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These two types of ratios are used because there exist algorithms where the difference between these two is significant.
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In the literature, an approximation ratio for a maximization (minimization) problem of "c" - ϵ (min: "c" + ϵ) means that the algorithm has an approximation ratio of "c" ∓ ϵ for arbitrary ϵ > 0 but that the ratio has not (or cannot) be shown for ϵ = 0.
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An example of this is the optimal inapproximability — inexistence of approximation — ratio of 7 / 8 + ϵ for satisfiable MAX-3SAT instances due to Johan Håstad.
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As mentioned previously, when "c" = 1, the problem is said to have a polynomial-time approximation scheme.
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An ϵ-term may appear when an approximation algorithm introduces a multiplicative error and a constant error while the minimum optimum of instances of size "n" goes to infinity as "n" does.
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In this case, the approximation ratio is "c" ∓ "k" / OPT = "c" ∓ o(1) for some constants "c" and "k".
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Given arbitrary ϵ > 0, one can choose a large enough "N" such that the term "k" / OPT < ϵ for every "n ≥ N".
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For every fixed ϵ, instances of size "n < N" can be solved by brute force, thereby showing an approximation ratio — existence of approximation algorithms with a guarantee — of "c" ∓ ϵ for every ϵ > 0.
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Sabri Brothers
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The Sabri Brothers (Punjabi, ) is a music band from Pakistan who are performers of Sufi qawwali and are closely connected to the Chishti Order, They are referred to as "Roving ambassadors of Pakistan".
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The band was initially founded and led by Ghulam Farid Sabri and his brother Maqbool Ahmed Sabri.
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They were the first exponents of qawwali to the West, when they performed at New York's Carnegie Hall in 1975.
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Sabri Brothers have given a number of soulful qawwali performances globally, and their stature in the Sub-continent is colossal.
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The Sabri brothers learned music from their father, Inayat Hussain Sabri.
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He trained his sons in qawwali and Indian classical music.
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Ghulam Farid's public performance was at the annual Urs festival of Mubarak Shah in Kalyana in 1946.
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The family moved from Kalyana to Karachi, Pakistan following the Partition of India in 1947.
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Ghulam Farid Sabri, Kamal Ahmed Sabri, And Maqbool Ahmed Sabri furthered their knowledge of music under Ustad Fatehdin Khan, Ustad Ramzan Khan, and Ustad Latafat Hussein Khan Bareilly Sharif, They even furthered their knowledge of Poetry under Hazrat Hairat Ali Shah Warsi who was their spiritual master (Daada Peer) too...
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Ghulam Farid Sabri was performing as a supporting lead in the group of his teacher Ustad Kallan Khan Qawwal.
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Later, With the help of his father Maqbool Ahmed Sabri formed a Qawwali group at the age of eleven and named it "Bacha Qawwal Party".
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Soon afterwards after insistence by their father, Ghulam Farid Sabri joined him and became the leader of the party, which soon came to be known as "Ghulam Farid Sabri – Maqbool Ahmed Sabri Qawwal Party".
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During their 1975 American tour, their promoter Beate Gordon suggested the band name was too long, so they changed it to "The Sabri Brothers and ensemble from Pakistan".
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Sabri Brothers initially started their career by performing at Sufi Shrines and private gatherings, Their first recording was officially released in 1958 under the EMI Pakistan label, was the Urdu qawwali titled "Mera Koi Nahi Hai Tera Siwa" (There Is No For Me One But You), which was later appeared in the 1965 Pakista...
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1970s witnessed the rise of the Sabri Brothers.
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They are the only qawwali troupe which has a "first class" status on the Pakistan Television Corporation.
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In 1970 the Government of Pakistan sent them to Nepal as representatives for a royal wedding.
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They released their blockbuster hits included "Bhar Do Jholi Meri Ya Muhammad" (Fulfill my needs O Prophet Muhammad !, 1972).
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"Tajdar-E-Haram" (King of the Kaaba, 1970), " O Sharabi Chord De Peena " (Hey, Alcoholic, Stop Drinking, 1976), "Khwaja Ki Deewani" (Lover of Khawaja, 1976) and "Balaghal Ula Be Kamalehi" (Reaching the Highest Heights Through Perfection, 1971).
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Popular film and recording artists in Pakistan, their several qawwalis featured in films.
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"Mohabbat Karne Walo Hum Mohabbat Iss Ko Kehtain Hain" in the 1970 film "Chand Suraj", "Aaye Hain Tere Dar Pe Tau Kucch Lay Ke Jaen Gay" in the 1972 film "Ilzam", "Bhar Do Jholi Meri Ya Muhammad" in the 1975 film "Bin Badal Barsaat", "Bawa Farid Sarkar" in the 1974 film "Sasta Khoon Mehnga Pani", "Teri Nazr-e-Karam Ka ...
