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# Ford

[fawrd, fohrd]
Ford, Bill: see under Ford, Henry.
Ford, Edsel Bryant: see under Ford, Henry.
Ford, Ford Madox, 1873-1939, English author; grandson of Ford Madox Brown. He changed his name legally from Ford Madox Hueffer in 1919. The author of over 60 works including novels, poems, criticism, travel essays, and reminiscences, Ford also edited the English Review (1908-11) and the Transatlantic Review (1924, Paris); among his contributors were Thomas Hardy, James Joyce, and D. H. Lawrence. Ford's most important fictional works are The Good Soldier (1915), a subtle and complex novel about the relationship of two married couples, and a tetralogy (1924-28): Some Do Not, No More Parades, A Man Could Stand Up, and The Last Post (pub. together as Parade's End, 1950). These works reveal the collapse of the Tory-Christian virtues under the violence and social hypocrisy that culminated in World War I. Ford collaborated with Joseph Conrad on The Inheritors (1901), Romance (1903), and other works. His memoir of Conrad (1924) discusses the narrative techniques that the two writers evolved. Toward the end of his life, Ford lived in France and the United States and was a member of the faculty of Olivet College in Michigan.

See his letters (ed. by R. M. Ludwig, 1965); biographies by F. MacShane (1965) and A. Mizener (1971, repr. 1985); studies by F. MacShane, ed. (1972), S. Stand, ed. (1981), A. B. Snitow (1984), and R. A. Cassell, ed. (1987).

Ford, Gerald Rudolph, 1913-2006, 38th president of the United States (1974-77), b. Omaha, Nebr. He was originally named Leslie Lynch King, Jr., but his parents were divorced when he was two, and when his mother remarried he assumed the name of his stepfather. Admitted to the Michigan bar in 1941, he was a member (1949-73) of the U.S. House of Representatives, where he served as the Republican minority leader (1965-73). Ford gained a reputation as a loyal Republican who supported his party on virtually all issues. A consistent proponent of a large defense budget, he led the Republican opposition to the Great Society programs of President Lyndon B. Johnson. He was permanent chairman of the Republican National Convention in 1968 and 1972.

In Oct., 1973, Ford was nominated by President Richard Nixon to succeed the disgraced Spiro T. Agnew as vice president of the United States; on Dec. 6, 1973, he was sworn in, becoming the first person to be appointed to the office under the procedures specified by the 25th Amendment. As vice president, Ford traveled widely around the country, attempting to rally for the Nixon administration the support that had eroded as a result of the Watergate affair. His tenure as vice president was short, however; when Nixon resigned on Aug. 9, 1974, Ford became president. He pledged to continue Nixon's foreign policy and to work to curb inflation. One month later he issued a complete pardon to Nixon for all criminal acts perpetrated by Nixon while he was president. In the 1974 congressional elections the Republicans suffered substantial losses, attributable both to Watergate and to the economy. To deal with the economic recession, Ford proposed (1975) tax cuts, limited social spending (with continued high defense expenditure), and heavy taxation on imported oil. The Democratic Congress opposed many elements of the program. Ford was defeated by Democrat Jimmy Carter in the 1976 presidential election.

See Ford's Selected Speeches, ed. by M. V. Doyle (1973); C. Fitzgerald, ed., Gerald R. Ford (1988).

Ford, Henry, 1863-1947, American industrialist, pioneer automobile manufacturer, b. Dearborn, Mich.

## The Inception of the Ford Motor Company

Ford showed mechanical aptitude at an early age and left (1879) his father's farm to work as an apprentice in a Detroit machine shop. He soon returned to his home, but after considerable experimentation with power-driven vehicles, he went (1890) to Detroit again and worked as a machinist and engineer with the Edison Company. Ford continued working in his spare time as well, and in 1896 he completed his first automobile. Resigning (1899) from the Edison Company he launched the Detroit Automobile Company.

