Jacob, Max, 1876-1944, French writer and painter, b. Brittany. His dream-inspired verse, plays, novels, and paintings bridged and gave impetus to the symbolist and surrealist schools. His conversion (1914) from Judaism to Roman Catholicism had great impact on his work. Among Jacob's novels are Saint Matorel (1911) and Filibuth; ou La Montre en or (1922); his verse, usually light and ironic, includes Fond de l'eau (1927) and Rivages (1932). Prose and poetry are combined in his Défense de Tartufe (1919) and the play Le Siège de Jérusalem: drame céleste (1912-14). His critical study, Art poétique (1922), had wide influence. One-man shows of Jacob's paintings were held in New York in 1930 and 1938. He died in a Nazi concentration camp.

See study of his paintings by G. Kamber (1971); study of his religious poetry by J. Schneider (1978).

Aub, Max, 1903-72, Spanish author, b. Paris. He was educated in Spain where he lived until 1942, when he emigrated to Mexico. His style combines realism with fantasy. He used the Spanish civil war and its consequences as the theme for his most important work, a trilogy of novels—Campo cerrado [closed field] (1943), Campo de sangre [bloody field] (1945), and Campo abierto [open field] (1951). His other works include Jusep Torres Campalans (1958) and La calle de Valverde [Valverde street] (1961).
Liebermann, Max, 1847-1935, German genre painter and etcher. He went to Paris in 1873, where he was impressed by the Barbizon school of painters. In Holland he was influenced by Frans Hals and Jozef Israëls. His early works were realistic, but beginning about 1890 he developed a style closely related to impressionism. As leader of the Berlin secession group (1898-1910), he was instrumental in bringing French impressionism to Germany, where younger artists were already moving toward expressionism. Liebermann depicted the life of the working classes, landscapes, outdoor group studies, and painted more than 200 portraits. A secular Jew and one of his country's most honored artists, he was president of the Prussian Academy of Arts (1920-32) during the Weimar Republic. In his last year, however, he was forbidden to paint by the Nazis and his works were removed from museums and private collections. His painting The Ropewalk in Edam (1904) is in New York's Metropolitan Museum of Art.

See B. C. Gilbert, ed., Max Liebermann: From Realism to Impressionism (2005).

Eastman, Max, 1883-1969, American author, b. Canandaigua, N.Y., grad. Williams, 1905. For many years a Communist and a leader of American liberal thought, he edited the left-wing periodicals The Masses (1913-17) and the Liberator (1918-23). His eventual disillusionment with Communism is reflected in such works as Marxism, Is It Science? (1940), Stalin's Russia (1940), and Reflections on the Failure of Socialism (1955). His other works include Enjoyment of Poetry (1913), his most popular work; Enjoyment of Laughter (1936); and Poems of Five Decades (1954). Among his autobiographical works is Love and Revolution (1965).
Brod, Max, 1884-1968, Israeli writer and composer, b. Prague. Brod is best known for his historical novels, written in German, notably The Redemption of Tycho Brahe (1916, tr. 1928) and Reubeni, Prince of the Jews (1925, tr. 1928). A lifelong friend of Franz Kafka, he wrote an excellent biography of Kafka (1937, tr. 1947) and also edited Kafka's writings. Brod's numerous other works include a biography of Heine (1934, tr. 1956), an autobiography (1960), and plays, poems, novels, and essays. His musical compositions include works for orchestra, notably Requiem Hebraicum, and for voice and piano. Long an active Zionist, Brod left Prague for Palestine in 1939 where he directed the Habima Theater.
Bruch, Max, 1838-1920, German composer. He conducted the Liverpool Philharmonic Orchestra (1880-83) and taught at the Berlin Hochschule (1892-1910). His Violin Concerto in G Minor (1868) and his variations on the Kol Nidre (1881) for cello and orchestra are his best-known compositions. Bruch also wrote three symphonies.
Planck, Max, 1858-1947, German physicist. Seeking to explain the experimental spectrum (distribution of electromagnetic energy according to wavelength) of black body radiation, he introduced the hypothesis (1900) that oscillating atoms absorb and emit energy only in discrete bundles (called quanta) instead of continuously, as assumed in classical physics. The success of his work and subsequent developments by Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, and others established the revolutionary quantum theory of modern physics, of which Planck is justly regarded as the father. In 1918, Planck received the Nobel Prize in physics for his work on black body radiation. He was professor at the Univ. of Berlin (1889-1928) and president (1930-35) of the Kaiser Wilhelm Society for the Advancement of Science, Berlin, which after World War II was reconstituted as part of the Max Planck Institutes. He was an editor of the Annalen der Physik and member of the Royal Society (London) and the American Physical Society. His name is honored in Planck's constant. English translations of his works include A Survey of Physics (1925, new ed. 1960), Introduction to Theoretical Physics (5 vol., 1932-33), Treatise on Thermodynamics (3d rev. ed. 1945), and Scientific Autobiography and Other Papers (1949).
Born, Max, 1882-1970, British physicist, b. Germany, Ph.D. Univ. of Göttingen, 1907. He was head of the physics department at the Univ. of Göttingen from 1921 to 1933. When Nazi policies forced him to leave Germany, he went to England; he was a lecturer at Cambridge, then became (1936) a professor of natural philosophy at the Univ. of Edinburgh. Born was made a British citizen in 1939. In 1953 he retired to West Germany. Known for his research in quantum mechanics, he shared the 1954 Nobel Prize in Physics with Walter Bothe. Born's writings include Problems of Atomic Dynamics (1926, tr. 1960).