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The Sabri Brothers troupe has toured Europe, Asia and the Middle East.
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They were the first exponents of qawwali to the West, when they performed at New York's Carnegie Hall in 1975 promote and sponsored by Beate Gordon.
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They performed in the United States and Canada under the auspices of The Performing Arts Program of The Asia Society and recorded a program at Brooklyn College Television Center.
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In 1972, they performed a charity concert for the construction of Pakistani Children School in Abu Dhabi.
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In the same year, with the co-operation of Star Agencies they performed in various cities of England such as London, Bradford, Birmingham and Manchester which become very popular.
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The amount of these programs donated to Earthquake relief fund Pakistan.
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In 1975, Ahmed Suleiman co-operated to arrange concert in Durban, South Africa.
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Inspired with these programs Chevrolet Company gifted an automatic car to Brothers, which they donated for the development of poor children.
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In 1977, they recorded the album 'Pakistan: The Music of the Qawal' for the UNESCO Collection of Traditional Music which was later released in CD form by Auvidis in 1990.
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They performed at Royal Albert Hall on 20 June 1976 in "World of Islam Festival".
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In April 1978, the album "Qawwali" was recorded in the United States, while the Sabri Brothers were on tour.
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The New York Times review described the album as, "The Aural Equivalent of Dancing Dervishes" and the, "Music of Feeling."
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In 1977, The Sabri Brothers toured India, their concerts were attended by many celebrities of Bollywood.
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During that tour they appeared in Indian film "Sultan E Hind - Khwaja Gharib Nawaz" and recorded "Aftaab E Risalat Madine Mei Hai" which was a blockbuster hit.
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In 1979, they performed a charity concert for the construction of Karachi School of Art which was later released in album "Sabri Brothers Live in Concert at Ali Bhai Auditorium".
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In June 1981, they performed at the Royal Tropical Institute in Amsterdam which was released in album "Tasleem".
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In 1982, they appeared in film "Sahaaray" with their famous qawwali "Tajdar-e-Haram".
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The same year they performed in Midway Hotel for the construction of Al Shifa Hosiptal near Karachi Airport.
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In 1983, they recorded the album "Nazre Shah Karim" to commemorate the Golden Jubilee of His Highness Prince Aga Khan, sponsored by Tajico Group.
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The income of this album was donated to Aga Khan Hospital, Karachi.
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On 3 August 1985, a stupendous program was held at Sheraton Hotel, Karachi and collect Rs.
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141,500/- for Bangladesh Flood Fund.
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In 1985, Maqbool Ahmed Sabri recorded a solo Ghazal album in front of live audience in Karachi, the album was released as "Awargi" which was a blockbuster, The album was a collection of light playful compositions by poet Farhat Shahzad.
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In 1988 "EMI Pakistan" released another solo ghazal album of Maqbool Ahmed Sabri titled "Tere Ghungroo Toot Gaye to Kiya" which was a hit.
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In the same year he recorded qawwali for music director Anu Malik in the Indian movie Gangaa Jamunaa Saraswati which was picturesque on Mithun Chakraborty.
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In 1989, The Sabri Brothers performed at WOMAD festival held in UK and France.
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They recorded an album in UK which was released as the album "Ya Habib" in 1990.
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"Ya Habib" consists of four long songs, each combining powerful, sensitive, often improvised vocals with rhythmic percussion, thudding tabla and mesmeric harmonium drones.
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In 1989 and 1992, they performed at various South Asian Association for Regional Cooperation festivals.
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In 1992, they performed at The Hope Theater in Melbourne, Australia.
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Their tour of Australia was sponsored by Friends of the University of Wollongong.
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In the same year they performed at fundraiser in Dubai to benefit Shaukat Khanum Memorial Cancer Hospital.
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In 1992 they also performed in various countries for SAARC festival.
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During April 1994, Sabri brothers were set to begin on a tour across Europe.
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They were to perform at around 30–35 shows, covering almost the entire continent.
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On 5 April 1994, Ghulam Farid Sabri suddenly complained of chest pain, He suffered a massive heart attack and was rushed to hospital but en route to the hospital, he died in the arms of his brother Maqbool Ahmed Sabri who was left heartbroken after his death but still carried on his and his elder brother's mission.
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After the passing away of Ghulam Farid Sabri, the group was led by Maqbool Ahmed Sabri and "Mehmood Ghaznavi Sabri" completed the tour with his name.
