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Computational problem in graph theory
maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can
Maximum_flow_problem
Mathematical optimization problem
minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network
Minimum-cost_flow_problem
Class of computational problems
Specific types of network flow problems include: The maximum flow problem, in which the goal is to maximize the total amount of flow out of the source terminals
Network_flow_problem
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
Directed graph where edges have a capacity
Commons has media related to Flow networks. Maximum Flow Problem Real graph instances Lemon C++ library with several maximum flow and minimum cost circulation
Flow_network
Computational problem in graph theory
polynomial time using a reduction to the maximum flow problem. It may be used to model various application problems of choosing an optimal subset of tasks
Closure_problem
Network flow problem (mathematics)
multi-commodity flow problem is a network flow problem with multiple commodities (flow demands) between different source and sink nodes. Given a flow network
Multi-commodity_flow_problem
Graph which remains connected when fewer than k edges are removed
would perform O ( n 2 ) {\displaystyle O(n^{2})} iterations of the Maximum flow problem, which can be solved in O ( n 3 ) {\displaystyle O(n^{3})} time.
Edge_connectivity
Algorithm in mathematical optimization
(alternatively, preflow–push algorithm) is an algorithm for computing maximum flows in a flow network. The name "push–relabel" comes from the two basic operations
Push–relabel maximum flow algorithm
Push–relabel_maximum_flow_algorithm
Sequence of operations for a task
problem in this category, such as the popular simplex algorithm. Problems that can be solved with linear programming include the maximum flow problem
Algorithm
Generalization of network flow problems
circulation problem and its variants are a generalisation of network flow problems, with the added constraint of a lower bound on edge flows, and with flow conservation
Circulation_problem
Algorithm for maximum cardinality matching
matching problem is closely related to the augmenting paths arising in maximum flow problems, paths along which one may increase the amount of flow between
Hopcroft–Karp_algorithm
Mathematical propositions in network flow theory
multi-commodity flow problems. The classic max-flow min-cut theorem states that for networks with a single type of flow (single-commodity flows), the maximum possible
Approximate max-flow min-cut theorem
Approximate_max-flow_min-cut_theorem
American computer scientist and mathematician
for industrial and Applied Mathematics 1988: A new approach to the maximum-flow problem, V Goldberg, RE Tarjan, Journal of the ACM (JACM) 35 (4), 921-940
Robert_Tarjan
American mathematician
Ford–Fulkerson algorithm, one of the best-known algorithms for solving the maximum flow problem in networks. D. R. Fulkerson was born in Tamms, Illinois, the third
D._R._Fulkerson
Directed graph with no directed cycles
to the same problem on the condensation of the graph. It may be solved in polynomial time using a reduction to the maximum flow problem. Some algorithms
Directed_acyclic_graph
Path-finding using high-weight graph edges
maximum flows. A closely related problem, the minimax path problem or bottleneck shortest path problem asks for the path that minimizes the maximum weight
Widest_path_problem
Graph theory problem: find a matching containing the most edges
algorithm solves the more general problem of computing the maximum flow. A bipartite graph (X + Y, E) can be converted to a flow network as follows. Add a source
Maximum-cardinality_matching
American mathematician (1927–2017)
in network flow problems. He was the son of mathematician Lester R. Ford Sr. Ford's paper with D. R. Fulkerson on the maximum flow problem and the Ford–Fulkerson
L._R._Ford_Jr.
