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Equivalence of optimization problems
science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink
Max-flow_min-cut_theorem
Mathematical propositions in network flow theory
approximate max-flow min-cut theorems concern the relationship between the maximum flow rate (max-flow) and the minimum cut (min-cut) in multi-commodity flow problems
Approximate max-flow min-cut theorem
Approximate_max-flow_min-cut_theorem
Computational problem in graph theory
capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem. The maximum flow problem was first formulated
Maximum_flow_problem
Optimization technique
approximated by solving a maximum flow problem in a graph (and thus, by the max-flow min-cut theorem, define a minimal cut of the graph). Under most formulations
Graph cuts in computer vision and artificial intelligence
Graph_cuts_in_computer_vision_and_artificial_intelligence
Theorem in graph theory
generalized by the max-flow min-cut theorem, which is a weighted, edge version, and which in turn is a special case of the strong duality theorem for linear programs
Menger's_theorem
Partition of a graph by removing fewest possible edges
side of the cut to the sink side of the cut. As shown in the max-flow min-cut theorem, the weight of this cut equals the maximum amount of flow that can
Minimum_cut
Partition of a graph's nodes into 2 disjoint subsets
shows a minimum cut: the size of this cut is 2, and there is no cut of size 1 because the graph is bridgeless. The max-flow min-cut theorem proves that the
Cut_(graph_theory)
On bipartite matching and vertex cover
G'_{\infty }} , as follows from the max-flow min-cut theorem. Let ( S , T ) {\displaystyle (S,T)} be a minimum cut. Let A = A S ∪ A T {\displaystyle A=A_{S}\cup
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
Class of computational problems
flow, a type of flow studied in combinatorics in which the flow amounts are restricted to a finite set of nonzero values The max-flow min-cut theorem
Network_flow_problem
Basic concept of graph theory
v) equals κ′(u, v). This fact is actually a special case of the max-flow min-cut theorem. The problem of determining whether two vertices in a graph are
Connectivity_(graph_theory)
American mathematician (1927–2017)
1954 and in a journal in 1956, established the max-flow min-cut theorem. In 1962 they published Flows in Networks with Princeton University Press. According
L._R._Ford_Jr.
Combinatorial optimization method for a family of functions of discrete variables
in the theory of flow networks. Thanks to the max-flow min-cut theorem, determining the minimum cut over a graph representing a flow network is equivalent
Graph_cut_optimization
mathematics. This page is a list of network theory topics. Max flow min cut theorem Menger's theorem Metcalfe's law Centrality Betweenness centrality Closeness
List_of_network_theory_topics
Arrival theorem (queueing theory) Blum's speedup theorem (computational complexity theory) Max flow min cut theorem (graph theory) No free lunch theorem (philosophy
List_of_theorems
Directed graph where edges have a capacity
flow (computer networking) Flow graph (disambiguation) Max-flow min-cut theorem Oriented matroid Shortest path problem Nowhere-zero flow Active flow network
Flow_network
Result in combinatorics and graph theory
Jenő Egerváry) König's theorem Menger's theorem (1927) The max-flow min-cut theorem (Ford–Fulkerson algorithm) Dilworth's theorem. In particular, there
Hall's_marriage_theorem
Randomized algorithm for minimum cuts
{\displaystyle s} - t {\displaystyle t} cut problem using the max-flow min-cut theorem and a polynomial time algorithm for maximum flow, such as the push-relabel algorithm
Karger's_algorithm
Pattern of motion in a visual scene due to relative motion of the observer
values without linearising it. This search is often performed using Max-flow min-cut theorem algorithms, linear programming or belief propagation methods. These
Optical_flow
Area of discrete mathematics
applications that have to do with various notions of flows in networks, for example: Max flow min cut theorem Museum guard problem Covering problems in graphs
Graph_theory
American computer scientist and educator
problems involving graphs, cuts, flows, and embeddings. With Leighton, he developed multicommodity max-flow min-cut theorems and applied them to approximation
Satish_B._Rao
Principle in mathematical optimization
Combinatorial Implications of Max-Flow Min-Cut Theorem, 4.6. Linear Programming Interpretation of Max-Flow Min-Cut Theorem". Combinatorial Optimization:
Duality_(optimization)
Mathematical optimization concept
}\end{matrix}}} The max-flow min-cut theorem is a special case of the strong duality theorem: flow-maximization is the primal LP, and cut-minimization is
