NP (complexity)

In computational complexity theory, NP is a complexity class used to classify decision problems. NP is the set of decision problems for which the problem instances, where the answer is "yes", have proofs verifiable in polynomial time by a deterministic Turing machine. An equivalent definition of NP is the set of decision problems solvable in polynomial time by a non-deterministic Turing machine; this definition is the basis for the abbreviation NP. These two definitions are equivalent because the algorithm based on the Turing machine consists of two phases, the first of which consists of a guess about the solution, generated in a non-deterministic way, while the second phase consists of a deterministic algorithm that verifies if the guess is a solution to the problem. Decision problems are assigned complexity classes based on the fastest known algorithms. Therefore, decision problems may change classes, it is easy to see that the complexity class P is contained in NP, because if a problem is solvable in polynomial time a solution is verifiable in polynomial time by solving the problem.

But NP contains many more problems. An algorithm solving such a problem in polynomial time is able to solve any other NP problem in polynomial time; the most important P versus NP problem, asks whether polynomial time algorithms exist for solving NP-complete, by corollary, all NP problems. It is believed that this is not the case; the complexity class NP is related to the complexity class co-NP for which the answer "no" can be verified in polynomial time. Whether or not NP = co-NP is another outstanding question in complexity theory; the complexity class NP can be defined in terms of NTIME as follows: N P = ⋃ k ∈ N N T I M E. where N T I M E is the set of decision problems that can be solved by a non-deterministic Turing machine in O time. Alternatively, NP can be defined using deterministic Turing machines as verifiers. A language L is in NP if and only if there exist polynomials p and q, a deterministic Turing machine M, such that For all x and y, the machine M runs in time p on input For all x in L, there exists a string y of length q such that M = 1 For all x not in L and all strings y of length q, M = 0 Many computer science problems are contained in NP, like decision versions of many search and optimization problems.

In order to explain the verifier-based definition of NP, consider the subset sum problem: Assume that we are given some integers, we wish to know whether some of these integers sum up to zero. Here, the answer is "yes", since the integers corresponds to the sum + + 5 = 0; the task of deciding whether such a subset with zero sum exists is called the subset sum problem. To answer if some of the integers add to zero we can create an algorithm which obtains all the possible subsets; as the number of integers that we feed into the algorithm becomes larger, both the number of subsets and the computation time grows exponentially. But notice that if we are given a particular subset we can efficiently verify whether the subset sum is zero, by summing the integers of the subset. If the sum is zero, that subset is a proof or witness for the answer is "yes". An algorithm that verifies whether a given subset has sum zero is a verifier. Summing the integers of a subset can be done in polynomial time and the subset sum problem is therefore in NP.

The above example can be generalized for any decision problem. Given any instance I of problem P and witness W, if there exists a verifier V so that given the ordered pair as input, V returns "yes" in polynomial time if the witness proves that the answer is "yes" and "no" in polynomial time otherwise P is in NP; the "no"-answer version of this problem is stated as: "given a finite set of integers, does every non-empty subset have a nonzero sum?". The verifier-based definition of NP does not require an efficient verifier for the "no"-answers; the class of problems with such verifiers for the "no"-answers is called co-NP. In fact, it is an open question whether all problems in NP have verifiers for the "no"-answers and thus are in co-NP. In some literature the verifier is called the "certifier" and the witness the "certificate". Equivalent to the verifier-based definition is the following characterization: NP is the class of decision problems solvable by a non-deterministic Turing machine that runs in polynomial time.

That is to say, P is in NP whenever P is recognized by some polynomial-time non-deterministic Turing machine M with an existential acceptance condition, meaning that w ∈ P if and only if some computation path of M leads to an accepting state. This definition is equivalent to the verifier-based definition becau

Tabs of the United States Army

In the United States Army, "tabs" are small cloth and/or metal arches displaying a word or words signifying a special skill that are worn on U. S. Army uniforms. On the Army Combat Uniform, the tabs are worn above a unit's shoulder patch and are used to identify a unit's or a soldier's special skill or are worn on shoulder patches as part of a unit's unique heritage. Individual tabs are worn as small metal arches above or below medals or ribbons on the Army Service Uniform. Tabs are valued uniquely in the U. S. Army because images rather than words are traditionally used for the symbolism of the shoulder patch worn to identify a soldier's unit, it is only to identify an individual soldier's or a whole unit's special skill that an additional shoulder patch is worn that uses words rather than images to symbolize this skill. For example, while any member of a special forces unit will wear the unit identifying patch that includes an arrowhead, sword and Airborne Tab, only soldiers who have completed special forces training will have been awarded and wear an additional tab containing the words "SPECIAL FORCES".

Some tabs are awarded to recognize an individual soldier's combat related skills or marksmanship and are worn by a soldier permanently. These tabs are considered special skill badges and have metal equivalents that are worn on the soldier's chest if their uniform does not have a place for shoulder patches. Other tabs recognize a whole unit's special skill and are considered to be part of a specific unit's shoulder sleeve patch and are worn by a soldier only while they belong to that unit; the Jungle Expert Tab is unique in that while it is awarded to recognize an individual soldier's skill, it is only worn by soldiers while they belong to certain units. Tabs awarded at the state level by the U. S. Army National Guard can only be worn by soldiers. There are four permanent individual skill/marksmanship tabs authorized for wear by the U. S. Army. In order of precedence, they are the Special Forces Tab, the Ranger Tab, the Sapper Tab, the President's Hundred Tab. Only three skill tabs may be worn at one time.