A disagreement with his associates led Ford to organize (1903) the Ford Motor Company in partnership with Alexander Malcomson, James Couzens (who devised and oversaw the company's successful early business and accounting procedures), the Dodge brothers, and others. In 1907 he purchased the stock owned by most of his associates, and thereafter the Ford family remained in control of the company. By cutting the costs of production, by adapting the conveyor belt and assembly line to automobile production, and by featuring an inexpensive, standardized car, Ford was soon able to outdistance all his competitors and become the largest automobile producer in the world. He came to be regarded as the apostle of mass production. In 1908 he guided his chief engineer Harold Wills in the design of the Model T; nearly 17 million cars were produced worldwide before the model was discontinued (1928) and a new design—the Model A—was created to meet growing competition. Highly publicized for paying wages considerably above the average, Ford began in 1914—the year he created a sensation by announcing that in future his workers would receive \$5 for an 8-hr day—a profit-sharing plan that would distribute up to \$30 million annually among his employees.

## Later Years

In 1915, in an effort to end World War I, he headed a privately sponsored peace expedition to Europe that failed dismally, but after the American entry into the war he was a leading producer of ambulances, airplanes, munitions, tanks, and submarine chasers. In 1918 he ran unsuccessfully for the U.S. Senate on the Democratic ticket. After weathering a severe financial crisis in 1921, he began producing high-priced motor cars along with other vehicles and founded branch firms in England and in other European countries. Strongly opposed to trade unionism, Ford—who incurred considerable antagonism because of his paternalistic attitude toward his employees and his statements on political and social questions—stubbornly resisted union organization in his factories by the United Automobile Workers until 1941. A staunch isolationist before World War II, Ford again converted his factories to the production of war material after 1941. In 1945 he retired.

## Other Accomplishments and Controversies

His numerous philanthropies, in addition to the Ford Foundation, included \$7.5 million for the Henry Ford Hospital in Detroit and \$5 million for a museum in Dearborn, where in 1933 he established Greenfield Village—a reproduction of an early American village. Ford also wrote, in collaboration with Samuel Crowther, My Life and Work (1923), Today and Tomorrow (1926), Moving Forward (1931), and Edison as I Knew Him (1930).

Ford's international reputation made him a natural target for journalists. His libel suit against the Chicago Tribune in 1919 led to an examination by the Tribune attorney, intended to show Ford's lack of education. Anti-Semitic articles in Ford's Dearborn Independent brought further legal controversy; he was forced to apologize for the articles. In the 1930s, Ford was widely attacked for employing Harry Bennett, a former boxer who established a squad of thugs to spy, beat up, and otherwise intimidate union organizers.

Ford was also a poor manager who failed to capitalize on his company's early success. In the 1920s he failed to respond to consumer tastes by introducing new models and the company fell far behind General Motors. By the time of his retirement, the company's accounting procedures were so primitive that Ford's managers were unable to accurately tell how much it cost to manufacture a car and the company was losing \$9.5 million a month.

## Later Generations

Henry Ford's son, Edsel Bryant Ford, 1893-1943, b. Detroit, shared in the control of the vast Ford industrial interests. He was president of the Ford Motor Company from 1919 until his death, when his father once more became (1943) president of the company. The eldest Ford soon retired again when his grandson, Henry Ford 2d, 1917-87, b. Detroit, succeeded him in 1945. The younger Henry Ford moved quickly to restructure and modernize the company, which had slipped from the world's largest automobile manufacturer in 1920 to number three in the U.S. market in 1945. He removed a number of long-time Ford executives, such as Bennett, and for the first time in company history, recruited outsiders for positions of responsibility. The company spent \$1 billion between 1945 and 1955 to expand its operations, introduced successful new models, and raised \$690 million in capital by offering stock to the public (1956). Although Ford modernized and revitalized the company, his tenure also saw the introduction of the Edsel, which lost the company \$250 million, and Ford's autocratic management style forced a number of top executives, such as Lee Iacocca, to quit. In 1960, Ford became chief executive officer and chairman of the corporation, offices he held until retiring as CEO in 1979 and as chairman in 1980.