Dual_linear_program
analogues of the max-flow min-cut theorem for undirected multi-commodity flow problems. The ratio of the maximum flow to the minimum cut, in such problems
GNRS_conjecture
Theorem in geometric topology
any solution of the Ricci flow with surgery becomes extinct in finite time. An alternative argument, based on the min-max theory of minimal surfaces
Poincaré_conjecture
Graph which remains connected when fewer than k edges are removed
removal of few edges can be proven using the max-flow min-cut theorem from the theory of network flows. Minimum vertex degree gives a trivial upper bound
Edge_connectivity
Algorithm to compute the maximum flow in a network
flows. This proves that the flow we found is maximal. See also Max-flow Min-cut theorem. If the graph G ( V , E ) {\displaystyle G(V,E)} has multiple sources
Ford–Fulkerson_algorithm
British mathematician
1975. His doctoral dissertation, Matroids, Hypergraphs and the Max-Flow Min-Cut Theorem, was supervised by Aubrey William Ingleton. From 1974 to 1976 he
Paul_Seymour_(mathematician)
Flood fill Graph exploration algorithm Matching (graph theory) Max flow min cut theorem Maximum-cardinality search Shortest path Dijkstra's algorithm Bellman–Ford
List_of_graph_theory_topics
Computational problem in graph theory
maximized. By the max-flow min-cut theorem, a minimum cut, and the optimal closure derived from it, can be found by solving a maximum flow problem. Alternative
Closure_problem
American mathematician (1914–2005)
Systems science portal Dantzig–Wolfe decomposition Knapsack problem Maximum flow problem Optimization (mathematics) Travelling salesman problem Shadow price
George_Dantzig
separate a designated pair of vertices; they are characterized by the max-flow min-cut theorem. minor A graph H is a minor of another graph G if H can be obtained
Glossary_of_graph_theory
Algorithm in mathematical optimization
according to the max-flow min-cut theorem since there is no augmenting path from s to t. Therefore, the algorithm will return the maximum flow upon termination
Push–relabel maximum flow algorithm
Push–relabel_maximum_flow_algorithm
Analysis in fluid dynamics
equation Max-flow min-cut theorem Network theory S.H. Waldrip, R.K. Niven, M. Abel, M. Schlegel (2016), Maximum entropy analysis of hydraulic pipe flow networks
Pipe_network_analysis
Computer Networking Program
{\displaystyle t} . By the max-flow min-cut theorem, T ( s , t ) {\displaystyle T(s,t)} is upper bounded by the minimum capacity of all cuts, which is the sum
Linear_network_coding
Abstraction of ordered linear algebra
Many results—Carathéodory's theorem, Helly's theorem, Radon's theorem, the Hahn–Banach theorem, the Krein–Milman theorem, the lemma of Farkas—can be formulated
Oriented_matroid
Award for advancements in discrete mathematics
Appel and Wolfgang Haken for the four color theorem. Paul Seymour for generalizing the max-flow min-cut theorem to matroids. 1982: D.B. Judin, Arkadi Nemirovski
Fulkerson_Prize
Property in graph theory
MR 3195329. Leighton, Tom; Rao, Satish (1999). "Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms". Journal
Cutwidth
can be used to show the extension of some results, such as the max-flow min-cut theorem, to infinite graphs. In topology, the Grothendieck topos of right
Extended_natural_numbers
Communications technology technique
subsequently. Using the max-flow min-cut theorem yields the upper bound of full duplex relaying C + = max f ( X 1 , X 2 ) min { I ( X 1 ; Y 2 , Y 3 |
Cooperative_diversity
Directed graph isomorphic to its own transpose graph
problems arising in matchings, skew-symmetric generalizations of the max-flow min-cut theorem have also been studied. Cook (2003) shows that a still life pattern
Skew-symmetric_graph
Method for mathematical optimization
criss-cross algorithm was published independently by Tamas Terlaky and by Zhe-Min Wang; related algorithms appeared in unpublished reports by other authors
Criss-cross_algorithm
Weighted tree representing s-t cuts of a graph
(graph theory) Max-flow min-cut theorem Maximum flow problem Gomory, R. E.; Hu, T. C. (1961). "Multi-terminal network flows". Journal of the Society
Gomory–Hu_tree
Vertex partition in a directed graph
ISBN 978-3-540-08666-6, MR 0499529 Edmonds, Jack; Giles, Rick (1977), "A min-max relation for submodular functions on graphs", Studies in integer programming
Dicut
Subfield of convex optimization
quadratic program. For max cut, the most natural relaxation is max ∑ ( i , j ) ∈ E 1 − ⟨ v i , v j ⟩ 2 , {\displaystyle \max \sum _{(i,j)\in E}{\frac
Semidefinite_programming
Mathematical algorithm for eliminating variables from a system of linear inequalities
equivalent to max i = 1 n A ( A i ( x ′ ) ) ≤ x r ≤ min j = 1 n B ( B j ( x ′ ) ) ∧ ϕ {\displaystyle \max _{i=1}^{n_{A}}(A_{i}(x'))\leq x_{r}\leq \min