Recursive algorithm in graph theory
cut. In practice, the minimum cut problem is always discussed with the maximum flow problem, to explore the maximum capacity of a network, since the minimum
Stoer–Wagner_algorithm
Polynomial-time algorithm for the assignment problem
running time. Ford and Fulkerson extended the method to general maximum flow problems in the form of the Ford–Fulkerson algorithm. In this simple example
Hungarian_algorithm
American mathematician (1914–2005)
Dantzig–Wolfe decomposition Knapsack problem Maximum flow problem Optimization (mathematics) Travelling salesman problem Shadow price Gass, Saul I. (2011)
George_Dantzig
Method of estimating the parameters of a statistical model, given observations
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed
Maximum_likelihood_estimation
Optimization technique
these energy minimization problems can be approximated by solving a maximum flow problem in a graph (and thus, by the max-flow min-cut theorem, define a
Graph cuts in computer vision and artificial intelligence
Graph_cuts_in_computer_vision_and_artificial_intelligence
Combinatorial optimization problem
dj. An integral maximum flow of minimum cost can be found in polynomial time; see network flow problem. Every integral maximum flow in this network corresponds
Assignment_problem
Algorithm for computing the maximal flow of a network
algorithm is a strongly polynomial algorithm for computing the maximum flow in a flow network, conceived in 1970 by Israeli (formerly Soviet) computer
Dinic's_algorithm
Algorithm to compute the maximum flow in a network
Ford–Fulkerson algorithm (FFA) is a greedy algorithm that computes the maximum flow in a flow network. It is sometimes called a "method" instead of an "algorithm"
Ford–Fulkerson_algorithm
Overview of and topical guide to algorithms
algorithm Maximum flow problem Ford–Fulkerson algorithm Edmonds–Karp algorithm Push–relabel maximum flow algorithm Minimum-cost flow problem Bipartite
Outline_of_algorithms
Russian mathematician (born 1947)
preflow-push based algorithms for the maximum flow problem, and the co-inventor of the Hopcroft–Karp–Karzanov algorithm for maximum matching in bipartite graphs
Alexander_V._Karzanov
Least-weight tree connecting graph vertices
problem (which is equivalent in the single-terminal case to the maximum flow problem), and approximating the minimum-cost weighted perfect matching. Other
Minimum_spanning_tree
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
of maximum flow and minimum cost flow problems yield a coefficient matrix with these properties (and with empty C). Thus, such network flow problems with
Unimodular_matrix
American mathematician
travelling salesman problem. In 1971 he co-developed with Jack Edmonds the Edmonds–Karp algorithm for solving the maximum flow problem on networks, and in
Richard_M._Karp
Optimization problem in mathematics
{\displaystyle R_{i,j},A_{i,j}} the above integer program is the dual of a maximum flow problem and therefore solvable in polynomial time. Not all choices for these
Rectangle_packing
Class of computational problem
Flow-shop scheduling is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. In a general job-scheduling
Flow-shop_scheduling
Framework in mathematics
DTW-equivalent shortest path problem to the maximum flow problem in the dual graph, which can be solved by most max-flow algorithms. However, when the
Graphical_time_warping
American computer scientist
on the maximum flow problem[GT88][CG97][GR98] and shortest path problem,[CGR96][GH05] including the discovery of the push–relabel maximum flow algorithm
Andrew_V._Goldberg
Numerical analysis of electric power flow
power-flow study is a numerical analysis of the flow of electric power in an interconnected system. It is also known as power-flow analysis, load-flow study
Power-flow_study
Abstract computer for designing parallel algorithms
PRAM algorithm for the maximum flow problem can provide strong speedups relative to the fastest serial program for the same problem. The article Ghanim,
Parallel_RAM
are consistent in all respects with those given in a discussion of the maximum-flow minimum-cut theorem. Cederbaum's theorem applies to a particular type
Cederbaum's maximum flow theorem
Cederbaum's_maximum_flow_theorem
Set of edges without common vertices
a largest matching in a bipartite graph can be treated as a network flow problem. Finding a largest matching in a general graph is much more difficult;
Matching_(graph_theory)
Partition of a graph by removing fewest possible edges
weighted min-cut problem allowing both positive and negative weights can be trivially transformed into a weighted maximum cut problem by flipping the sign
Minimum_cut
Combinatorial optimization method for a family of functions of discrete variables
computing the maximum flow over the network. Given a pseudo-Boolean function f {\displaystyle f} , if it is possible to construct a flow network with positive
Graph_cut_optimization
Data structure for representing a forest
be used for the same purpose is Euler tour tree. In solving the maximum flow problem, link/cut trees can be used to improve the running time of Dinic's
Link/cut_tree
Disproved conjecture in network flow theory
statement about single-source unsplittable flows. It concerns the problem of converting a fractional flow, which may split each terminal's demand across
Dinitz–Garg–Goemans conjecture
Dinitz–Garg–Goemans_conjecture
time by transforming the problem into an instance of the maximum flow problem. For unit disks with bounded ply (the maximum number of disks that have
Barrier_resilience
(CGO'03), 91-104, 2003. Xue, J. and Knoop, J. A Fresh Look at PRE as a Maximum Flow Problem. International Conference on Compiler Construction (CC'06), pages