Although family shareholders continued to have voting control of the company, nonfamily members headed Ford until 1999, when Bill Ford (William Clay Ford, Jr.), 1957-, became chairman. Working at Ford Motor Company from 1979, Bill Ford became vice president of the commercial truck vehicle center in 1994, chairman of the finance committee in 1995, and chairman of the board in 1999. In 2001 he also became chief executive officer of Ford, but the company's difficulties led him to resign that post in 2006.

## Bibliography

See biographies by A. Nevins and F. E. Hill (3 vol., 1954-62), B. Herndon (1969), R. Lacey (1986), and S. Watts (2005); R. M. Wik, Henry Ford and Grass-Roots America (1970); P. Collier and D. Horowitz, The Fords (1987); N. Baldwin, Henry Ford and the Jews (2001); D. Brinkley, Wheels for the World (2003).

Ford, Henry, 2d: see under Ford, Henry.
Ford, John, 1586-c.1640, English dramatist, b. Devonshire. He went to London to study law but was never called to the bar. The early part of his playwriting career was taken up with collaborations, primarily with Dekker. His three major tragedies, 'Tis Pity She's a Whore, The Broken Heart, and Love's Sacrifice, and a historical play, Perkin Warbeck appeared between 1627 and 1634. Ford was the most important playwright during the reign of Charles I. His plays are characterized by a sympathetic treatment of thwarted love, and they stress the conflict between the power of human passion and the laws of conscience and society. They are intense, melancholy, and violent, often revealing his interest in abnormal psychology and taboo subjects—'Tis Pity She's a Whore deals with incest.

See biography by D. K. Anderson (1972); studies by M. Stavig (1968), F. Ali (1974), and D. Anderson (1986).

Ford, John, 1895-1973, American film director, b. Cape Elizabeth, Maine, as John Martin Feeney. Ford began directing in 1917 after an apprenticeship with his brother Francis. Over the next 50 years, he brought a painterly eye, an appreciation of his actors' strengths, and a deep love of Americana to over 200 feature films. Although Ford set films in other parts of the country or world, including several in Ireland, he returned to the Western repeatedly throughout his career. These films merge a beautiful pictorial style, using the buttes and mesas almost as architectural features, with stories that frequently deal with the nature of military command. Among his films are The Iron Horse (1924), The Informer (1935), Stagecoach (1939), The Grapes of Wrath (1940), How Green Was My Valley (1941), Fort Apache (1947), The Quiet Man (1952), The Searchers (1957), and The Man Who Shot Liberty Valance (1962). Ford won six Academy Awards. During World War II he served in the U.S. navy and made the acclaimed documentary June 7th (1944).

See biographies by A. Sinclair (1979), S. Eyman (1999), and J. McBride (2001); studies by P. Bogdanovich (1968), J. McBride and M. Wilmington (1974), and T. Gallagher (1986).

Ford, Paul Leicester, 1865-1902, American historian and novelist, b. Brooklyn, N.Y. His father, Gordon L. Ford, then possessed probably the best library of Americana in the country; Paul edited, with his brother Worthington Chauncey Ford, rare material from this library in Winnowings in American History (15 vol., 1890-91) and made valuable bibliographies, including ones on Hamilton (1886) and Franklin (1889). He was the editor of many documents of the early Republic. His True George Washington (1896, repr. 1970) was the first to present Washington as human and fallible. Ford also wrote several novels.
Ford, Richard, 1944-, American novelist, b. Jackson, Miss.; grad. Michigan State Univ. (B.A., 1966), Univ. of California, Irvine (M.F.A., 1970). Ford's concerns are those of a moralist who displays a deeply felt sympathy toward his often struggling, sometimes down-at-the-heels characters; his prose style is straightforward, even spare. His literary reputation was established with The Sportswriter (1986), a widely acclaimed novel that is still his best known. It is a first-person account of a weekend in the life of novelist-turned-sportswriter Frank Bascombe, a tough-minded yet thoughtful, alienated yet acutely observant character whose reflections on his own life reveal much about contemporary America. Bascombe, who has gone from sportswriter to successful real estate agent, reappears in two sequels published in succeeding decades, Independence Day (1995, Pulitzer Prize) and The Lay of the Land (2006). Ford's other novels include A Piece of My Heart (1976), The Ultimate Good Luck (1981), and Wildlife (1990). He has also written numerous short stories, many dealing with life in the rural West and collected in Rock Springs (1987), the three novellas of Women with Men (1997), and later stories focused on marital infidelity in the collection A Multitude of Sins (2002). Other works include essays, a play (1983), and a screenplay (1991).