See his autobiography, My Life and My Views (1968).

Hoffmann, Max, 1869-1927, German general in World War I. A brilliant strategist, he contributed to the German victory over the Russians at Tannenberg and in 1916 became chief of staff of the eastern armies. As military representative he helped negotiate the Treaty of Brest-Litovsk with Russia. In his war diary and later books he bitterly criticized the German high command.
Dessoir, Max, 1867-1947, German philosopher. He earned doctorates from the universities of Berlin (philosophy, 1889) and Würtzburg (medicine, 1892). He was a professor at Berlin from 1897 until 1933, when the Nazis forbade him to teach. He worked mainly in the area of aesthetics, trying to foster a general science of great art. Dessoir understood an aesthetic object to be one occurring either in nature or in art, the parts of which are related to each other with an intensity beyond that of normal experience. He defined five primary forms of aesthetic experience: beauty, ugliness, comedy, tragedy, and the sublime. He saw the role of art as moral an social and regarded "Art for art's sake" as a futile and fatuous maxim. Dessoir was also interested in parapsychology. Among his few works translated into English are Outlines of the History of Psychology (tr. 1912) and Aesthetics and Theory of Art (tr. 1970).
Abramovitz, Max: see Harrison, Wallace Kirkman.
Frisch, Max, 1911-91, Swiss writer. He obtained a diploma in architecture in 1941, and his designs included the Zürich Recreation Park. After 1955 he became recognized as one of Europe's major literary voices. In the novels Stiller (1954, tr. I'm Not Stiller, 1958), Homo faber (1957, tr. 1959), and Mein Name sei Gantenbein (1964, tr. A Wilderness of Mirrors, 1965), Frisch was essentially concerned with the human search for personal identity. His best-known plays are Biedermann und die Brandstifter (1953, tr. The Firebugs, 1963), and Andorra (1961, tr. 1962), a study of mass psychology.

See his autobiographical Montauk (tr. 1976), his Sketchbooks (1974, 1977); studies by M. Butler (1983) and W. Koepke (1990); biographies by U. W. Weisstein (1967) and C. Petersen (tr. 1972).

Reger, Max, 1873-1916, German composer; he studied with Hugo Riemann in Wiesbaden. Through his sensitive interpretations of Mozart and Bach he won acclaim as a pianist. In 1901 he settled in Munich, where he taught composition and organ, and from 1907 until his death he taught at the Leipzig Conservatory. In 1911 he became conductor of the court orchestra at Meiningen. He was highly esteemed in Germany for his organ music, which exhibits extreme polyphonic complexity and a consummate technique. Among his important compositions for the organ are Fantasy and Fugue in C Minor (1898) and Fantasy and Fugue on Bach (1900). His enormous output also includes Improvisation (1898), for pianos; the Symphonic Prologue to a Tragedy, for orchestra; and more than 300 songs.
Reinhardt, Max, 1873-1943, Austrian theatrical producer and director, originally named Max Goldmann. After acting under Otto Brahm at the Deutsches Theater in Berlin, he managed (1902-5) his own theater, where he produced more than 50 plays. He was director of the Deutsches Theater after 1905 and of the smaller Kammerspiele, which he built in 1906. Reinhardt often used the entire auditorium for a production, seeking to bridge the gap between actor and audience by placing the spectator within the action. He staged gigantic productions, full of pageantry and color, and was especially noted for his direction of mob scenes. His settings, which incorporated the ideas of Appia and Craig, were masterfully executed. Among his world-famous productions were The Lower Depths, A Midsummer Night's Dream, Faust, Oedipus Rex, and The Miracle. He was also one of the first to stage the plays of the expressionists after World War I. In 1919 he opened an enormous arena theater, the Grosses Schauspielhaus ("Theatre of the Five Thousand"), and in 1920 he was among the founders of the Salzburg Festival, where he annually staged Everyman with the Austrian Alps as his backdrop. In 1933 he was forced by the Nazis to flee Germany. In the United States he directed a movie version of A Midsummer Night's Dream (1935) and a stage pageant with music by Kurt Weill, The Eternal Road (1934, produced 1937). He became a U.S. citizen in 1940.