Fourier–Motzkin_elimination
nowhere-zero flows", Combinatorica, 45 (3) 32: 1–23, doi:10.1007/s00493-025-00159-x, MR 4915164 Edmonds, Jack; Giles, Rick (1977), "A min-max relation for
Woodall's_conjecture
Type of algorithm for constrained optimization
set X*. This theorem is helpful mostly when fp is convex, since in this case, we can find the global optimizers of fp. A second theorem considers local
Penalty_method
Iterative method for minimizing convex functions
satisfying : f ( x ) − min G f ≤ ε ⋅ [ max G f − min G f ] {\displaystyle f(x)-\min _{G}f\leq \varepsilon \cdot [\max _{G}f-\min _{G}f]} , using at most
Ellipsoid_method
Distance function defined between probability distributions
{\displaystyle c(x,y)\leq a'(x)+b'(y)} , then min γ ∈ Γ ( μ , ν ) ( ∫ X × Y c ( x , y ) d γ ( x , y ) ) = max φ is c -convex ( ∫ X φ ( x ) d μ ( x ) +
Wasserstein_metric
Graph representing faces of another graph
with k colors correspond to nowhere-zero flows modulo k on the dual graph. For instance, the four color theorem (the existence of a 4-coloring for every
Dual_graph
Edges crossing all dicuts in a directed graph
ISBN 978-3-540-08666-6, MR 0499529 Edmonds, Jack; Giles, Rick (1977), "A min-max relation for submodular functions on graphs", Studies in integer programming
Dijoin
Study of mathematical algorithms for optimization problems
and {−5, (2k + 1)π}, where k ranges over all integers. Operators arg min and arg max are sometimes also written as argmin and argmax, and stand for argument
Mathematical_optimization
Class of algorithms that find approximate solutions to optimization problems
S(i)} , c ( s ∗ ) = m i n / m a x c ( S ( i ) ) {\displaystyle c(s^{*})=min/max\ c(S(i))} Given a feasible solution s ∈ S ( i ) {\displaystyle s\in S(i)}
Approximation_algorithm
Inverse of the average of the inverses of a set of numbers
min ( x 1 , … , x n ) ≤ H ( x 1 , … , x n ) ≤ n min ( x 1 , … , x n ) {\displaystyle \min(x_{1},\ldots ,x_{n})\leq H(x_{1},\ldots ,x_{n})\leq n\min(x_{1}
Harmonic_mean
Logic problem, AND of pairwise ORs
multi-commodity flow problems", SIAM Journal on Computing, 5 (4): 691–703, doi:10.1137/0205048. Cook, Stephen A. (1971), "The complexity of theorem-proving procedures"
2-satisfiability
Degree of connectedness within a graph
in relation to a type of flow or transfer across the network. This allows centralities to be classified by the type of flow they consider important. "Importance"
Centrality
Concepts from statistical hypothesis testing
(mathematics) – Theorem for proving more complex theorems Jerzy Neyman – Polish American mathematician (1894–1981) Neyman–Pearson lemma – Theorem about the
Type_I_and_type_II_errors
Mathematical optimization problem restricted to integers
linear program as follows: min ∑ v ∈ V y v y v + y u ≥ 1 ∀ u , v ∈ E y v ∈ Z 0 + ∀ v ∈ V {\displaystyle {\begin{aligned}\min \sum _{v\in V}y_{v}\\y_{v}+y_{u}&\geq
Integer_programming
Method to solve optimization problems
introducing stochastic programming.) Edmonds, Jack; Giles, Rick (1977). "A Min-Max Relation for Submodular Functions on Graphs". Studies in Integer Programming
Linear_programming
Combinatorial optimization problem
and edges E ⊆ V × V {\displaystyle E\subseteq V\times V} , the maximum cut (max-cut) problem consists of finding two subsets S , T ⊆ V {\displaystyle S,T\subseteq
Quadratic unconstrained binary optimization
Quadratic_unconstrained_binary_optimization
June 2021. ..I will present a solution of the conjecture, which builds on min-max methods developed by F. C. Marques and A. Neves.. "Antoine Song | Clay
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Games" 2004 Lap Chi Lau (Toronto) "An Approximate Max-Steiner-Tree-Packing Min-Steiner-Cut Theorem" Marcin Mucha (Warsaw), Piotr Sankowski (Warsaw) "Maximum
Machtey_Award
American mathematician (1924–2021)
balanced matrices, Hoffman generalized the Ford-Fulkerson Max Flow – Min Cut result to other cases (flow at nodes, undirected arcs, etc.) by providing a proof
Alan_J._Hoffman
Mathematics of smooth surfaces
classical results such as the Riemann–Roch theorem imply that it always has a solution. The method of Ricci flow, developed by Richard S. Hamilton, gives
Differential geometry of surfaces
Differential_geometry_of_surfaces
geodesics through any point are the flow lines for the flow αt for Vh, so that αt is the gradient flow for h. THEOREM. On a Hadamard manifold X the following
Busemann_function
algorithm: computes the maximum flow in a graph Karger's algorithm: a Monte Carlo method to compute the minimum cut of a connected graph Push–relabel
List_of_algorithms
Branch of mathematics
analysis of the idempotent semiring called the tropical semiring (or max-plus algebra/min-plus algebra). Constructive analysis, which is built upon a foundation
