Wednesday, August 11, 2010
Vinay Deolalikar, a mathematician who works for HP Labs, claims to have proven that P is not equal to NP. The problem is the greatest unsolved problem in theoretical computer science and is one of seven problems in which the Clay Mathematics Institute has offered million dollar prizes to the solutions.
The question of whether P equals NP essentially asks whether there exist problems which take a long time to solve but whose solutions can be checked quickly. More formally, a problem is said to be in P if there is a program for a Turing machine, an ideal theoretical computer with unbounded amounts of memory, such that running instances of the problem through the program will always answer the question in polynomial time — time always bounded by some fixed polynomial power of the length of the input. A problem is said to be in NP, if the problem can be solved in polynomial time when instead of being run on a Turing machine, it is run on a non-deterministic Turing machine, which is like a Turing machine but is able to make copies of itself to try different approaches to the problem simultaneously.
Mathematicians have long believed that P does not equal NP, and the question has many practical implications. Much of modern cryptography, such as the RSA algorithm and the Diffie-Hellman algorithm, rests on certain problems, such as factoring integers, being in NP and not in P. If it turned out that P=NP, these methods would not work but many now difficult problems would likely be easy to solve. If P does not equal NP then many natural, practical problems such as the traveling salesman problem are intrinsically difficult.
In 2000, the Clay Foundation listed the “Clay Millenium Problems,” seven mathematical problems each of which they would offer a million dollars for a correct solution. One of these problems was whether P equaled NP. Another of theseseven, the Poincaré conjecture, was solved in 2002 by Grigori Perelman who first made headlines for solving the problem and then made them again months later for refusing to take the prize money.
On August 7, mathematician Greg Baker noted on his blog that he had seen a draft of a claimed proof by Deolalikar although among experts a draft had apparently been circulating for a few days. Deolalikar’s proof works by connecting certain ideas in computer science and finite model theory to ideas in statistical mechanics. The proof works by showing that if certain problems known to be in NP were also in P then those problems would have impossible statistical properties. Computer scientists and mathematicians have expressed a variety of opinions about Deolalikar’s proof, ranging from guarded optimism to near certainty that the proof is incorrect. Scott Aaronson of the Massachusetts Institute of Technology has expressed his pessimism by stating that he will give $200,000 of his own money to Deolalikar if the proof turns out to be valid. Others have raised specific technical issues with the proof but noted that the proof attempt presented interesting new techniques that might be relevant to computer science whether or not the proof turns out to be correct. Richard Lipton, a professor of computer science at Georgia Tech, has said that “the author certainly shows awareness of the relevant obstacles and command of literature supporting his arguments.” Lipton has listed four central objections to the proof, none of which are necessarily fatal but may require more work to address. On August 11, 2010, Lipton reported that consensus of the reviewers was best summarized by mathematician Terence Tao, who expressed the view that Deolalikar’s paper probably did not give a proof that P!=NP even after major changes, unless substantial new ideas are added.