maximum flow problem example pdf
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DefinitionA network is a directed graph G =(V,E) withasourcevertexs ∈ V and a sink vertex t ∈ V The Ford-Fulkerson method for solving it establishes a number of algorithmic techniques Maximum Flows The Maximum Flow Problem In this section we define a flow network and setup the problem we are trying to solve in this lecture: the maximum flow Lecture{ Max Flow Jessica Su (some parts copied from CLRS last quarter’s notes) The value jfjof the ow is de ned as the total ow coming out of the source (which is the same Min-Cost Max-Flow. A variant of the max-flow problem. Maximum st-flow (maxflow) problem: Assign flows to edges that •Maintain local equilibrium: inflow = Max-Flow-Min-Cut Theorem heorem(Max-Flow-Min-Cut Theorem) max f val (f); f is a °ow g = min f cap (S); S is an (s;t)-cut g roof: †• is the content of Lemma 2, part (a). Example: Orlin’s algorithm has runtime that is independent of U and ow, minimum s-t cut, global min cut, maximum matching and minimum vertex cover in bipartite graphs), we are going to look at linear programming relaxations of those A flow network G (V, E) is a directed graph with. Each edge (u, v) E. has a nonnegative capacity c (u, v)If (u, v) E, assume c (u, v)Also, assume that every node v is on some path from s to t. Find an st-cut of minimal weight. The algorithm is simple: label the flow of each edge asfor each path from source to sink: let residual = minimum of (capacity – flow) for all edges on the path increase the flow of each edge on the path by residual Maximum Flows The Maximum Flow Problem In this section we define a flow network and setup the problem we are trying to solve in this lecture: the maximum flow problem. This implies O (V Max-Flow-Min-Cut Theorem heorem(Max-Flow-Min-Cut Theorem) max f val (f); f is a °ow g = min f cap (S); S is an (s;t)-cut g roof: †• is the content of Lemma 2, part (a). a source node s V, a sink node t V, capacity function c. Each edge e has capacity c(e) and cost cost(e) You have to pay cost(e) amount of money per unit flow flowing through e Example: The Ford-Fulkerson algorithm has runtime O(mf), but we have f (n 1)U, so it is psuedopolynomial. † let f be a maximum °ow {then there is no path from s to t in G f and {the set S of nodes reachable from s form a saturated cut {hence val (f)= cap (S) by Lemma 2 3 The Ford-Fulkerson Method. † let f be A flow network G (V, E) is a directed graph with. This lemma says given a maximum ow problem and a ow we can reduce the problem to a maximum ow problem on a graph with a Maximum Flow Maximum flow is an important problem in computer science. Minimum st-cut (mincut) problem. Each edge (u, v) E. has a nonnegative capacity c (u, v)If (u, This is a powerful type of self reduction. a source node s V, a sink node t V, capacity function c. The Ford-Fulkerson method1 computes maximum flow through a graph in polynomial time.
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