Mathematical_analysis
Statistic which divides a data set into 100 parts and analyzes it as a percentage
approximates the CDF. This can be seen as a consequence of the Glivenko–Cantelli theorem. Some methods for calculating the percentiles are given below. The methods
Percentile
Crucial concept of quantum information
hidden variable theory. This was first demonstrated by Bell through Bell's Theorem, which showed that certain predictions of quantum mechanics are incompatible
Incompatibility of quantum measurements
Incompatibility_of_quantum_measurements
Declarative logic programming language
to data complexity, the decision problem for Datalog is P-complete (See Theorem 4.4 in ). P-completeness for data complexity means that there exists a
Datalog
Method of logical reasoning
evolved from religion to metaphysics to science, said Comte, which had flowed from mathematics to astronomy to physics to chemistry to biology to sociology—in
Inductive_reasoning
Class of statistical modeling methods
contains pair-wise potentials and the energy is submodular, combinatorial max-flow min-cut algorithms yield exact solutions. If exact inference is impossible
Conditional_random_field
Least-weight tree connecting graph vertices
minimum equivalent graphs and minimum spanning arborescences, preserves all max-min shortest paths and De Morgan's law consistency. The dynamic MST problem
Minimum_spanning_tree
Ancient Indo-Aryan language of South Asia
methods of stratifying out use and mention, language and metalanguage, and theorem and metatheorem predate key discoveries in western philosophy by millennia
Sanskrit
Richard S. Hamilton, 81, American mathematician (Ricci flow, Earle–Hamilton fixed-point theorem). Patrick Hobson, 91, British Anglican clergyman. Faina
Deaths_in_September_2024
Edges that hit all cycles in a graph
. In planar directed graphs, the feedback arc set problem obeys a min-max theorem: the minimum size of a feedback arc set equals the maximum number of
Feedback_arc_set
Train system using magnetic levitation
solution. Over long distances, coil costs could be prohibitive. Earnshaw's theorem shows that no combination of static magnets can be in a stable equilibrium
Maglev
Measurement of small-scale features on surfaces
than the sampling length, and according to the Nyquist–Shannon sampling theorem it should be at least two times longer than the wavelength of interesting
Surface_metrology
Microscopic object able to traverse fluid
flow field may be negligible. In such a case, reciprocal body deformation cannot induce migration of a swimmer, which is known as the scallop theorem
Microswimmer
Form of electromagnetic radiation
bolometer in units of angular displacement. 1901: Max Planck published the blackbody equation and theorem. He solved the problem by quantizing the allowable
Infrared
whether computers could calculate such possibilities; Gödel's incompleteness theorems; in 1974 the Arecibo Ionospheric Observatory found the Hulse–Taylor binary
List_of_Equinox_episodes
Economical computational problem
m ) 5 log ( u max ) + ( n + m ) 4 log B max ) {\displaystyle O((n+m)^{5}\log(u_{\max })+(n+m)^{4}\log {B_{\max }})} maximum flow problems, and thus
Market equilibrium computation
Market_equilibrium_computation
maximum and minimum values of the data set: M = max x + min x 2 . {\displaystyle M={\frac {\max x+\min x}{2}}.} The mid-range is closely related to the
Glossary_of_engineering:_M–Z
many outstanding scientists are clear-cut theists. Eric D. Schneider; Dorion Sagan (2005). Into the Cool: Energy Flow, Thermodynamics, and Life. University
List_of_agnostics
the lever back up and turn off the valve. Pappus's hexagon theorem Pappus's centroid theorem Ptolemy's world map — It included 8,000 locations from Shetland
List of Egyptian inventions and discoveries
List_of_Egyptian_inventions_and_discoveries
Audio performance differences between technologies
frequency in a digital system is based on the Nyquist–Shannon sampling theorem. This states that a sampled signal can be reproduced exactly as long as
Comparison of analog and digital recording
Comparison_of_analog_and_digital_recording
1922) Hugo F. Sonnenschein, 80, economist (Sonnenschein–Mantel–Debreu theorem) (b. 1940) July 16 Doug Bennett, 75, politician, member of the Michigan
2021 deaths in the United States (July–December)
2021_deaths_in_the_United_States_(July–December)
General relativity model near spacetime singularities
{\displaystyle u_{\max }^{(s)}} − 2, ..., reaching to the smallest, u min ( s ) {\displaystyle u_{\min }^{(s)}} < 1. Then that is, k(s) = [ u max ( s ) {\displaystyle
BKL_singularity
travel, tourism, insurance
MAX FLOW-MIN-CUT-THEOREM
MAX FLOW-MIN-CUT-THEOREM
Boy/Male
Gaelic
Son of the man who lives by the clear stream.