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To devote an album entirely to the Persian poetry of Jami, a luminary of the Sufi Tradition, was an ambition of Ghulam Farid Sabri which he had always cherished.
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He did the recordings in July 1991 at the SFB studios in Berlin, but the CD was not released while he was still alive.
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Thus, it becomes a memorial not only to the Persian poet, but also to Ghulam Farid Sabri.
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The same year In 1994, Maqbool Ahmed Sabri & Mehmood Ghaznavi Sabri Led The Sabri Brothers and performed at New Jazz Festival Moers, Germany.
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In 1995, Maqbool Ahmed Sabri was about to perform in Meltdown Festival Event held in the UK, A week before the event, he suffered a very major accident in Lahore.
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Maqbool Ahmed Sabri was in a very serious and critical condition during that time, prayers were made on national radio For Maqbool Sabri's health.
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He recovered, though he had a damaged leg after the accident and had to undergo multiple knee surgeries.
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In 1996, they performed at the Brooklyn Academy of Music- Next Wave Festival, as part of a double-bill with alternate-rockers Corner Shop, and released a live album "Ya Mustapha (Or Ya Mustafa)" of their performance.
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The album became one of their greatest hits.
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In 1997, they once again performed at the Royal Albert Hall in front of the Prince of Wales, Prince Charles for the Celebration of 50 Years of Independence of India and Pakistan together with Sri Lanka and Bangladesh, They were one of the few artists who have performed at the royal albert hall multiple times.
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In 1998, the group toured Australia and also performed at Sydney Opera House.
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Shortly after the program in Opera House, Kamal Ahmed Sabri (second eldest of the brothers) the supporting vocalist and instrumentalist of the group suffered a heart attack, though he survived he could not take part in qawwali due to bad health conditions.
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In 1999, they participated in the 'Voices of God' festival in Marrakesh, Morocco in May.
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The group also performed in various qawwali concerts held in Morocco in 1999.
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Sabri Brothers performed at Musica Sacra International Mktoberdorf Event in the year 2000 at Germany.
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In 2001, Kamal Ahmed Sabri second eldest of the Sabri brothers, died after suffering from a second heart attack, after the death of Kamal Sabri, instruments such as Swarmandal and Flexatone came to end in the ensemble.
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Later, The same year the group performed and recorded live in DOM at "On The Carpet Oriental Culture Festival" on 17 November 2001 in Moscow which was later released in 2003 As "Live in Moscow – Diwani"
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Maqbool Ahmed Sabri along with Mehmood Ghaznavi Sabri toured India in 2004 and recorded an album titled "Rabb E Akbar".
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which was his last official release.
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In 2006, he re created and re released his old hits like "Khwaja Ki Deewani", "Mera Koi Nahi Hai Tere Siwa", "Mann Bole Mann", and "Hum Panjatani Hai" which proved to be a big hit.
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Maqbool Ahmed Sabri again toured India in 2006 and performed in Ajmer Sharif and in Pune, Maharashtra.
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The Sabri Brothers led by Maqbool Ahmed Sabri & Mehmood Ghaznavi Sabri continued to tour and performed in Russia in the year 2007,
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Maqbool Ahmed Sabri did various international and national tours till his passing away, He performed in Hyderabad India in 2008.
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His last concert was organized by Hilton Hotel in South Africa which was during 2010.
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Since then his heath conditions were not good.
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On 21 September 2011, Maqbool Ahmed Sabri died in South Africa due to a cardiac arrest after being treated for two months for health problems.
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He was buried near his elder brother Ghulam Farid Sabri.
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The group is now led by the youngest brother Mehmood Ghaznavi Sabri.
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While other family members and disciples perform in their own separate groups to carry on the legacy of Ghulam Farid Sabri and Maqbool Ahmed Sabri.
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Many consider the Sabri Brothers instrumentally more adventurous, than Nusrat Fateh Ali Khan.
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Aziz Mian mastered in presenting intoxication as closeness to God, and said more than 3,000 couplets in that metaphor, and even Nusrat Fateh Ali Khan was a perfectionist in discussing the beauty of the Creator of feminine attractiveness.
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although The Sabri Brothers even though recited many famous couplets and poetries in presenting intoxication and closeness to God as a part of Sufi culture, They mostly focused on "Hamd (Praise of God)", "Naat (Praising of the Holy Prophet)", And "Manqabat (Praising of Holy People And Saints)".
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During 1970s a squabble started between the two greatest Sufi singers when Sabri Brothers released one of their biggest hits "Bhar do Jholi Meri", while Aziz Mian released his biggest hit "Main Sharaabi".