See H. Carter, The Theatre of Max Reinhardt (1914, repr. 1964); J. L. Styan, Max Reinhardt (1982).

Pechstein, Max, 1881-1955, German expressionist painter and graphic artist. Early contact with the art of Van Gogh stimulated his development toward expressionism. In 1906, Pechstein joined the Brücke group. His work, less intense and more decorative than that of the other members of the group, gained wide popularity. In 1914 he went to the South Seas, a trip which inspired his Palau Triptych and other works.
Roach, Max (Maxwell Lemuel Roach), 1924-2007, African-American jazz drummer, b. Newland, N.C. Raised in Brooklyn, N.Y., he was playing jazz in Harlem clubs by 1943. Roach had an important role in the genesis of bop (see jazz), providing jagged, layered rhythms to groups led by Dizzy Gillespie (1944) and Charlie Parker (1945-53), and elevating drums to the status of solo instruments. An innovative virtuoso who mingled power with subtlety, Roach became (1954) co-leader with trumpeter Clifford Brown of a hard-bop jazz quintet that also included Sonny Rollins. After Brown's death (1956), Roach led a variety of jazz small groups, and in the early 1960s he was an early public jazz champion of racial equality, particularly in his We Insist! Freedom Now Suite (1960). He founded M'Boom, an all-percussion group, in the 1970s and the Max Roach Double Quartet, in which strings played an important part, in the 80s, and later led the So What Brass Quintet. Roach also composed music for the theater and for dances by Alvin Ailey.
Scheler, Max, 1874-1928, German philosopher. He taught at the universities of Jena (1901-7) and Munich (1907-10), where he was influenced by Franz Brentano and the followers of Edmund Husserl. From 1910 he concentrated on writing, but he returned to university teaching at Cologne and Frankfurt after World War I. Scheler was concerned with the permanent values in human personality and human action; this concern brought him to important work in phenomenology, which spread beyond Germany, chiefly through his influence. In his early thought, for which he is best known, Scheler taught that love is the great principle of human association, and he regarded God as the source of all love. His most basic work is Formalism in Ethics and Non-Formal Ethics of Values (2 vol., 1913-16; tr. 1973); other important works include On the Eternal in Man (1921; tr. 1960) and Man's Place in Nature (1928; tr. 1961).

See his Selected Philosophical Essays, tr. with an introd. by D. R. Lachterman (1973); biography by J. R. Staude (1967); studies by E. W. Ranly (1966), A. R. Luther (1972), and A. Deeken (1974), and J. H. Nota (1983).

Schmeling, Max (Maximilian Schmeling), 1905-2005, German boxer. He debuted as a professional fighter in 1924 and came to the United States in 1928. Two years later the methodical slugger beat heavyweight champion Jack Sharkey (by a foul) to become Europe's first world champ. He lost the title to James J. Braddock in 1932. In his greatest upset, a 1936 bout, Schmeling knocked out Joe Louis, then an unbeaten 22-year-old contender, and was lauded in Hitler's Germany as an Aryan idol, though he was neither political nor a racist. When they met again in a hugely hyped 1938 match, Louis, by then world's champion, was hailed as America's conquering hero. Louis knocked out Schmeling in the first round after 124 seconds. Schmeling had a career total of 70 fights, 56 of which he won, 40 by knockout. He returned to Germany, where after World War II he became a successful businessman and philanthropist.

