A paper posted online in January takes theoretical computer scientists halfway toward proving one of the biggest conjectures in their field. The new study, when combined with three other recent papers, offers the first tangible progress toward proving the Unique Games Conjecture since it was proposed in 2002 by Subhash Khot, a computer scientist now at New York University.

Over the past decade and a half, the conjecture — which asks whether you can efficiently color networks in a certain way — has inspired discoveries in topics as diverse as the geometry of foams and the stability of election systems. And if the conjecture can be proved, its implications will reach far beyond network-coloring: It will establish what is the best algorithm for every problem in which you’re trying to satisfy as many as possible of a set of constraints — the rules in a sudoku puzzle, or the seating preferences of a collection of wedding guests, for instance.

“For a whole huge family of problems, their complexity is beautifully explained if the conjecture is true,” said Irit Dinur, a theoretical computer scientist at the Weizmann Institute of Science in Rehovot, Israel.

Yet until this recent round of papers, all attempts to prove the conjecture had fizzled out. Meanwhile, computer scientists had come up with tantalizing hints that the conjecture might in fact be false.

“Some people felt that it is only a matter of time until the conjecture is refuted,” said Dor Minzer, a graduate student at Tel Aviv University in Israel and a co-author, along with Khot and Muli Safra of Tel Aviv University, of the four new papers (they are joined on the two middle papers by Dinur and Guy Kindler of the Hebrew University of Jerusalem).

With the new work, the prevailing winds appear to have shifted. “This is very strong evidence that the Unique Games Conjecture is true,” said Boaz Barak, a theoretical computer scientist at Harvard University who had previously been one of the conjecture’s most vocal doubters. “It’s a very exciting piece of work.”

One Beautiful Explanation

When Khot formulated the Unique Games Conjecture as a graduate student, he had a specific goal in mind: to better understand the computational complexity of the “graph coloring” problem, in which the goal is to color the nodes of a network (or “vertices” of a “graph,” as mathematicians put it) so that no two connected vertices are the same color. Theoretical computer scientists already knew that for graphs that require three or more colors, the problem of finding a coloring that uses the fewest possible colors is “NP-hard,” meaning that it belongs to a giant collection of computational problems, all of which researchers believe are beyond the reach of any efficient algorithm. (You can always find a valid coloring simply by trying every possible way to color the vertices, but for large graphs that brute force algorithm is very inefficient.)

But what if your palette holds more colors than are strictly necessary? Perhaps you’re considering graphs that require four colors, but you have 100 colors at your disposal. Can you efficiently color these graphs using that wider palette? Khot suspected that the answer was still no, but he couldn’t prove it. “This was my dream, since essentially the first year of my Ph.D.,” he said.

Khot figured out that the key to solving this problem lay in understanding the complexity of another problem, in which the goal is again to color a graph, but now there are rules that tell you, whenever you color a vertex, what color you must use on each vertex connected to it. The rules in this Unique Games problem might not all be compatible with one another (like conflicting seating requests at a wedding), so the goal is to find a coloring that satisfies as many rules as possible. (The name Unique Games comes from an equivalent formulation of the problem in terms of games rather than graph coloring.)

At first glance, it might seem that if the rules on your graph are almost perfectly compatible — say there’s some coloring that satisfies 99 percent of them — then it shouldn’t be too hard to find a not-utterly-abysmal coloring that satisfies, say, 1 percent of the rules. But Khot suspected that in this setting, finding even a 1 percent coloring is so tricky that there’s no efficient algorithm that can accomplish it for every graph. And the problem remains hard, he conjectured, even if your graph can satisfy 99.9999 percent of its rules and you’d be content with a coloring that satisfies 0.0001 percent of them — or any other pair of percentages, no matter how far apart.

Khot’s motivation in formulating this conjecture was tightly tied to graph coloring. But as he and other theoretical computer scientists started studying the ramifications of the conjecture, they found that it encodes a massive amount of information about problems far removed from graph coloring. The conjecture “completely took on a life of its own,” Khot said.

In particular, in 2008, Prasad Raghavendra of the University of California, Berkeley showed that if the conjecture is true then a certain simple algorithm called semidefinite programming offers the best possible approximate solutions to all “constraint satisfaction” problems, in which you’re trying to satisfy as many of a set of rules as possible.

“A priori, you can think of algorithm design as something very creative, where for every problem you have to figure out a new and different algorithm,” Barak said. But Raghavendra’s result meant that if the Unique Games Conjecture is true, then for a host of different problems, no ingenuity is needed — semidefinite programming is the one-size-fits-all solution.

“In science and math in particular, this is the kind of thing you really love — to have one beautiful explanation for a lot of things,” Dinur said. “From that perspective, you really want this conjecture to be true.”

After Raghavendra’s result, theoretical computer scientists’ degree of faith in the Unique Games Conjecture “became all about whether we believe semidefinite programming is so powerful,” said Dana Moshkovitz, of the University of Texas at Austin. For many theoretical computer scientists, this seemed a dubious proposition.