Researcher claims solution to P vs NP math problem
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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.
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Oil spill hits Australia’s Sunshine coastline
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Sunday, March 15, 2009
200,000 litres of oil leaked into waters off the coast of Brisbane from the Pacific Adventurer when their fuel tanks were damaged in rough seas on Wednesday. The figure is about ten times higher than the original estimate of twenty thousand litres of oil. The devastating diesel oil spill has spread along 60 kilometres (37 miles) of the Queensland coast. In addition, 31 containers with 620 tonnes of ammonium nitrate fertiliser flew overboard during the violent storm.
Questions are being asked why the Hong Kong cargo ship was out in seas with nine meter waves caused by Cyclone Hamish, a Category 5 tropical cyclone, as well as why the fertiliser containers were not properly secured. One of the overboard containers ruptured the hull of the Pacific Adventurer, causing between 30 to 100 tonnes of oil to spew from the severely damaged ship.
If the ammonium nitrate mixes with the heavy oil, an explosion could occur. None of the containers have been recovered. Some of these may float, but it is believed that they may have sunk which then may cause algal blooms.
Disaster zones have been declared at Bribie and Moreton Islands, and along the Sunshine coast.
The vessel’s owner, Swire Shipping, reported that a second leak began on Friday, when the ship began listing after docking at Hamilton for repairs. “As full soundings of the vessel’s tanks were being taken at the port to determine how much oil had leaked from the vessel, a small quantity of fuel oil escaped from the Pacific Adventurer,” it stated. The ship was brought upright, and a recovery vessel was used to suck up the oil from the water. The leak produced a 500m-long oil slick down the Brisbane River. Booms were placed around this oil spill so that a skimmer could clean up the second spill.
Swire Shipping could face clean up costs of AU$100,000 a day as well as fines up to AU$1.5million (US$977,000; £703,000) if found guilty of environmental breaches or negligence.
Sunshine Coast beaches are slowly starting to be reopened. The beach of Mooloolaba was still closed following reports of burning sensations from swimmers. 12 beaches remain closed; however, 13 have been reopened.
Over 300 state government and council workers are using buckets, rakes and spades in the clean up effort. Sunshine Coast Mayor Bob Abbott says the majority will be gone by Sunday afternoon. The full environmental impact on wildlife is not yet known. One turtle and seven pelicans have been found covered in oil.
There are concerns that the drinking water of Moreton Island is at risk, as the island uses water from the underground water table near the oil spill site.
“Every bucketload of contaminated sand has to be removed from the island by barge, and each bucketload from a front-end loader weighs about one tonne. It’s just an impossible task,” said Mr Trevor Hassard of the Tangalooma Dolphin Education Centre.
The commercial fishing industry has suffered from the incident. Trawlers won’t resume operations until Sunday evening, and any catches will be tested for human consumption.