See studies by H. Guagliardo, ed. (2000) and E. A. Walker (2000).

Ford, William Clay, Jr.: see Bill Ford under Ford, Henry.
Ford, Worthington Chauncey, 1858-1941, American historian and editor, b. Brooklyn, N.Y. He was joint editor, with his brother Paul Leicester Ford, of Winnowings in American History (15 vol., 1890-91). While chief of the manuscripts division of the Library of Congress (1902-9), Ford edited the first 15 volumes of the new edition of the Journals of the Continental Congress. From 1909 until his retirement in 1929 he was editor of the Massachusetts Historical Society and lectured on historical manuscripts at Harvard. Ford edited The Writings of George Washington (14 vol., 1889-93) and wrote a standard biography of Washington (1899). He also edited volumes of correspondence of writings of the Adams family.
ford, shallow place in a body of water, especially a river, that may be crossed by wading. Around the crossings habitually forded, cities sprang up; hence fords came to be the sites of numerous river towns. They have been of particular importance in migrations and in the deployment of armies in campaigns and have therefore been frequently fortified.

(born Feb. 3, 1926, Cleveland, Ohio, U.S.) American dancer, choreographer, and ballet director. He trained in modern dance with Hanya Holm and Martha Graham. Between 1946 and 1962 he danced in Broadway musical productions as well as with dance companies such as the American Ballet Theatre. He became a choreographer in 1962, forming his own company and creating Pierrot Lunaire. Subsequently he staged works with most of the major dance companies throughout the world. Tetley's work helped to bring about a synthesis of modern dance and classical ballet. His creative staging and daring, often sexual, subject matter were sometimes controversial, but he was praised for the passion and strong physicality of his work.

(born Feb. 16, 1944, Jackson, Miss., U.S.) U.S. novelist and short-story writer. His first novel, A Piece of My Heart (1976), showed the influence of William Faulkner. The Sportswriter (1986) and its sequel, Independence Day (1995, Pulitzer Prize), drew on his experience as a writer for a sports magazine in the 1980s. His story collection Rock Springs (1987) examines the lives of the lonely and alienated.

(baptized April 17, 1586, Ilsington, Devon, Eng.—died 1639?) British dramatist. Early in his career he studied law and wrote collaboratively with several other playwrights, but little more is known of his life, and the dating of many of his works is uncertain. His revenge tragedies are characterized by scenes of austere beauty, insight into human passions, and poetic diction of a high order. His reputation rests on the first four plays he wrote alone, only one of which can be dated with certainty: The Broken Heart; The Lover's Melancholy (1628); Perkin Warbeck; and 'Tis Pity She's a Whore, an eloquently sympathetic story of incestuous lovers that is his best-known work.

(born July 30, 1863, Wayne county, Mich., U.S.—died April 7, 1947, Dearborn, Mich.) U.S. industrialist and pioneer automobile manufacturer. Ford worked his way up from a machinist's apprentice (at age 15) to the post of chief engineer at the Edison Company in Detroit. He built his first experimental car in 1896. In 1903, with several partners, he formed the Ford Motor Company. In 1908 he designed the Model T; demand became so great that Ford developed new mass-production methods, including the first moving assembly line in 1913. He developed the Model A in 1928 to replace the Model T, and in 1932 he introduced the V-8 engine. He observed an eight-hour workday and paid his workers far above the average, holding that well-paid labourers become the consumers that industrialists require, but strenuously opposed labour unions. As the first to make car ownership affordable to large numbers of Americans, he exerted a vast and permanent influence on American life. Seealso Ford Foundation.