The Special Forces Tab is a service school qualification tab of the United States Army, awarded to any soldier completing either the Special Forces Qualification Course, or the Special Forces Detachment Officer Qualification Course. Soldiers who are awarded the Special Forces Tab are authorized to wear it for the remainder of their military careers when not serving in a Special Forces command; the Special Forces Tab can be revoked by the Chain of Command for significant violations of conduct considered contrary to the high standards expected of a Special Forces soldier. The Special Forces Tab was created in 1983 and is an embroidered quadrant patch worn on the upper left sleeve of a military uniform; the cloth tab is teal blue with yellow embroidered letters. The Ranger Tab is a qualification tab authorized upon completion of the U. S. Army's Ranger School by a member of the U. S. military, civilian personnel, or non-U. S. military personnel. The Ranger Tab was approved by the Chief of Staff, Army, on 30 October 1950.

The Ranger Tab can be revoked Section 1-31, para. 13. The full color tab is worn 1⁄2 inch below the shoulder seam on the left sleeve of the Army green coat; the subdued tab is worn 1⁄2 inch below the shoulder seam on the left sleeve of utility uniforms, field jackets and the desert battle dress uniform. The full color tab is 2 3⁄8 inches long, 11⁄16 inch wide, with a 1⁄8 inch yellow border and the word "RANGER" inscribed in yellow letters 5⁄16 inch high; the subdued tab is identical, except the background is olive drab and the word "RANGER" is in black letters. The Sapper Tab is a qualification tab, authorized for graduates of the U. S. Army's Sapper School; the Sapper Tab was approved by the Chief of Staff, Army, on 28 June 2004. The Sapper tab can be revoked by the Engineer Commanding Officer of Ft. Leonard Wood, MO for misconduct, or not upholding the standard as an Engineer. Any requests will be processed through USASC; the full color tab is worn 1⁄2 inch below the shoulder seam on the left sleeve of the Army green coat.

The subdued tab is worn 1⁄2 inch below the shoulder seam on the left sleeve of utility uniforms, field jackets and the desert battle dress uniform. The full color tab is 2 3⁄8 inches long, 11⁄16 inch wide, with a 1⁄8 inch red border and the word "SAPPER" inscribed in white letters 5⁄16 inch high; the woodland subdued tab is identical, except the background is olive drab and the word "SAPPER" is in black letters and the desert subdued tab has a khaki background with the word "SAPPER" in spice brown letters. The President's Hundred Tab is a marksmanship tab, authorized for soldiers who qualify among the top 100 scoring competitors in the President's Match held annually at the National Rifle Matches at Camp Perry, Ohio; this is a permanent award. Most competitors will compete each year to ensure that less qualified individuals do not receive the tab. On 27 May 1958, The National Rifle Association requested the Deputy Chief of Staff for Personnel's approval of a tab for presentation to each member of the "President's Hundred."

NRA's plan was to award the cloth tab together with a metal tab during the 1958 National Matches. The cloth tab was of high level interest and approved for wear on the uniform on 3 March 1958. A full-color embroidered tab of yellow 4 1⁄4 inches (11

Surajit Kar Purkayastha

Surajit Kar Purkayastha is and retired senior Indian Police Service officer of 1985 West Bengal cadre, serving as the State Security Advisor of West Bengal since 1 June 2018. Prior to that,he served as Director General of Police of West Bengal and was the 36th Police Commissioner of Kolkata. Purkayastha holds an Electrical engineering degree from Indian Institute of Technology Kharagpur and after completion of his studies he joined Calcutta Electric Supply Corporation, he is an officer of 1985 West Bengal cadre. His first posting was at Bolpur subdivision as SDPO in Birbhum district, he has worked as Additional Superintendent of Police in South 24 Parganas district and as Additional Deputy Commissioner Asansol–Durgapur Police Commissionerate and Barrackpore Police Commissionerate. He worked in Kolkata Police in various posts like DC Security Control-2, DCDD, DC Traffic etc. Atter that he became the Superintendent of police of Howrah and went to Iran as a Research and Analysis Wing officer and stayed there for 5 years.

He hold the ranks of DIG-Vigilance, IG-South Bengal, IG-Law and Order. He became ADG- Law and Order before assuming the office of Police Commissioner of Kolkata. On 14 February 2013 he was appointed as the 36th Police Commissioner of Kolkata replacing Ranjit Kumar Panchananda and he served as the Commissioner for 3 years and on 13 February he was succeeded by Rajeev Kumar. After serving as Police Commissioner of Kolkata for 3 years, he was appointed as the DGP of the state succeeding Shri. G. M. P. Reddy and served as the State Police chief for 2 years until 31 May 2018 and he was succeeded by Virendra; the state government, did not let Kar Purkayastha go after his retirement. The Government has created a post State Security Advisor on the lines of the National Security Advisor and Purkayastha was supposed to fill the post