Award for his Unique games Conjecture in Computational Complexity Theory

The 36 year-old IIT Bombay alumnus Subhash Khot, an Indian-American theoretical computer scientist at the Courant Institute of Mathematical Sciences of New York University, has been chosen for the International Mathematical Union’s Nevanlinna Prize, which is given “for outstanding contributions in mathematical aspects of information sciences”.

The award is given once every four years during the International Congress of Mathematicians (ICM). The ICM2014 began on August 13 at Seoul, Republic of Korea. Khot’s research has to do with a field in computer science called ‘Computational Complexity’, which seeks to understand the power and limits of efficient computation with standard computers.

“Khot's prescient definition of the "Unique Games" problem,” said the award citation, “and his leadership in the effort to understand its complexity and its pivotal role in the study of efficient approximation of optimization problems, have produced breakthroughs in algorithmic design and approximation hardness, and new exciting interactions between computational complexity, analysis and geometry.”

Unlike the normal practice of awarding major prizes in mathematics for groundbreaking results, the Nevalninna Prize this time is for a conjecture and that too when the opinion on its truth is divided within the world computer science community. But, by posing the right questions that have enabled great insights into the nature of computational complexity and approximations to computationally hard problems, the conjecture, called the Unique Games Conjecture (UGC), has already proved its great value. More pertinently, Khot has also used the conjecture to prove major results that will remain valid regardless of the truth of the UGC.

“I do believe whether it is eventually shown to be false or not changes little about the brilliance of the conjecture,” wrote Richard Lipton of Georgia Institute of Technology, Atlanta, in a blog post on the UGC. “It takes insight, creativity of the highest order, and a bit of ‘guts’ to make such a conjecture,” Lipton said.

The central question in computational complexity is: How hard are problems to solve? More precisely, if one has found the cleverest possible way to solve a particular problem, how fast will a computer find the answer using it? It is now a truism that some problems are so intractably difficult that computers cannot reliably find the answer at all, at least not in any reasonable amount of time (such as before the end of the universe).

A typical optimization problem, to quote an example given by Meena Mahajan, a theoretical computer scientist from the Institute of Mathematical Sciences, Chennai, is: What is the minimum number of tea stalls that should be put up in a large university campus so that no one needs to walk, say, more than 200 metres along a road to reach one? As the size of the campus increases, the computational time to find the minimum number of stalls grows exponentially fast, even with the best current algorithms.

This is what underlies the famous P ≠ NP conjecture, which is one of the seven $1 million Millennium Problems posed by the Clay Mathematics Institute (CMI). A problem that is tractable and the number of algorithmic steps that its solution requires is at most some power of the problem size is said to belong to P class (where P stands for 'polynomial time'). A problem belongs to NP class (NP stands for 'non-deterministic polynomial time') if the computer can efficiently verify a proposed solution for it is correctness but does not have the resources – time or the number of algorithmic steps or memory size -- as a function of the "input size" of the problem to obtain the solution.

Formulated by the American-Canadian computer scientist Stephen Cook in 1971, the conjecture states that these two classes are distinct. This means that there are computational problems whose solutions are beyond the reach of any computer algorithm. Most computer scientists believe that the conjecture is true but even after four decades it is yet to be proved, not want of attempts though.

Such “computationally intractable” or “NP-hard” problems have profound consequences. For instance, they limit our ability to tackle large-scale problems in science and engineering, such as optimal design of protein folding or figuring out the best design for a chip or the best train schedule. Conversely, however, computational intractability enables computer security against hackers attempting to access on-line confidential data.

So, computer scientists asked: If a problem is too hard to be computationally solved quickly and precisely, can we at least find a good approximation? “Counter-intuitive though it may seem,” Mahajan points out returning to her tea stall example, “while we do not know how to find the minimum efficiently, we can find a number that is no more than twice the minimum efficiently! That is, we can efficiently find an approximate solution. Unfortunately, there are many optimization problems for which even this may not be possible.”

The UGC essentially addresses the question of solving NP-hard problems even approximately. It thus complements the P ≠ NP conjecture. In the initial years after P vs, NP conjecture was made, many computer scientists believed that that good approximations must be easier than finding the exact answer to an NP-hard problem. But they soon discovered that, while they could come up with good approximation algorithms for some NP-hard problems (like our tea-stall problem), for most of them even finding a good approximation was not possible. There was no prescriptive way of determining whether approximation was possible. That is, approximation itself was an NP-hard problem.

The Unique Games problem, a remarkably simple problem, encapsulates the elements that make many hard problems hard to solve even approximately. The problem is simply about finding an efficient way of assigning colours to the nodes of a network such that any two connected nodes have different colours (Fig).

If one has only two colours (say yellow and green), the problem is easy. The problem becomes trickier even when you add just one more colour (say blue). When you colour the first node, say with Y, you don’t know what colour the connected nodes should have, G or B. If you choose one and get to a node that cannot be coloured without violating the condition, you have no way of knowing if a different selection would have solved the problem.

It is not the method that was faulty. In fact, no other method will be able to solve it reliably and efficiently. The problem is NP-hard, meaning effectively impossible. But Khot asked the related question: Which colouring scheme breaks the fewest rules possible? That is, which colouring is the best approximation. The conjecture basically is that if you have lots of colours, even an efficient method to colour the nodes anywhere close to the best one is impossible.

The UGC, which Khot enunciated in 2002, can be stated as follows: It is not just hard but impossible reliably to find an approximate answer to Unique Games quickly. That it is, the problem is NP-hard even to solve approximately. Thus, if the conjecture is true, the problem Unique Games problem, in a technical sense, sets a benchmark for NP-hard problems.

“Khot's work attempts to give a unified explanation for why so many problems seem hard to approximate,” points out Mahajan. “What makes this so wonderful is that if the UGC is true, it explains in one shot why a host of other problems have resisted solutions so far; they are all at least as hard as Unique Games. All the difficulties encountered in tackling many different optimization problems get distilled into one problem,” she adds.

A couple of years later after Khot made his conjecture, computer scientists realized the real power and importance of the conjecture. They found that, if the UGC was indeed true, then they could set firm limits on how well many other problems could be approximated. For instance, in our tea-stall example, it turns out that twice the minimum is the best one can do under the assumption that P ≠ NP. If one tried to do better than a factor-of-2 approximation, say with a more sophisticated algorithm, it would imply that P = NP. The simple algorithm that gave the factor-of-2 approximation is the best one can do. According to the UGC, an efficient approximation for the Unique Games problem would imply P = NP.

Independent of its truth, the conjecture, however, has proved to be remarkably powerful. In the process of determining how well NP-hard problems could be approximated, Khot and others have proved several significant results in other areas, which seem far removed from computational complexity, such as geometry of different ways of measuring distances, some new theorems in Fourier analysis, better understanding algorithms based on linear and semi-definite programming and structure of ‘foams’. The last connection, which is essentially a tiling problem, came as a surprise even to Khot, according to Mahajan.

While there is a significant group of researchers working to prove the conjecture, there is an equally significant set working to disprove it. Although scientists are yet to find an algorithm that can efficiently find a good approximate solution to Unique Games, finding one such would mean a significant algorithmic breakthrough. Such a new approximation algorithm is most likely to be very different from the approximation algorithms that are known today. Indeed, the process has already thrown up some excellent new algorithmic methods for other situations. In any case, the UGC is likely to keep theoretical computer scientists busy for some years to come.

This article has been edited for a factual error