(born April 7, 1939, Detroit, Mich., U.S.) U.S. film director, screenwriter, and producer. He worked under Roger Corman before achieving his first success with the low-budget but stylish You're a Big Boy Now (1967). He wrote or cowrote screenplays for several films, including Patton (1970, Academy Award). He won acclaim for writing and directing the Mafia epic The Godfather (1972, Academy Awards for best picture and screenplay). His other films include The Conversation (1974), The Godfather, Part II (1974, Academy Awards for best director, picture, and screenplay), Apocalypse Now (1979), Peggy Sue Got Married (1986), The Godfather, Part III (1990), The Rainmaker (1997), and Youth Without Youth (2007).

(born Feb. 16, 1944, Jackson, Miss., U.S.) U.S. novelist and short-story writer. His first novel, A Piece of My Heart (1976), showed the influence of William Faulkner. The Sportswriter (1986) and its sequel, Independence Day (1995, Pulitzer Prize), drew on his experience as a writer for a sports magazine in the 1980s. His story collection Rock Springs (1987) examines the lives of the lonely and alienated.

(baptized April 17, 1586, Ilsington, Devon, Eng.—died 1639?) British dramatist. Early in his career he studied law and wrote collaboratively with several other playwrights, but little more is known of his life, and the dating of many of his works is uncertain. His revenge tragedies are characterized by scenes of austere beauty, insight into human passions, and poetic diction of a high order. His reputation rests on the first four plays he wrote alone, only one of which can be dated with certainty: The Broken Heart; The Lover's Melancholy (1628); Perkin Warbeck; and 'Tis Pity She's a Whore, an eloquently sympathetic story of incestuous lovers that is his best-known work.

(born July 30, 1863, Wayne county, Mich., U.S.—died April 7, 1947, Dearborn, Mich.) U.S. industrialist and pioneer automobile manufacturer. Ford worked his way up from a machinist's apprentice (at age 15) to the post of chief engineer at the Edison Company in Detroit. He built his first experimental car in 1896. In 1903, with several partners, he formed the Ford Motor Company. In 1908 he designed the Model T; demand became so great that Ford developed new mass-production methods, including the first moving assembly line in 1913. He developed the Model A in 1928 to replace the Model T, and in 1932 he introduced the V-8 engine. He observed an eight-hour workday and paid his workers far above the average, holding that well-paid labourers become the consumers that industrialists require, but strenuously opposed labour unions. As the first to make car ownership affordable to large numbers of Americans, he exerted a vast and permanent influence on American life. Seealso Ford Foundation.

orig. Ford Hermann Hueffer

(born Dec. 17, 1873, Merton, Surrey, Eng.—died June 26, 1939, Deauville, France) English novelist, editor, and critic. Ford collaborated with Joseph Conrad on The Inheritors (1901) and Romance (1903). As the founder of the English Review (1908), he generously encouraged younger writers. He was gassed and shell-shocked in World War I; after the war he changed his name to Ford. Of more than 70 published works, his best known are The Good Soldier (1915), a novel about the demise of aristocratic England; and the tetralogy Parade's End—Some Do Not (1924), No More Parades (1925), A Man Could Stand Up (1926), and Last Post (1928)—which explores the breakdown of Edwardian culture and the emergence of new values.

orig. Ford Hermann Hueffer

(born Dec. 17, 1873, Merton, Surrey, Eng.—died June 26, 1939, Deauville, France) English novelist, editor, and critic. Ford collaborated with Joseph Conrad on The Inheritors (1901) and Romance (1903). As the founder of the English Review (1908), he generously encouraged younger writers. He was gassed and shell-shocked in World War I; after the war he changed his name to Ford. Of more than 70 published works, his best known are The Good Soldier (1915), a novel about the demise of aristocratic England; and the tetralogy Parade's End—Some Do Not (1924), No More Parades (1925), A Man Could Stand Up (1926), and Last Post (1928)—which explores the breakdown of Edwardian culture and the emergence of new values.