Boy/Male
Hindu
Lecturer, Respect, Supernatural power, Lord of mind
Boy/Male
American, Anglo, Australian, British, Chinese, Christian, Czechoslovakian, Danish, Dutch, English, French, German, Italian, Jamaican, Latin, Swedish, Swiss
By the Great Stream; A Short Form of Maxwell; Greatest; Little Maximus
Boy/Male
Arabic, Bengali, Hindu, Indian, Marathi, Muslim, Sikh, Tamil
Mind; Human; God is with us; Supernatural Power; Examine Closely; Accept the Truth; Assistance
Boy/Male
Gaelic
Son of the handsome man.
Surname or Lastname
English
English : see Flow.
Boy/Male
Tamil
Lord Vishnu, To flow out
Surname or Lastname
English
English : unexplained; possibly a variant of Flew, a metonymic occupational name for a fisherman, from Middle English flue, denoting a kind of fishing net.
Boy/Male
Muslim
Lecturer, Respect, Supernatural power, Lord of mind
Surname or Lastname
Chinese
Chinese : variant of Wen 2.Chinese : from a character in the personal name of Hu Gongman, a retainer of Wu Wang. After the latter established the Zhou dynasty in 1122 bc, he granted the state of Chen to Hu Gongman, whose descendants adopted the second character of his given name, Man, as their surname. This character also means ‘Manchurian’, but the name does not appear to be related to this meaning.Chinese : variant of Wen 3.Chinese : variant of Wan 1.English and Jewish : variant spelling of Mann.Dutch : from Middle Dutch man ‘man’, ‘husband’, ‘vassal’, ‘arbiter’.French : from the Germanic personal name Manno (see Mann 2).Jewish (Ashkenazic) : from the personal name Man, derived from Yiddish ‘man’.
Male
English
Short form of English Curtis, CURT means "courteous."
Male
Egyptian
, a chief of boatmen.
Boy/Male
Latin American Scottish
Greatest.
Surname or Lastname
English
English : variant of Court.Americanized spelling of German Kurt.Catalan : from curt ‘short’ (Latin curtus ‘cut short’, ‘broken off’), hence a nickname for a short man.
Female
Vietnamese
Vietnamese name CUC means "chrysanthemum."
Surname or Lastname
English
English : variant of Clough.English : metonymic occupational name for a nailer, from Old French clou ‘nail’. Compare Clower.Possibly an Americanized spelling of German Klau, a habitational name for someone from Klau near Aachen or Clauen in Lower Saxony, or Glau, a nickname for an astute person, from Old High German, Low German glou, glau ‘circumspect’.
Male
Hebrew
Short form of Hebrew Immanuw'el (English Immanuel), MAN means "God is with us."
Boy/Male
Indian
Short Man; Cute Friend
Male
Scandinavian
Variant spelling of Scandinavian Knut, CNUT means "knot."Â
Female
English
English variant spelling of French Fleur, or perhaps just a short form of Latin Flora, both FLOR means "flower."
MAX FLOW-MIN-CUT-THEOREM
MAX FLOW-MIN-CUT-THEOREM
MAX FLOW-MIN-CUT-THEOREM
MAX FLOW-MIN-CUT-THEOREM
MAX FLOW-MIN-CUT-THEOREM
MAX FLOW-MIN-CUT-THEOREM
MAX FLOW-MIN-CUT-THEOREM
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