In computational complexity theory, a problem is NP-complete when it can be solved by a restricted class of brute force search algorithms and it can be used to simulate any other problem with a similar algorithm. More each input to the problem should be associated with a set of solutions of polynomial length, whose validity can be tested such that the output for any input is "yes" if the solution set is non-empty and "no" if it is empty; the complexity class of problems of this form is called NP, an abbreviation for "nondeterministic polynomial time". A problem is said to be NP-hard if everything in NP can be transformed in polynomial time into it, a problem is NP-complete if it is both in NP and NP-hard; the NP-complete problems represent the hardest problems in NP. If any NP-complete problem has a polynomial time algorithm, all problems in NP do; the set of NP-complete problems is denoted by NP-C or NPC. Although a solution to an NP-complete problem can be verified "quickly", there is no known way to find a solution quickly.
That is, the time required to solve the problem using any known algorithm increases as the size of the problem grows. As a consequence, determining whether it is possible to solve these problems called the P versus NP problem, is one of the fundamental unsolved problems in computer science today. While a method for computing the solutions to NP-complete problems remains undiscovered, computer scientists and programmers still encounter NP-complete problems. NP-complete problems are addressed by using heuristic methods and approximation algorithms. NP-complete problems are in NP, the set of all decision problems whose solutions can be verified in polynomial time. A problem p in NP is NP-complete if every other problem in NP can be transformed into p in polynomial time, it is not known whether every problem in NP can be solved—this is called the P versus NP problem. But if any NP-complete problem can be solved then every problem in NP can, because the definition of an NP-complete problem states that every problem in NP must be reducible to every NP-complete problem.
Because of this, it is said that NP-complete problems are harder or more difficult than NP problems in general. A decision problem C is NP-complete if: C is in NP, Every problem in NP is reducible to C in polynomial time. C can be shown to be in NP by demonstrating that a candidate solution to C can be verified in polynomial time. Note that a problem satisfying condition 2 is said to be NP-hard, whether or not it satisfies condition 1. A consequence of this definition is that if we had a polynomial time algorithm for C, we could solve all problems in NP in polynomial time; the concept of NP-completeness was introduced in 1971, though the term NP-complete was introduced later. At the 1971 STOC conference, there was a fierce debate between the computer scientists about whether NP-complete problems could be solved in polynomial time on a deterministic Turing machine. John Hopcroft brought everyone at the conference to a consensus that the question of whether NP-complete problems are solvable in polynomial time should be put off to be solved at some date, since nobody had any formal proofs for their claims one way or the other.
This is known as the question of whether P=NP. Nobody has yet been able to determine conclusively whether NP-complete problems are in fact solvable in polynomial time, making this one of the great unsolved problems of mathematics; the Clay Mathematics Institute is offering a US$1 million reward to anyone who has a formal proof that P=NP or that P≠NP. The Cook–Levin theorem states that the Boolean satisfiability problem is NP-complete. In 1972, Richard Karp proved that several other problems were NP-complete. Since the original results, thousands of other problems have been shown to be NP-complete by reductions from other problems shown to be NP-complete. An interesting example is the graph isomorphism problem, the graph theory problem of determining whether a graph isomorphism exists between two graphs. Two graphs are isomorphic if one can be transformed into the other by renaming vertices. Consider these two problems: Graph Isomorphism: Is graph G1 isomorphic to graph G2? Subgraph Isomorphism: Is graph G1 isomorphic to a subgraph of graph G2?
The Subgraph Isomorphism problem is NP-complete. The graph isomorphism problem is suspected to be neither in P nor NP-complete, though it is in NP; this is an example of a problem, thought to be hard, but is not thought to be NP-complete. The easiest way to prove that some new problem is NP-complete is first to prove that it is in NP, to reduce some known NP-complete problem to it. Therefore, it is useful to know a variety of NP-complete problems; the list below contains some well-known problems that are NP-complete when expressed as decision problems. To the right is a diagram of some of the problems and the reductions used
Anthony L. Krotiak was a United States Army soldier and a recipient of the United States military's highest decoration—the Medal of Honor—for his actions in World War II. Krotiak joined the Army from his birth city of Chicago, Illinois in November 1941, by May 8, 1945 was serving as a private first class in Company I, 148th Infantry Regiment, 37th Infantry Division. On that day, in the Balete Pass, the Philippines, he smothered the blast of a Japanese-thrown grenade with his body, sacrificing himself to protect those around him. For these actions, he was posthumously awarded the Medal of Honor the next year, on February 13, 1946. Krotiak, aged 29 at his death, was buried in Holy Sepulchre Cemetery, Illinois. Private First Class Krotiak's official Medal of Honor citation reads: He was an acting squad leader, directing his men in consolidating a newly won position on Hill B when the enemy concentrated small arms fire and grenades upon him and 4 others, driving them to cover in an abandoned Japanese trench.