See his autobiography (1977, tr. 1998); L. A. Erenberg, The Greatest Fight of Our Generation: Louis vs. Schmeling (2005); D. Margolick, Beyond Glory: Joe Louis vs. Max Schmeling, and a World on the Brink (2005).

Beckmann, Max, 1884-1950, German painter. A member of the Berlin secession from 1908 to 1911, he was impressionistic in his early style. A subsequent expressionistic phase was altered c.1917 by the savage new objectivity of George Grosz. Beckmann developed a richer, more personal, more dramatic, and more symbolic art in the 1920s. The power of his allegorical expressionism increased through the war years, which, after fleeing Nazi Germany in 1937, he spent in Amsterdam. Beckmann lived his last three years in New York City, where he taught at the Brooklyn Museum School. His well-known triptych, Departure (1932-35; Mus. of Modern Art, N.Y.C.) is one of 18 powerfully monumental triptychs that culminated in The Argonauts (1950).
Horkheimer, Max, 1895-1973, German philosopher and sociologist. As director (1930-58) of the Institute for Social Research in Frankfurt, he played an important role in the development of critical theory and Western Marxism. In Eclipse of Reason (1947) and Dialectic of Enlightenment (1947, written with Theodor Adorno), he developed a critique of scientific positivism, whose "instrumental rationality" had become a form of domination in both capitalist and socialist countries. Against an older, deterministic Marxism, he argued that culture and consciousness are partly independent of economics, and his ideas about liberation and consumer society continue to influence contemporary empirical sociologists.
Klinger, Max, 1857-1920, German painter, sculptor, and etcher. Before 1886 he produced cycles of original and somewhat morbidly imaginative etchings, such as Deliverances of Sacrificial Victims Told in Ovid and Brahms-Phantasie. From 1886 to 1894 Klinger devoted himself primarily to painting, usually on a grandiose scale. Among his paintings are Judgment of Paris and Christ on Olympus (both: Vienna). After 1894 he worked predominantly in sculpture, his most successful medium. Notable examples are Salome, Cassandra, and the dramatic polychromed statue of Beethoven (all: Leipzig) and the bust of Nietzsche (Weimar).
Ophüls, Max, 1902-57, German-born French film director, b. Saarbrücken as Maximilian Oppenheimer. He started his career in the 1920s as an stage actor and director and began directing films in Berlin during the early 1930s. His early works include Liebelei (1933), made in Austria, and La Signora di Tutti (1933), filmed in Italy. A Jew, he fled Nazi Germany for France (1933), became a French citizen (1938), and after the fall of France settled in California (1941). There Ophüls made four now-classic Hollywood films: The Exile (1947), Letter from an Unknown Woman (1948), Caught (1949), and The Reckless Moment (1949). In 1949 he returned to France, where he directed his final and finest films: La Ronde (1950), Le Plaisir (1951), Madame de… (1953), and Lola Montès (1955). Typically, his films are sophisticated and memory-laden tales of love-obsessed women and the complexities of romance gone awry, portrayed with a fluid gliding camera technique. His son, Marcel Ophüls, 1929-, also a French film director, is known for his searing documentaries. In his most acclaimed film, The Sorrow and the Pity (1970), he explored the World War II collaboration of French citizens and their complicity in the Holocaust. His other films include A Sense of Loss (1972), about Ireland's political conflicts; Hotel Terminus (1987; Academy Award), a look at the life of Nazi Klaus Barbie; and The Troubles We've Seen (1994), focusing on wartime journalism.

See A. L. Williams, Max Ophüls and the Cinema of Desire (1977, repr. 1980, 1992); S. M. White, The Cinema of Max Ophüls (1995); L. Bacher, Max Ophüls in the Hollywood Studios (1996).