(born April 16, 1821, Calais, Fr.—died Oct. 6, 1893, London, Eng.) British painter. He studied in Bruges, Antwerp, Paris, and Rome. In Italy (1845) he met Peter von Cornelius, a member of the Nazarenes, who influenced his palette and style. His use of brilliant colour, meticulous handling, and taste for literary subjects had a strong effect on the Pre-Raphaelites, most notably Dante Gabriel Rossetti. His most famous paintings are The Last of England (1852–55), a poignant tribute to emigration, and Work (1852–63), a Victorian social commentary. In 1861 he became a founding member of William Morris's company, for which he designed stained glass and furniture.

U.S. philanthropic foundation. It was established in 1936 with gifts and bequests from Henry Ford and his son, Edsel (1893–1943). By the early 21st century its assets exceeded \$10 billion. Its chief concerns have been international affairs (particularly population control and alleviation of food shortages), humanities and the arts, communications (especially public television), and, in later years, resources and the environment.

(born April 7, 1939, Detroit, Mich., U.S.) U.S. film director, screenwriter, and producer. He worked under Roger Corman before achieving his first success with the low-budget but stylish You're a Big Boy Now (1967). He wrote or cowrote screenplays for several films, including Patton (1970, Academy Award). He won acclaim for writing and directing the Mafia epic The Godfather (1972, Academy Awards for best picture and screenplay). His other films include The Conversation (1974), The Godfather, Part II (1974, Academy Awards for best director, picture, and screenplay), Apocalypse Now (1979), Peggy Sue Got Married (1986), The Godfather, Part III (1990), The Rainmaker (1997), and Youth Without Youth (2007).

(born April 16, 1821, Calais, Fr.—died Oct. 6, 1893, London, Eng.) British painter. He studied in Bruges, Antwerp, Paris, and Rome. In Italy (1845) he met Peter von Cornelius, a member of the Nazarenes, who influenced his palette and style. His use of brilliant colour, meticulous handling, and taste for literary subjects had a strong effect on the Pre-Raphaelites, most notably Dante Gabriel Rossetti. His most famous paintings are The Last of England (1852–55), a poignant tribute to emigration, and Work (1852–63), a Victorian social commentary. In 1861 he became a founding member of William Morris's company, for which he designed stained glass and furniture.

The Ford-Fulkerson algorithm (named for L. R. Ford, Jr. and D. R. Fulkerson) computes the maximum flow in a flow network. It was published in 1956. The name "Ford-Fulkerson" is often also used for the Edmonds-Karp algorithm, which is a specialization of Ford-Fulkerson.

The idea behind the algorithm is very simple: As long as there is a path from the source (start node) to the sink (end node), with available capacity on all edges in the path, we send flow along one of these paths. Then we find another path, and so on. A path with available capacity is called an augmenting path.

## Algorithm

Given is a graph $G\left(V,E\right)$, with capacity $c\left(u,v\right)$ and flow $f\left(u,v\right)=0$ for the edge from $u$ to $v$. We want to find the maximum flow from the source $s$ to the sink $t$. After every step in the algorithm the following is maintained:

• $f\left(u,v\right) leq c\left(u,v\right)$. The flow from $u$ to $v$ does not exceed the capacity.
• $f\left(u,v\right) = - f\left(v,u\right)$. Maintain the net flow between $u$ and $v$. If in reality $a$ units are going from $u$ to $v$, and $b$ units from $v$ to $u$, maintain $f\left(u,v\right)=a-b$ and $f\left(v,u\right)=b-a$.
• $sum_v f\left(u,v\right) = 0 Longleftrightarrow f_\left\{in\right\}\left(u\right) = f_\left\{out\right\}\left(u\right)$ for all nodes $u$, except $s$ and $t$. The amount of flow into a node equals the flow out of the node.