A grenade thrown from above landed in the center of the group. Pushing his comrades aside and jamming the grenade into the earth with his rifle butt, he threw himself over it, making a shield of his body to protect the other men; the grenade exploded under him, he died a few minutes later. By his extraordinary heroism in deliberately giving his life to save those of his comrades, Pfc. Krotiak set an inspiring example of utter devotion and self-sacrifice which reflects the highest traditions of the military service. List of Medal of Honor recipients for World War II This article incorporates public domain material from websites or documents of the United States Army Center of Military History. "Anthony L. Krotiak". Claim to Fame: Medal of Honor recipients. Find a Grave. Retrieved 2008-01-21
The Mining industry of Ghana accounts for 5% of the country's GDP and minerals make up 37% of total exports, of which gold contributes over 90% of the total mineral exports. Thus, the main focus of Ghana's mining and minerals development industry remains focused on gold. Ghana is Africa's largest gold producer, producing 80.5 t in 2008. Ghana is a major producer of bauxite and diamonds. Ghana has 23 large-scale mining companies producing gold, diamonds and manganese, there are over 300 registered small scale mining groups and 90 mine support service companies. Other mineral commodities produced in the country are natural gas, petroleum and silver. Export earnings from minerals averaged 35%, the sector is one of the largest contributors to Government revenues through the payment of mineral royalties, employee income taxes, corporate taxes. In 2005, gold production accounted for about 95% of total mining export proceeds; the extractive mining industry of Ghana is expected to generate an annual revenue of GH₵75.7 billion in 2014 and other than industrial minerals and exports from South Ghana such as timber, diamonds and manganese, Ashanti Region has a great deposit of barites.
Relevant institutions include: Ministry of Lands and Natural Resources – overall responsibility for the mining industry Minerals Commission – recommends mineral policy. Within the Ministry, the Minerals Commission has responsibility for administering the Mining Act, recommending mineral policy, promoting mineral development, advising the government on mineral matters, serving as a liaison between industry and the government; the Ghana Geological Survey Department conducts geologic studies. The Ghana National Petroleum Corporation is the government entity responsible for petroleum exploration and production; the Precious Minerals Marketing Corporation is the government entity responsible for promoting the development of small-scale gold and diamond mining in Ghana and for purchasing the output of such mining, either directly or through licensed buyers. The Mines Department has authority in mine safety matters. All mine accidents and other safety problems must be reported to the Ghana Chamber of Mines, the private association of operating mining companies.
The Chamber provides information on Ghana's mining laws to the public negotiates with the mine labor unions on behalf of its member companies. The overall legislative framework for the mining sector in Ghana is provided by the Minerals and Mining Act of 2006. Under the Law, mining companies must pay royalties. Other legislation that affects mining and mineral exploration in Ghana includes the Minerals Commission Law of 1986; the Petroleum Law, 1984, sets out the policy framework and describes the role of the Ministry of Mines and Energy, which regulates the industry. The Ghana National Petroleum Corporation, empowered to undertake petroleum exploration and production on behalf of the government, is authorized to enter joint ventures and production-sharing agreements with commercial organizations; the regulation of artisanal gold mining is set forth in the Small-Scale Gold Mining Law, 1989. The Precious Minerals Marketing Corporation Law, 1989, set up the Precious Minerals Marketing Corporation to promote the development of small-scale gold and diamond mining in Ghana and to purchase the output of such mining, either directly or through licensed buyers.
In the gold sector, Gold Fields Limited of South Africa held a 71.1% interest in the Tarkwa and the Damang gold mines in a joint venture with Toronto-based IAMGOLD Corp. and the Government of Ghana. AngloGold Ashanti Ltd. of South Africa operated the Bibiani and the Iduapriem open pit gold mines and the Obuasi underground gold mine. The Bibiani and the Obuasi mines were 100% owned by AngloGold Ashanti and the Iduapriem mine was 80% owned by AngloGold Ashanti and 20% by the International Finance Corporation. Golden Star Resources Ltd. held a 90% interest in the Bogoso/Prestea and the Wassa open pit mines and a 90% interest in the idled Prestea underground mine. Newmont Mining Corporation of the United States held a 100% interest in the Ahafo gold property and an 85% interest i