Theiler, Max, 1899-1972, South African-American research physician, b. Pretoria, educated at the Univ. of Cape Town, St. Thomas's Hospital (London), and the London School of Tropical Medicine. Theiler's research on yellow fever, begun while he was connected with the department of tropical medicine of Harvard Medical School (1922-30), was continued at the Rockefeller Foundation, of which he became a staff member in 1930. He became known for his researches on yellow fever, encephalomyelitis, and other viruses associated with the tropics. For his work in developing a vaccine for yellow fever he was awarded the 1951 Nobel Prize in Physiology or Medicine.
Max, Gabriel, 1840-1915, German painter and illustrator, b. Prague; son and pupil of the sculptor Josef Max (1803-54). A student of psychology and anthropology, Gabriel Max is best known as a painter of mystical subjects. Characteristic of his ethereal style is The Last Token (Metropolitan Mus.).
Max, Peter, 1937-, American artist, b. Berlin. Max is noted for his undulating graphic designs in bright, vibrating colors. His style has influenced much commercial art. It is reminiscent of art nouveau and comic strip art, incorporating psychedelic colors in floral and celestial motifs.
Black, Max, 1909-88, American analytical philosopher, b. Baku, Russia (now Bakı, Azerbaijan), grad. Cambridge, Ph.D. Univ. of London, 1939. He taught at the Univ. of Illinois (1940-46) before going to Cornell (1946). Influenced by Ludwig Wittgenstein, he wrote A Companion to Wittgenstein's Tractatus (1964). His concern with clear language was expressed in Language and Philosophy (1949), Models and Metaphors (1962), The Labyrinth of Language (1968), and Margins of Precision: Essays in Logic and Language (1970).
Wertheimer, Max, 1880-1943, German psychologist, b. Prague. He studied at the universities of Prague, Berlin, and Würzburg (Ph.D., 1904). His original researches, while he was a professor at Frankfurt and Berlin, placed him in the forefront of contemporary psychology. Wertheimer came to the United States in 1933, shortly before the Nazis seized power in Germany. He immediately joined the graduate faculty of the New School for Social Research (1933-43). Wertheimer's discovery (1910-12) of the phi phenomenon (concerning the illusion of motion) gave rise to the influential school of Gestalt psychology. His early experiments, in collaboration with Wolfgang Köhler and Kurt Koffka, introduced a new approach (macroscopic as opposed to microscopic) to the study of psychological problems. In the latter part of his life he directed much of his attention to the problem of learning; this research resulted in a book, posthumously published, called Productive Thinking (1945, repr. 1978).
Ernst, Max 1891-1976, German painter. After World War I, Ernst joined the Dada movement in Paris and then became a founder of surrealism. Apart from the medium of collage, for which he is well known, Ernst developed other devices to express his fantastic vision. In frottage he rubbed black chalk on paper held against various materials such as leaves, wood, and fabrics to achieve bizarre effects. He was also the author of several volumes of collage novels. A note of whimsy often characterizes his dreamlike landscapes while other works reveal an allegorical imagination. Two Children Are Threatened by a Nightingale and several other works are in the Museum of Modern Art, New York City.

See his Beyond Painting (1948); studies by J. Russell (1967) and U. M. Schneede (1973); R. Rainwater, Max Ernst, Beyond Surrealism: An Exhibition of the Artist's Books and Prints (1986); W. A. Camfield, ed., Max Ernst: Dada and the Dawn of Surrealism (1993); W. Spies, ed., Max Ernst: A Retrospective (2005).

Slevogt, Max, 1868-1932, German painter. Slevogt, together with Max Liebermann and Lovis Corinth, was among the principal exponents of German impressionism and was influenced by Millet and Courbet. A prolific painter, he attempted to capture movement through broad, informal brush work. His portrait of the singer Francisco d'Andrade as Don Giovanni (1902) is in the Staatsgalerie, Stuttgart.
Müller, Max (Friedrich Maximilian Müller, Friedrich Max Müller, or Friedrich Max-Müller), 1823-1900, German philologist and Orientalist, b. Dessau; son of the poet Wilhelm Müller. After specializing in Sanskrit in Germany, he went to Oxford, where he lived for the remainder of his life. Müller did more than any other scholar to popularize philology and mythology, particularly in his lectures Science of Language (1861, 1863). He advanced the theory that myths originated from metaphors describing natural pnenomena. Greatly interested in comparative religion, he wrote works on Indian religion and philosophy, including the standard edition of the Rig-Veda with Commentary (6 vol., 1849-73). From c.1875 until his death Müller was engaged in his greatest work, the editing of Sacred Books of the East (51 vol.), being translations of important Asian religious writings.

See his memoirs (tr. 1906); studies by J. H. Voigt (1967) and R. Neufeldt (1980).

