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Topic: Max flow min cut theorem

	BackgroundMaterial (Site not responding. Last check: 2007-10-22)
		The capacity of a cut (S, T) is the sum of the capacity of all edges from S to T. It is an easy exercise to show that the value of any flow cannot exceed the capacity of any cut.
		Furthermore the sum of the flow on the forward edges is the value of the flow.
		Theorem [ Menger] If x and y are nonadjacent vertices in a graph G, then the minimum number k(x, y) of vertices required to separate x and y equals the maximum number m(x, y) of vertex disjoint paths from x to y.
	www.math.gatech.edu /~trotter/Section2-LP.htm (2739 words)

	Maximum Flow Problem (Site not responding. Last check: 2007-10-22)
		The maximum flow problem is again structured on a network; but here the arc capacities, or upper bounds, are the only relevant parameters.
		The maximum flow from node 1 to node 8 is 30 and the flows that yield this flow are shown on the figure.
		The fact that the sum of the capacities of the arcs on the minimal cut equals the maximum flow is a famous theorem of network theory called the max flow min cut theorem.
	www.me.utexas.edu /~jensen/models/network/net11.html (178 words)

	MaxFlow - PineWiki
		The essential idea is that any flow that is not maximum can be improved by adjusting flows along some augmenting path, which will be a path that may include both edges in the graph with unused capacity and backwards edges with some flow on them already.
		The first is that the size of any s-t flow is bounded by the capacity of any s-t cut, and in particular that the size of the max flow is less than or equal to the capacity of the min cut.
		Because the Ford-Fulkerson flow saturates some cut, its size equals the capacity of some cut---so the size of the maximum flow is at least as big as the size of the minimum cut.
	pine.cs.yale.edu /pinewiki/MaxFlow (1605 words)

	Max-flow min-cut theorem - Wikipedia, the free encyclopedia
		The maximal amount of a flow is equal to the capacity of a minimal cut.
		For the cut S = {s,p},T = {o,q,r,t}, the capacity across the cut is c(s,o) + c(p,r) = 3 + 2 = 5.
		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.
	en.wikipedia.org /wiki/Max_flow_min_cut_theorem (534 words)

	Mathematical Programming Glossary (Site not responding. Last check: 2007-10-22)
		The problem is to maximize the flow from s to t subject to conservation of flow constraints at each node and flow bounds on each arc.
		The max flow labeling algorithm provides a constructive proof of the Max flow - Min cut theorem.
		The maximum flow through a (single commodity) capacitated network from a specified node, called the source, to another node, called the sink, equals the value of the minimum cutset.
	glossary.computing.society.informs.org /index.php?page=M.html (2376 words)

	Reading Group - Stanford High Performance Networking Group
		If we want to find what is the maximum flow that can be flowing from a source s to a destination t in a network, where links (called edges) are directional and have a maximum transmission capacity (see figure), there is a well-known set of theorems and algorithms that apply.
		The Max-Flow and Min-Cut Theorem basically says that f is a max-flow <=> there is a cut C such that <=> there are no augmentating paths in the residual path Gf.
		Basically one uses non feasible flows (where the flow conservation law is not observed), where we push this unrealistic flow.
	klamath.stanford.edu /reading/MaxFlow.html (577 words)

	0540515
		To show how network flow problems arise in real life; to show how network flow problems may be solved; and to see the significance of solutions for the original real-life problems.
		To ensure that the students are able (i) to construct network flow models which represent problems in real life, (ii) to solve certain problems posed within these models, and (iii) to appreciate the possible impact of the solution on the real-life problem.
		Flows on networks which arise in real life; and how these may be modelled.
	www.york.ac.uk /depts/maths/ugrad/courses/nextyear/0540515.htm (302 words)

	Menger's Theorems and Max-Flow-Min-Cut (Site not responding. Last check: 2007-10-22)
		A {0,1}-flow is function from the set of arcs to the set {0,1), such that sum of the flow on arcs into any vertex is the same as the sum of the flow on arcs out of the vertex, except for the two special vertices: the source, u, and the sink, v.
		Clearly, F is the set of edges of a flow f', where f' is one on arcs of F and zero on arcs not in F.
		The sum of the capacities on the arcs of A
	www.math.fau.edu /locke/Menger.htm (599 words)

	Brown CS: Tech Report CS-93-30
		The best-known theorem in combinatorial optimization is the classical max-flow min-cut theorem of Ford and Fulkerson.
		We prove a polylogarithmic bound on the worst case ratio between the minimum multicut and the value of the maximum multicommodity flow in the special case when the demands are symmetric.
		Computation of minimum cuts in directed multicommodity flow problems with symmetric demand is a basic step for approximation algorithms for a number of NP-complete problems.
	www.cs.brown.edu /publications/techreports/reports/CS-93-30.html (267 words)

	MS&E 217/317, Combinatorial Optimization, Autumn 2003-04
		Assume that the current flow x is given to your, and you have to find the path to augment on as well as the amount of flow to send (i.e.
		Here n is the number of vertices, m is the number of edges, v_j, w_j are the endpoints of the j-th edge, x_j is the current flow on the j-th edge, and u_j is its capacity.
		König’s Theorem: The size of the largest matching in an undirected bipartite graph is the same as the size of the smallest vertex cover.
	www.stanford.edu /~ashishg/msande217/handouts/hw3.html (461 words)