Partial-redundancy elimination
Partial-redundancy_elimination
On bipartite matching and vertex cover
Kőnig (1931), describes an equivalence between the maximum matching problem and the minimum vertex cover problem in bipartite graphs. It was discovered independently
Kőnig's theorem (graph theory)
Kőnig's_theorem_(graph_theory)
Weighted tree representing s-t cuts of a graph
pairs in the graph. The Gomory–Hu tree can be constructed in |V| − 1 maximum flow computations. It is named for Ralph E. Gomory and T. C. Hu. Let G = (
Gomory–Hu_tree
"5-flow conjecture". Open Problem Garden. Archived from the original on November 26, 2018. mdevos (March 31, 2010). "4-flow conjecture". Open Problem Garden
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
American mathematician (1943–2024)
were in the field of geometric flows. In 1986, Peter Li and Shing-Tung Yau discovered a new method for applying the maximum principle to control the solutions
Richard_S._Hamilton
Method to solve optimization problems
linear programming problems. Certain special cases of linear programming, such as network flow problems and multicommodity flow problems, are considered
Linear_programming
Algebra whose elements are stable matchings
order. The closure problem can, in turn, be solved in polynomial time by transforming it into an instance of the maximum flow problem. Dan Gusfield has
Lattice_of_stable_matchings
Experimental work reported in Caragea & Vishkin (2011) for the Maximum flow problem, and in two papers by Edwards and Vishkin (2012a, 2012b) for the
Explicit_multi-threading
Fulkerson Flows in Networks. Prentice-Hall, 1962. Presents the Ford–Fulkerson algorithm for solving the maximum flow problem, along with many ideas on flow-based
List of publications in mathematics
List_of_publications_in_mathematics
the Fermi–Pasta–Ulam–Tsingou problem. In network theory, Ford & Fulkerson compute a solution to the maximum flow problem. Householder invents his eponymous
Timeline of computational mathematics
Timeline_of_computational_mathematics
Computational problem of graph theory
flow problem typically involves a directed graph where each edge represents a pipe, wire, or road, and each edge has a capacity, which is the maximum
Shortest_path_problem
Algorithm in graph theory
algorithm. The algorithm is usually formulated in terms of a minimum-cost flow problem. The network simplex method works very well in practice, typically 200
Network_simplex_algorithm
Method of analyzing variables in software
block starts in a state with a value less than the maximum. The details depend on the data-flow problem. If the minimum element represents totally conservative
Data-flow_analysis
Number used to indicate probability of winning tournaments
whether a team has been eliminated by use of the algorithm for the maximum flow problem. The addition of a second Wild Card team makes the reverse scenario
Magic_number_(sports)
Partial differential equation
geometric analysis, the Ricci flow (/ˈriːtʃi/ REE-chee, Italian: [ˈrittʃi]), sometimes also referred to as Hamilton's Ricci flow, is a certain partial differential
Ricci_flow
Edges that hit all cycles in a graph
feedback arc set and its removal leaves a maximum acyclic subgraph; weighted versions of these optimization problems are also used. If a feedback arc set is
Feedback_arc_set
Partition of a graph's nodes into 2 disjoint subsets
max in the objective function. The max-flow problem is the dual of the min-cut problem. The sparsest cut problem is to bipartition the vertices so as to
Cut_(graph_theory)
Global warming about 55 million years ago
thermal maximum (PETM), alternatively "Eocene thermal maximum 1 (ETM1)" and formerly known as the "Initial Eocene" or "Late Paleocene thermal maximum", was
Paleocene–Eocene thermal maximum
Paleocene–Eocene_thermal_maximum
Flow where fluid particles follow smooth paths in layers
flow is in the smooth flow of a viscous liquid through a tube or pipe. In that case, the velocity of flow varies from zero at the walls to a maximum along
Laminar_flow
Type of diagram
the maximum number of vehicles can pass by a point in a given time period. The flow and capacity at which this point occurs is the optimum flow and optimum
Fundamental diagram of traffic flow
Fundamental_diagram_of_traffic_flow
Algorithm to compute the maximum flow in a flow network
an implementation of the Ford–Fulkerson method for computing the maximum flow in a flow network in O ( | V | | E | 2 ) {\displaystyle O(|V||E|^{2})} time
Edmonds–Karp_algorithm
Study of interactions between travellers and infrastructure
movement of traffic and minimal traffic congestion problems. The foundation for modern traffic flow analysis dates back to the 1920s with Frank Knight's
Traffic_flow
Study of mathematical algorithms for optimization problems
on a compact set attains its maximum point or view. One of Fermat's theorems states that optima of unconstrained problems are found at stationary points
Mathematical_optimization
Optimization algorithm
Sequential ordering problem (SOP) Job-shop scheduling problem (JSP) Open-shop scheduling problem (OSP) Permutation flow shop problem (PFSP) Single machine
Ant colony optimization algorithms
Ant_colony_optimization_algorithms
{\displaystyle O((n+m)^{5}\log(u_{\max })+(n+m)^{4}\log {B_{\max }})} maximum flow problems, and thus it runs in time O ( ( n + m ) 8 log ( u max ) + ( n
Fisher_market
Concept in graph theory
In graph theory, a nowhere-zero flow or NZ flow is a network flow that is nowhere zero. It is intimately connected (by duality) to coloring planar graphs
Nowhere-zero_flow
Volume of fluid which passes per unit time
engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes
Volumetric_flow_rate
Branch of fluid mechanics