Weber, Max, 1864-1920, German sociologist, economist, and political scientist. At various times he taught at Berlin, Freiburg, Munich, and Heidelberg. One of Weber's chief interests was in developing a methodology for social science, and his works had a considerable influence on 20th-century social scientists. As a technique of sociological analysis, he devised the concept of "ideal types," generalized models of historical situations that could be used as a basis for comparing societies. He opposed the orthodox Marxian view of the time that economics was the preeminent determining factor in social causation and instead stressed the plurality and interdependence of causes. Weber emphasized the role of religious values, ideologies, and charismatic leaders in shaping societies. In his Protestant Ethic and the Spirit of Capitalism (1920, tr. 1930) he developed a thesis concerning the intimate connection between the ascetic ideal fostered by Calvinism and the rise of capitalist institutions. A keen observer of politics in his own time, he first admired, then repudiated Otto von Bismarck, and he later advocated for Germany a democratic form of government somewhat on the American model. He has also been influential in using statistical sociology in the study of economic policy. Among his other books are Wirtschaft und Gesellschaft [economy and society] (4th ed. 1956) and General Economic History (1924, tr. 1927).

See From Max Weber: Essays in Sociology (with a biography and appraisal by H. H. Gerth and C. Wright Mills, 1946); studies by J. Freund (1968), A. Mitzman (1969), W. G. Runciman (1972), D. Beetham (1974), W. J. Mommsen (1974), G. Roth (1979), and J. Alexander (1983).

Weber, Max, 1881-1961, American painter, b. Russia. At 10 he accompanied his family to Brooklyn, N.Y. He studied art at Pratt Institute and in 1905 went abroad. In Paris he studied under J. P. Laurens, later visiting Spain and Italy and returning to New York in 1909. Weber's work in the following decade was fauvist and then cubist inspired. Characteristic of the latter trend is his well-known Chinese Restaurant (Whitney Mus., New York City). He began to introduce Jewish subjects into his work c.1917. During the 1920s, Weber alternated painting with teaching. Contemporary and social themes were his subjects in the 1930s, when his work became increasingly abstract and revealed a new energetic use of line. Weber is represented in leading galleries throughout the United States. He wrote several essays on art theory.

See study by L. Goodrich (1949).

The max-flow min-cut theorem is a statement in optimization theory about maximum flows in flow networks. It derives from Menger's theorem. It states that:
The maximum amount of flow is equal to the capacity of a minimal cut.

In other words, the theorem states that the maximum flow in a network is dictated by its bottleneck. Between any two nodes, the quantity of material flowing from one to the other cannot be greater than the weakest set of links somewhere between the two nodes.


Suppose G(V,E) is a finite directed graph and every edge (u,v) has a capacity c(u,v) (a non-negative real number). Further assume two vertices, the source s and the sink t, have been distinguished.

A cut is a split of the nodes into two disjoint sets S and T, such that s is in S and t is in T. Hence there are


possible cuts in a graph. The capacity of a cut (S,T) is

c(S,T) = sum_{u in S, v in T | (u,v) in E} c(u,v),

the sum of the capacity of all the edges crossing the cut, from the region S to the region T.

The following three conditions are equivalent:

  1. f is a maximum flow in G
  2. The residual network G_f contains no augmenting paths.
  3. |f| = c(S,T) for some cut (S,T).

Proof sketch: If there is an augmenting path, we can send flow along it, and get a greater flow, hence it cannot be maximal. If there is no augmenting path, divide the graph into S, the nodes reachable from s in the residual network, and T, those not reachable. Then c(S,T) in the residual network must be 0. If it is not, there is an edge (u,v) with c(u,v) > 0. But then v is reachable from s, and cannot be in T.

In particular this proves that max flow ge min cut, since max flow ge |f| = c(S,T) ge min cut.

We also have max flow le min cut because given any flow, f, and any cut, (S,T), we can sum the equations of conservation of flow at every point of S setminus {s} to show that |f| = the flow of f along edges joining S and T which is no more than c(S,T).

Linear Program

The max-flow min-cut problem can be easily expressed as a pair of primal-dual linear programs. The following linear program computes the max flow between nodes S and T in graph G=(V, E), where V is the set of nodes and E is the set of edges. The capacity of any edge overrightarrow{uv}in E is represented by c(overrightarrow{uv}). The linear program assumes the graph to be directed. The linear program for the undirected case can be easily derived from the same.