	Approximate Max-Flow Min-(multi)cut Theorems and Their Applications - Garg, Vazirani, Yannakakis (ResearchIndex)
		Abstract: Consider the multicommodity flow problem in which the object is to maximize the sum of commodities routed.
		We prove the following approximate max-flow min-multicut theorem: min multicut O(logk) max flow min multicut; where k is the number of commodities.
		Our proof is constructive; it enables us to find a multicut within O(log k) of the max flow (and hence also the optimal multicut).
	citeseer.ist.psu.edu /garg93approximate.html (647 words)

	[Networking Lab : seminar] California Institute of Technology (Site not responding. Last check: 2007-10-22)
		The famous max-flow min-cut theorem states that a source node s can send information through a network (V,E) to a sink node t at a rate determined by the min-cut separating s and t.
		From a theoretical perspective, both extremes of this problem are fundamental theorems in combinatorics.
		His research interests are in the theory of discrete event dynamical networks and in the modeling and performance evaluation of computer and communication systems.
	netlab.caltech.edu /seminar/spring03.htm (4177 words)

	Learning objectives and review questions: Lecture 15 (Site not responding. Last check: 2007-10-22)
		Given a flow network, use the Edmonds-Karp algorithm to determine the maximum flow, and a cut corresponding to the maximum flow.
		Prove properties of cuts and relationships between cuts and flows, such as the max-flow min-cut theorem.
		You may use facts that we have proved in class, such as the max-flow min-cut theorem, unless the question forbids it.
	www.cs.fsu.edu /~asriniva/courses/alg05/Rev.Lec15.html (336 words)

	[No title]
		Goals is to flow as much as possible from S to T. Looked at the greedy Ford-Fulkerson algorithm: find a path from source to sink; push as much flow as possible along it, compute residual graph, and repeat until no paths left.
		We proved: FF alg find a max flow (though number of iterations could be large).
		Cost of a flow is the sum over all edges of the flow on that edge times the cost of the edge.
	www.cs.cmu.edu /afs/cs/academic/class/15451-f99/www/lectures/lect1026 (679 words)

	[No title]
		When we updated the flows we increased the xij values on the forward arcs and decreased the xji values on the backward arcs.
		The maximum value of the flow from a source node s to a sink node t in a capacitated network equals the minimum capacity among all s-t cuts.
		If the capacity of an s-t cut [s, EMBED Equation.3 ] in a network with both lower and upper bounds on arc flows is defined by (6.7), the maximum value of flow from node s to node t equals the minimum capacity among all s-t cuts.
	www.pitt.edu /~banwork/3087/Ch6Notes01.doc (1250 words)

	CS 660: Graphs and Network Flows
		Let G = (V, E) be a flow network and f be a flow in G. Let Gf = (V, Ef) be the residual network of G induced by f.
		Let G = (V, E) be a flow network and f be a flow in G. Let p be an augmenting path in Gf.
		Let f be a flow in G = (V, E) be a flow network with source s and sink t.
	www.eli.sdsu.edu /courses/fall95/cs660/notes/Graph/Graph.html (1276 words)

	[No title]
		There is an s-t cutset (S, T) whose capacity is the flow value of x.
		The maximum flow value is the minimum value of a cut.
		But we shall soon see how to solve max flows in a polynomial number of operations, even if data can be irrational.ó-¨Summary and Extensions¨q1.
	web.mit.edu /jorlin/www/15.082/Lectures/08_Max_Flows_1.ppt (581 words)

	MIT EECS Event (Site not responding. Last check: 2007-10-22)
		The most well-known theorem in combinatorial optimization is the classical max-flow min-cut theorem of Ford and Fulkerson.
		This theorem can be viewed as special case of the linear programming duality theorem, and it also serves as the basis for finding efficient algorithms to solve these two problems.
		Efficient algorithms for many other combinatorial optimization problems are based on min-max theorems that are special cases of the linear programming duality theorem in a similar way.
	www.eecs.mit.edu /AY94-95/events/17.html (306 words)

	Learning objectives and review questions: Lecture 3 (Site not responding. Last check: 2007-10-22)
		Explain the meaning of the following terms: residual network, residual capacity, augmenting path, cut, net flow across a cut, capacity of a cut, and a minimum cut.
		Given a flow network and a flow, find an augmenting path, and determine a flow with a greater value.
		Determine the new flow that results from using this augmenting path to increase the flow.
	www.cs.fsu.edu /~asriniva/courses/alg05/Rev.Lec3.html (150 words)