This means that the flow through the network is a legal flow after each round in the algorithm. We define the residual network $G_f\left(V,E_f\right)$ to be the network with capacity $c_f\left(u,v\right) = c\left(u,v\right) - f\left(u,v\right)$ and no flow. Notice that it can happen that a flow from $v$ to $u$ is allowed in the residual network, though disallowed in the original network: if $f\left(u,v\right)>0$ and $c\left(v,u\right)=0$ then $c_f\left(v,u\right)>0$.

Algorithm Ford-Fulkerson

Inputs Graph $,G$ with flow capacity $,c$, a source node $,s$, and a sink node $,t$
Output A flow $,f$ from $,s$ to $,t$ which is a maximum
# $f\left(u,v\right) leftarrow 0$ for all edges $,\left(u,v\right)$
# While there is a path $,p$ from $,s$ to $,t$ in $,G_f$, such that $,c_f\left(u,v\right) > 0$ for all edges $\left(u,v\right) in p$:
## Find $,c_f\left(p\right) = min\left\{c_f\left(u,v\right) ;|; \left(u,v\right) in p\right\}$
## For each edge $\left(u,v\right) in p$
### $f\left(u,v\right) leftarrow f\left(u,v\right) + c_f\left(p\right)$ (Send flow along the path)
### $f\left(v,u\right) leftarrow f\left(v,u\right) - c_f\left(p\right)$ (The flow might be "returned" later)

The path in step 2 can be found with for example a breadth-first search or a depth-first search in $G_f\left(V,E_f\right)$. If you use the former, the algorithm is called Edmonds-Karp.

When no more paths in step 2 can be found, $s$ will not be able to reach $t$ in the residual network. If $S$ is the set of nodes reachable by $s$ in the residual network, then the total capacity in the original network of edges from $S$ to the remainder of $V$ is on the one hand equal to the total flow we found from $s$ to $t$, and on the other hand serves as an upper bound for all such flows. This proves that the flow we found is maximal. See also Max-flow Min-cut theorem.

## Complexity

By adding the flow augmenting path to the flow already established in the graph, the maximum flow will be reached when no more flow augmenting paths can be found in the graph. However, there is no certainty that this situation will ever be reached, so the best that can be guaranteed is that the answer will be correct if the algorithm terminates. In the case that the algorithm runs forever, the flow might not even converge towards the maximum flow. However, this situation only occurs with irrational flow values. When the capacities are integers, the runtime of Ford-Fulkerson is bounded by O(E*f), where E is the number of edges in the graph and f is the maximum flow in the graph. This is because each augmenting path can be found in O(E) time and increases the flow by an integer amount which is at least 1.

A variation of the Ford-Fulkerson algorithm with guaranteed termination and a runtime independent of the maximum flow value is the Edmonds-Karp algorithm, which runs in O(VE2) time.

## Example

The following example shows the first steps of Ford-Fulkerson in a flow network with 4 nodes, source A and sink D. This example shows the worst-case behaviour of the algorithm. In each step, only a flow of 1 is sent across the network. If you used breadth-first-search instead, you would only need two steps.

Path Capacity Resulting flow network
Initial flow network
$A,B,C,D$ $min\left(c_f\left(A,B\right), c_f\left(B,C\right), c_f\left(C,D\right)\right)=$
$min\left(c\left(A,B\right)-f\left(A,B\right) ,c\left(B,C\right)-f\left(B,C\right), c\left(C,D\right)-f\left(C,D\right)\right)=$
$min\left(1000-0, 1-0, 1000-0\right)=1$
$A,C,B,D$ $min\left(c_f\left(A,C\right), c_f\left(C,B\right), c_f\left(B,D\right)\right)=$
$min\left(c\left(A,C\right)-f\left(A,C\right), c\left(C,B\right)-f\left(C,B\right), c\left(B,D\right)-f\left(B,D\right)\right)=$
$min\left(1000-0, 0-\left(-1\right), 1000-0\right)=1$
After 1998 more steps …
Final flow network

Notice how flow is "pushed back" from C to B when finding the path A,C,B,D.