Primal (Max flow linear program)



Subject To

sum_{overrightarrow{vu}in E} f(overrightarrow{vu}) - sum_{overrightarrow{uv}in E} f(overrightarrow{uv}) = 0 qquad forall u in Vneq {S, T} qquad leftrightarrow p(u)

sum_{overrightarrow{vT}in E} f(overrightarrow{vT}) - sum_{overrightarrow{Tv}in E} f(overrightarrow{Tv}) - f(overrightarrow{TS}) = 0 qquad leftrightarrow p(T)

sum_{overrightarrow{vS}in E} f(overrightarrow{vS}) + f(overrightarrow{TS}) - sum_{overrightarrow{Sv}in E} f(overrightarrow{Sv}) = 0 qquad leftrightarrow p(S)

f(overrightarrow{uv}) leq c(overrightarrow{uv}) qquad forall overrightarrow{uv}in E qquad leftrightarrow x(overrightarrow{uv})


f(overrightarrow{uv}) geq 0 qquad forall overrightarrow{uv} in E, quad f(overrightarrow{TS}) geq 0

Primal Explanation: The linear program computes the max flow in the variable f(overrightarrow{TS}). overrightarrow{TS} is a conceptual link. As there is flow conservation (flow in equals flow out) at every node, the more flow that is on this conceptual link, the more will be the flow between the source and the destination. The first constraint specifies the flow conservation at every node except the source and the destination. The second and third constraints are special flow conservation constraints for the destination and the source respectively. The final constraint limits the maximum flow on the edge to its capacity. The variable shown right at the end of every constraint is the corresponding dual variable.
An interesting fact about the above linear program is that the constraint matrix is totally unimodular. This means that, if all the inputs are integral, there exists as an integral solution. So, if all the capacities in the graph are integral, the max flow is integral.

Dual (Min cut linear program)

mathbf{sum_{overrightarrow{uv} in E} c(overrightarrow{uv}) . x(overrightarrow{uv})}

Subject To

p(v) - p(u) + x(overrightarrow{uv}) geq 0 qquad forall overrightarrow{uv} in E qquad leftrightarrow f(overrightarrow{uv})

p(S) - p(T) geq 1 qquad leftrightarrow f(overrightarrow{TS})


p(u) free quad forall u, x(overrightarrow{uv}) geq 0 quad forall overrightarrow{uv}

Dual explanation: The dual can be written mechanically from the primal based on the duality theorem in linear programming. In the dual, the variable x(overrightarrow{uv}) decides whether an edge is a cut edge are not. The edges for which x(overrightarrow{uv})>0 are part of the min cut and the edges for which it is 0 are not part of the cut edge. The nodes in the graph that are along with the node S after the cut can be considered to be one partition and the nodes in the graph that are with the node T can be considered to be another partition. The constraints represent the relationship between the nodes in different sets. To illustrate this, consider an example where the solution is such that the value of p for the nodes in the partition containing S is 0 and 1 otherwise. It is now obvious to see that the value of x for the cut edges is 1 and for the rest it is 0. This will also help in understanding the meaning of the constraints. Finally, as in the primal, the variable shown right at the end of every constraint is the corresponding primal variable.
One of the interesting relationships between the primal and dual is that the objective value of optimal solutions for the primal and the dual are exactly same. Thus, since the objective value of the primal is integral, the dual will also have an integral objective value. Thus, if all the input capacities are integral, the min cut is integral.


Given to the right is a network with nodes V={s,o,p,q,r,t}, and a total flow from the source s to the sink t of 5, which is maximal in this network. (Incidentally, this is the only maximal flow you can assign to this network.)

There are three minimal cuts in this network:

Cut Capacity
S={s,p},T={o,q,r,t} c(s,o)+c(p,r)=3+2=5
S={s,o,p},T={q,r,t} c(o,q)+c(p,r)=3+2=5
S={s,o,p,q,r},T={t} c(q,t)+c(r,t)=2+3=5

Notice that S={s,o,p,r},T={q,t} is not a minimal cut, even though both (o,q) and (r,t) are saturated in the given flow. This is because in the residual network G_f, there is an edge (r,q) with capacity c_f(r,q) = c(r,q)-f(r,q)=0-(-1)=1.


The theorem was proved by P. Elias, A. Feinstein, and C.E. Shannon in 1956, and independently also by L.R. Ford, Jr. and D.R. Fulkerson in the same year. Determining maximum flows is a special kind of linear programming problem, and the max flow min cut theorem can be seen as a special case of the duality theorem for linear programming.

See also


External links

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