P versus NP problem

The P versus NP problem is a major unsolved problem in computer science. It asks whether every problem whose solution can be verified can be solved quickly, it is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute, each of which carries a US$1,000,000 prize for the first correct solution. The informal term used above, means the existence of an algorithm solving the task that runs in polynomial time, such that the time to complete the task varies as a polynomial function on the size of the input to the algorithm; the general class of questions for which some algorithm can provide an answer in polynomial time is called "class P" or just "P". For some questions, there is no known way to find an answer but if one is provided with information showing what the answer is, it is possible to verify the answer quickly; the class of questions for which an answer can be verified in polynomial time is called NP, which stands for "nondeterministic polynomial time". An answer to the P = NP question would determine whether problems that can be verified in polynomial time can be solved in polynomial time.

If it turned out that P ≠ NP, believed, it would mean that there are problems in NP that are harder to compute than to verify: they could not be solved in polynomial time, but the answer could be verified in polynomial time. Aside from being an important problem in computational theory, a proof either way would have profound implications for mathematics, algorithm research, artificial intelligence, game theory, multimedia processing, philosophy and many other fields. Consider Sudoku, a game where the player is given a filled-in grid of numbers and attempts to complete the grid following certain rules. Given an incomplete Sudoku grid, of any size, is there at least one legal solution? Any proposed solution is verified, the time to check a solution grows as the grid gets bigger. However, all known algorithms for finding solutions take, for difficult examples, time that grows exponentially as the grid gets bigger. So, Sudoku is in NP but does not seem to be in P. Thousands of other problems seem similar, in that they slow to solve.

Researchers have shown that many of the problems in NP have the extra property that a fast solution to any one of them could be used to build a quick solution to any other problem in NP, a property called NP-completeness. Decades of searching have not yielded a fast solution to any of these problems, so most scientists suspect that none of these problems can be solved quickly. This, has never been proven; the underlying issues were first discussed in the 1950s, in letters from John Forbes Nash Jr. to the National Security Agency, from Kurt Gödel to John von Neumann. The precise statement of the P versus NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" and is considered by many to be the most important open problem in computer science. Although the P versus NP problem was formally defined in 1971, there were previous inklings of the problems involved, the difficulty of proof, the potential consequences. In 1955, mathematician John Nash wrote a letter to the NSA, where he speculated that cracking a sufficiently complex code would require time exponential in the length of the key.

If proved this would imply what is now called P ≠ NP, since a proposed key can be verified in polynomial time. Another mention of the underlying problem occurred in a 1956 letter written by Kurt Gödel to John von Neumann. Gödel asked whether theorem-proving could be solved in quadratic or linear time, pointed out one of the most important consequences—that if so the discovery of mathematical proofs could be automated; the relation between the complexity classes P and NP is studied in computational complexity theory, the part of the theory of computation dealing with the resources required during computation to solve a given problem. The most common resources are space. In such analysis, a model of the computer for which time must be analyzed is required; such models assume that the computer is deterministic and sequential. In this theory, the class P consists of all those decision problems that can be solved on a deterministic sequential machine in an amount of time, polynomial in the size of the input.

P ⊆ NP. Arguably the biggest open question in theoretical computer science concerns the relationship between those two classes: Is P equal to NP? Since 2002, William Gasarch has conducted three polls of researchers concerning this and related questions. Confidence that P ≠ NP has been increasing - in 2019, 88% believed P ≠ NP, as opposed to 83% in 2012 and 61% in 2002; when restricted to experts, the 2019 answers became 99% P ≠ NP. To attack the P = NP question, the concept of NP-completeness is useful. NP-complete problems are a set of problems to each of which any other NP-problem can be reduced in polynomial time and whose solution may still be verified in polynomial time; that is, any NP problem can

2006 Women's Australian Hockey League

The 2006 Women's Australian Hockey League was the 14th edition women's field hockey tournament. The tournament was held between 7 April – 14 May 2006. WA Diamonds won the tournament for the third time after defeating QLD Scorchers 4–2 in the final. Canberra Strikers finished in third place after defeating NSW Arrows 2–1 in the third and fourth place playoff; the 2006 Women's Australian Hockey League consisted of a single round robin format, followed by classification matches. Teams from all 8 states and territories competed against one another throughout the pool stage. At the conclusion of the pool stage, the top four ranked teams progressed to the semi-finals, while the bottom four teams continued to the classification stage; the first four rounds of the pool stage comprised two-legged fixtures based on aggregate scores to determine point allocation. There were 229 goals scored in 52 matches, for an average of 4.4 goals per match. 13 goals 10 goals 8 goals 6 goals 5 goals 4 goals 3 goals 2 goals 1 goal Source: Clearing House


KWHT is a radio station licensed to serve Pendleton, United States. The station, which began broadcasting in 1984, is owned by Randolph and Debra McKone's Elkhorn Media Group and the broadcast license is held by EMG2, LLC. KWHT broadcasts a country music format to the greater Walla Walla, area; this includes select programming from the Westwood One Radio Network. Syndicated music programming includes America's Grand Ole Opry Weekend from Westwood One. KWHT shares a studio building with sister stations KTIX, KUMA, KWHT; this multi-station facility is located at the west end of Eastern Oregon Regional Airport. This station received its original construction permit from the Federal Communications Commission on May 15, 1980; the new station was assigned the call letters KFMT by the FCC. In September 1983, Faith Media, Inc. announced an agreement to sell this permit for this still-under construction station to AgPal Broadcasting, Inc. The deal was approved by the FCC on December 1, 1983, the transaction was consummated on February 7, 1984.

AgPal Broadcasting was owned by JoAnn Harle plus Cheryl and Jim McAnally. Under new ownership, the station was assigned the current KWHT call letters by the FCC on February 3, 1984. After several extensions, KWHT received its license to cover from the FCC on October 17, 1984. In September 1997, AgPal Broadcasting, Inc. reached an agreement to sell KWHT and its sister stations to Capps Broadcast Group through its KSRV, Inc. subsidiary. The deal was approved by the FCC on May 14, 1998, the transaction was consummated on August 27, 1998. Effective November 1, 2017, Capps Broadcast Group sold KWHT and nine other broadcast properties to Elkhorn Media Group for $1.75 million. Paul Bonnell, known on the air as Kaptain Kevin Cook, worked as a disc jockey at KWHT in the 1990s after serving in the United States Air Force. Bonnell co-hosted a morning show at a radio station in Sacramento, until his death in 2007. Ron Arp, now the general manager of the Portland, Oregon office of a public relations company known as Fleishman-Hillard, was a news broadcaster at KWHT in the mid-1980s.

Fleishman-Hilliard is a part of the Omnicom Group. Jeff Walker Program Director at KWHT, was co-host of the morning show and helped manage the station during the mid to late 80s into the early 90s. KWHT official website Query the FCC's FM station database for KWHT Radio-Locator information on KWHT Query Nielsen Audio's FM station database for KWHT