental stumbling block.”“His intuition was right,” Arora said. “He’s now probably the number one expert in that field.”. Suddenly, everyone was studying the implications of the Unique Games Conjecture. “You should see the number of mathematicians working on problems that emanated from this conjecture,” said of the Institute for Advanced Study in Princeton, N.J. The years following the Max Cut result witnessed a flood of approximation hardness results — theorems of the form, “If the Unique Games Conjecture is true, then it’s NP-hard to approximate the solutions of problem X any closer than Y percent.”“This conjecture suddenly became really interesting and important,” Wigderson said. It even seemed to help prove approximation hardness results about problems that on the surface seemed to have nothing to do with the problem at the heart of the Unique Games Conjecture, which involves assigning colors to the nodes of a network.
“What was special about his problem?” Wigderson asked. “It wasn’t clear.”In 2008, of the University of California, Berkeley, if the UGC is true, then it’s possible to establish the approximation hardness of an entire category of common computational problems called “constraint satisfaction” problems. These involve trying to find the solution to a problem that satisfies as many of a set of constraints as possible — for example, the wedding seating chart that places feuding family members at different tables as much as possible.“We understand immediately an infinite class of problems by relying only on the one problem that Khot postulated is hard, which is amazing,” Wigderson said. The conjecture “creates an understanding that one rarely expects — that’s why it’s so interesting and beautiful,” he said.“It has changed the way we think about a lot of problems in computer science,” said, a theoretical computer scientist at Carnegie Mellon University in Pittsburgh. True or FalseKhot is most comfortable thinking quietly — whether he’s alone in his office, on a Washington Square Park bench surrounded by strollers, buskers and chess hustlers, in a cafe packed with NYU students or at home in India among family and friends.“When we go to a movie he doesn’t like, he’s working on his own,” Ratnaparkhi said.
“That happens quite a lot.”If the Unique Games Conjecture is ever disproved, all the approximation hardness results that emanate from it will collapse. But certain other results will hold firm: For some mysterious reason, the proofs of the approximation hardness results and the attempts to prove the conjecture itself have led mathematicians to state — and then prove — an assortment of theorems about seemingly unrelated areas of mathematics, including the, the, and even the. “Out popped these very natural questions,” O’Donnell said. These results will hold up regardless of whether the Unique Games Conjecture turns out to be true or false.It remains to be seen whether computer scientists will be able to prove or disprove Khot’s Unique Games Conjecture.
A proof would be a boon to computer scientists, but a disproof might be even more exciting, Arora said. Researchers agree that disproving the conjecture would probably require innovative new algorithmic techniques that could unlock a host of different approximation problems. “If someone came up with an efficient algorithm for the Unique Games problem, we would have a very valuable new insight into how to design algorithms,” Arora said.Khot doesn’t expect someone to prove or disprove his conjecture any time soon.
“At this point, we can probably hope to just keep constructing pieces of evidence in one direction or another,” he said. He is working on proving the conjecture but is also exploring whether he can come up with different avenues toward proving approximation hardness results.
“That’s the real goal,” he said.Before his son was born nearly three years ago, Khot used to think about problems related to the Unique Games Conjecture all the time. But with fatherhood, he said, “you suddenly realize there are much more important things in life than what you thought before.”Heading from his office — where he had been discussing his work with long pauses and carefully chosen words — to the playground, where he was to pick up his son, Khot could barely hide his anticipation. As the little boy ran to meet him, Khot’s brow unfurrowed, and a broad smile swept over his face.“Of all people, he’s happiest when he’s with Neev,” Ratnaparkhi said.
“He talks all the time with Neev.”Khot said that wanting to spend more time with his son has made him more efficient at work. Before, he would do his research in between reading the news and browsing the Internet. Now, “I have 9 to 5 free to work when he is at day care,” Khot said. “I’m much more organized.”Math occasionally creeps in while he’s playing with his son, Khot said, but “if the guy is running around, what are you going to do?”This article was reprinted on.