flow, however, the gas density and temperature also become variables. This requires two more equations in order to solve compressible-flow problems:
Compressible_flow
Subfield of mathematical optimization
optimally Designing water distribution networks Earth science problems (e.g. reservoir flow-rates) There is a large amount of literature on polynomial-time
Combinatorial_optimization
flow problems. In fact the preflow-push algorithm for max-flow can be derived by applying the original 1979 auction algorithm to the max flow problem
Auction_algorithm
Electrical power regulating system
slack bus is crucial to a load flow problem since it will account for transmission line losses. In a load flow problem, conservation of energy results
Slack_bus
Concept in graph theory and network analysis
given size k {\textstyle k} with maximum group centrality arises from the fact that for many measures, this problem is NP-hard. Therefore, heuristics
Group_centrality
Streamlined body for generating lift
air flow close to the upper surface rapidly becomes turbulent past the maximum thickness point, which increases the skin friction drag. A laminar flow wing
Airfoil
Mathematical optimization problem restricted to integers
This problem can be formulated as an integer linear program in which binary variables indicate whether a frequency is assigned to an antenna. Cash flow matching
Integer_programming
Optimization problem
The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a
Vehicle_routing_problem
Analysis in fluid dynamics
adjustment doesn't solve the problem, since most networks have several loops. It is okay to use this adjustment, however, because the flow changes won't alter
Pipe_network_analysis
set problem, i.e., the dominating set problem in line graphs. NP-complete variants include the connected dominating set problem and the maximum leaf
List_of_NP-complete_problems
maximum of the distances multiplied by the corresponding flows. The problem is NP-hard, as it can be used to formulate the Hamiltonian cycle problem by
Quadratic bottleneck assignment problem
Quadratic_bottleneck_assignment_problem
System involved in supplying a product or service to a consumer
customers, while supply chain management focuses on the optimization of the flow of goods within the supply chain's distribution channels to ensure efficiency
Supply_chain
Motion characterized by chaotic changes in pressure and flow velocity
turbulent flow is fluid motion exhibiting chaotic changes in pressure and flow velocity. It is in contrast to laminar flow, which occurs when a fluid flows in
Turbulence
Engineering ratio
requires a turndown ratio of at least 10:1. If the meter had an advertised maximum flow of 2,000,000 m3 per day then the required turndown ratio would be 20:1
Turndown_ratio
Solving an optimization problem with a quadratic objective function
with G has a maximum that is function of the clique number of G. Computing the clique number of a graph is a well-known NP-hard problem; hence, solving
Quadratic_programming
Algorithm for two disjoint paths in a graph
second path. The problem of finding two disjoint paths of minimum weight can be seen as a special case of a minimum cost flow problem, where in this case
Suurballe's_algorithm
Problem in computer networking
window syndrome (SWS) is a problem in computer networking caused by poorly implemented TCP flow control. A serious problem can arise in the sliding window
Silly_window_syndrome
Algorithm for linear programming
has a maximum value on the feasible region, then it has this value on (at least) one of the extreme points. This in itself reduces the problem to a finite
Simplex_algorithm
Algorithm used to solve non-linear least squares problems
(DLS) method, is used to solve non-linear least squares problems. These minimization problems arise especially in least squares curve fitting. The LMA
Levenberg–Marquardt_algorithm
Type of rechargeable flow battery
(VRB), also known as the vanadium flow battery (VFB) or vanadium redox flow battery (VRFB), is a type of rechargeable flow battery which employs vanadium
Vanadium_redox_battery
rearrangement distances. The Maximum Breakpoint Graph Decomposition problem can be reduced to the Maximum Eulerian Cycle Decomposition problem in polynomial time
Cycle decomposition (graph theory)
Cycle_decomposition_(graph_theory)
Combinatorial optimization graph problem
In mathematics, the minimum k-cut is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph
Minimum_k-cut
Law describing the pressure drop in an incompressible and Newtonian fluid
zero, the original problem is recovered. Flow through pipes with an oscillating pressure gradient finds applications in blood flow through large arteries
Hagen–Poiseuille_equation
Soviet and Israeli computer scientist
December 2023. E. A. Dinic (1970). "Algorithm for solution of a problem of maximum flow in a network with power estimation" (PDF). Doklady Akademii Nauk
Yefim_Dinitz
quantity per flush. Some of these low-flow fixtures are faucets, showerheads, and toilets. In the United States a maximum water usage of conventional plumbing
Low-flow_fixtures
Scheduling policy
fully utilized (is saturated) and of all the flows sharing this link, the data flow i achieves overall maximum data rate. Note that this definition is substantially
Max-min_fairness
Software designed to support and optimize warehouse and distribution center management
warehouse, optimising the use of available space, or coordinating tasks for maximum efficiency. In the 20th century, the term 'warehouse management information
Warehouse_management_system
Subfield of mathematical optimization
optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently, maximizing
Convex_optimization
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