	Math 190 - For Graduate Students (Site not responding. Last check: 2007-10-22)
		Graph coloring (Brook's theorem, sketch of four-color theorem)
		Max flow - min cut theorem, integrality, Menger's theorem
		The marriage lemma, symmetric chains, and Sperner's theorem
	www.math.unc.edu /gradcourses/math190.html (64 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Network flows and augmenting path algorithm: - flow, summation of flows, several sources and sinks [CLR,E,AMO]; - interlude: [(A-to-B)-paths + (B-to-C)-paths] -> (A-to-C)-paths, via Stable Marriage algorithm [CHR]; - circulations, lower bounds on flows in edges: a reduction [E,AMO]; - augmenting path algorithm, max-flow min-cut theorem [CLR,E,AMO].
		Application to computing connectivity in a graph: - applications of flows to vertex-to-vertex edge-connectivity, in a graph [E,AMO]; - economical algorithm for finding the edge-connectivity of a graph, with amortized time analysis [nD]; - applications of flows to vertex-connectivity, in a graph [E,AMO].
		Advanced augmenting path algorithms: - thickest flow paths algorithm [AMO]; - shortest paths algorithm [CLR,E,AMO]; - shortest paths algorithm with Layered Network data structure; its amortized time analysis [D70,E,AMO]; - its complexity for networks with capacity 1 in edges, with applications to graph connectivity problems [E,AMO].
	www.cs.bgu.ac.il /~dinitz/Course/AA/planold (146 words)

	Max- Flow Min-Cut Theorem (via CobWeb/3.1 planetlab2.cs.umd.edu) (Site not responding. Last check: 2007-10-22)
		sinks is equal to the minimum of the capacities of cuts seperating the
		capacity of a vertex-cut is equal to the maximum of the flow from s to
		The total flow must pass through any cut the cut that shows the least
	sci4um.com.cob-web.org:8888 /about23102.html (574 words)

	esm_goodaire_discrete_2\|The Max Flow -- Min Cut Theorem\|Flows and Cuts\|True-or-False (via CobWeb/3.1 ... (Site not responding. Last check: 2007-10-22)
		A directed network is a digraph in which each arc is assigned an integer weight.
		If uv is an arc in a digraph then the flow goes from u towards v.
		The capacity of the arc uv is denoted c
	wps.prenhall.com.cob-web.org:8888 /esm_goodaire_discrete_2/0,5290,379887-,00.utf8.html (130 words)

	22C:137, 22M:152 Theory of Graphs: Schedule and Plans (Site not responding. Last check: 2007-10-22)
		Introduction to flow networks and flows, the max-flow problem.
		Introduction to the theorem of Tutte and Nash-Williams (1961) showing necessary and sufficient conditions on the existence of edge-disjoint spanning trees (pages 58-61).
		Thomassen's theorem on the 5-choosability of planar graphs.
	www.cs.uiowa.edu /~sriram/137/spring05/schedule.html (367 words)

	[No title]
		If arc (i,j) is saturated, then it is not admissible until flow is sent from j to i, and this cannot happen until d(j) increases.
		Next lecture: an algorithm that does not rely on augmenting paths.¡$)¤óc¨The Capacity Scaling Algorithm¨½For any fixed value D, let G(x,D) be the arcs in G(x) with capacity at least D. A flow x is called D-maximum if there is no augmenting path of size D or more.
		It takes a flow that is D-maximum and outputs a flow that is D/2-maximum.
	web.mit.edu /jorlin/www/15.082/Lectures/09_Max_Flows_2.ppt (795 words)

	Math 428, Section B1, Summer 2004 Home
		Topics covered: Konig-Egervary theorem, Hungarian algorithm, equivalence of Hall and Konig-Egervary theorems, matchings in general graphs.
		Topics covered: chromatic number, coloring algorithms, bounds on chromatic number, six- and five-color theorems for planar graphs, connection to map coloring; started proof of Brooks' theorem.
		Thursday 6/10: Distributed handout on definitions related to Matrix-Tree Theorem and planarity.
	www.math.rutgers.edu /~nweining/math428-s04/428_index.html (533 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Make sure you know how to find the max flow and the minimum cut of a given network.
		Understand the significance of the max flow - min cut theorem.
		You must also know how to model problems as network flow problems.
	www.cs.uwm.edu /classes/cs704/coverage2.html (55 words)

	Ford-Fulkerson algorithm (Site not responding. Last check: 2007-10-22)
		Ford-Fulkerson algorithm solving the maximum-flow probem.The Ford-Fulkerson algorithm depend on three important ideas :residual networks, augmenting paths, and cuts.These ideas are essential to the important max-flow min-cut theorem,which characterizes the value of a maximum flow in terms of cuts of the flow network.
		Start with f(u,v)=0 for all u,v contained in V,giving an initial flow of value 0.At each iteration,increase the flow value by finding an augmenting path.
		Residual networks: Intutively,given a flow network and a flow,the residual network consists of edges that can admit more net flow.
	www.cs.bgu.ac.il /~visproj/kalich/ford.htm (108 words)

	RAND \| Papers \| On the Max Flow Min Cut Theorem of Networks.
		On the Max Flow Min Cut Theorem of Networks.
		For fax or mail orders, use the normal cover pricing and this PDF form.
		RAND's publications do not necessarily reflect the opinions of its research clients and sponsors.
	www.rand.org /pubs/papers/P826 (216 words)

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