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2pt 0em CSCE423/823 Computer Science & Engineering 423/823 Introduction Flow Networks Design and Analysis of Algorithms Ford-Fulkerson Method Lecture 07 Maximum Flow (Chapter 26) Edmonds-Karp Algorithm Maximum Bipartite Matching

  1. 2pt 0em CSCE423/823 Computer Science & Engineering 423/823 Introduction Flow Networks Design and Analysis of Algorithms Ford-Fulkerson Method Lecture 07 — Maximum Flow (Chapter 26) Edmonds-Karp Algorithm Maximum Bipartite Matching Stephen Scott (Adapted from Vinodchandran N. Variyam) 1 / 35

  2. Introduction CSCE423/823 Can use a directed graph as a flow network to model: Data through communication networks, water/oil/gas through pipes, assembly lines, etc. Introduction A flow network is a directed graph with two special vertices: source s Flow Networks Ford-Fulkerson that produces flow and sink t that takes in flow Method Each directed edge is a conduit with a certain capacity (e.g. 200 Edmonds-Karp Algorithm gallons/hour) Maximum Vertices are conduit junctions Bipartite Matching Except for s and t , flow must be conserved: The flow into a vertex must match the flow out Maximum flow problem: Given a flow network, determine the maximum amount of flow that can get from s to t Other application: Bipartite matching 2 / 35

  3. Flow Networks CSCE423/823 A flow network G = ( V, E ) is a directed graph in which each edge ( u, v ) 2 E has a nonnegative capacity c ( u, v ) � 0 If ( u, v ) 62 E , c ( u, v ) = 0 Introduction Assume that every vertex in V lies on some path from the source Flow Networks Example vertex s 2 V to the sink vertex t 2 V More Notation Multiple Sources and Sinks Ford-Fulkerson Method Edmonds-Karp Algorithm Maximum Bipartite Matching 3 / 35

  4. Flows CSCE423/823 A flow in graph G is a function f : V ⇥ V ! R that satisfies: Capacity constraint: For all u, v 2 V , 0  f ( u, v )  c ( u, v ) (flow 1 should be nonnegative and not exceed capacity) Introduction Flow conservation: For all u 2 V \ { s, t } , 2 Flow Networks Example More Notation X X f ( v, u ) = f ( u, v ) Multiple Sources and Sinks v 2 V v 2 V Ford-Fulkerson Method (flow entering a vertex = flow leaving) Edmonds-Karp Algorithm The value of a flow is the net flow out of s (= net flow into t ): Maximum Bipartite X X | f | = f ( s, v ) � f ( v, s ) Matching v 2 V v 2 V Maximum flow problem: given graph and capacities, find a flow of maximum value 4 / 35

  5. Flow Example CSCE423/823 Introduction Flow Networks Example More Notation Multiple Sources and Sinks Ford-Fulkerson Method Edmonds-Karp Algorithm Maximum Bipartite Matching What is the value of this flow? 5 / 35

  6. Multiple Sources and Sinks CSCE423/823 Introduction Might have cases where there are multiple sources and/or sinks; e.g. Flow Networks if there are multiple factories producing products and/or multiple Example More Notation warehouses to ship to Multiple Sources and Sinks Can easily accommodate graphs with multiple sources s 1 , . . . , s k and Ford-Fulkerson Method multiple sinks t 1 , . . . , t ` Edmonds-Karp Add to G a supersource s with an edge ( s, s i ) for i 2 { 1 , . . . , k } and Algorithm a supersink t with an edge ( t j , t ) for j 2 { 1 , . . . , ` } Maximum Bipartite Matching Each new edge has a capacity of 1 6 / 35

  7. Multiple Sources and Sinks (2) CSCE423/823 Introduction Flow Networks Example More Notation Multiple Sources and Sinks Ford-Fulkerson Method Edmonds-Karp Algorithm Maximum Bipartite Matching 7 / 35

  8. Ford-Fulkerson Method CSCE423/823 A method (rather than specific algorithm) for solving max flow Introduction Multiple ways of implementing, with varying running times Flow Networks Core concepts: Ford-Fulkerson Method Residual network: A network G f , which is G with capacities reduced 1 Residual based on the amount of flow f already going through it Networks Flow Augmenting path: A simple path from s to t in residual network G f Augmentation 2 Augmenting Path ) If such a path exists, then can push more flow through network Max-Flow Min-Cut Cut: A partition of V into S and T where s 2 S and t 2 T ; can 3 Theorem Basic measure net flow and capacity crossing a cut Ford-Fulkerson Algorithm Ford-Fulkerson Method repeatedly finds an augmenting path in residual network, Example Analysis of adds in flow along the path, then updates residual network Ford-Fulkerson Edmonds-Karp Algorithm Maximum 8 / 35 Bipartite

  9. Ford-Fulkerson-Method( G, s, t ) CSCE423/823 Introduction Flow Networks Initialize flow f to 0 ; Ford-Fulkerson 1 while there exists augmenting path p in residual Method Residual network G f do Networks Flow augment flow f along p ; 2 Augmentation Augmenting 3 end Path Max-Flow 4 return f ; Min-Cut Theorem Basic Ford-Fulkerson Algorithm Ford-Fulkerson Example Analysis of Ford-Fulkerson Edmonds-Karp Algorithm Maximum 9 / 35 Bipartite

  10. Residual Networks CSCE423/823 Given flow network G with capacities c and flow f , residual network G f consists of edges with capacities showing how one can change flow in G Introduction Define residual capacity of an edge as Flow Networks Ford-Fulkerson 8 c ( u, v ) � f ( u, v ) if ( u, v ) 2 E Method < Residual c f ( u, v ) = f ( v, u ) if ( v, u ) 2 E Networks Flow Augmentation 0 otherwise : Augmenting Path Max-Flow E.g. if c ( u, v ) = 16 and f ( u, v ) = 11 , then c f ( u, v ) = 5 and Min-Cut Theorem c f ( v, u ) = 11 Basic Ford-Fulkerson Algorithm Then can define G f = ( V, E f ) as Ford-Fulkerson Example Analysis of Ford-Fulkerson E f = { ( u, v ) 2 V ⇥ V : c f ( u, v ) > 0 } Edmonds-Karp Algorithm So G f will have some edges not in G , and vice-versa Maximum 10 / 35 Bipartite

  11. Residual Networks (2) CSCE423/823 Introduction Flow Networks Ford-Fulkerson Method Residual Networks Flow Augmentation Augmenting Path Max-Flow Min-Cut Theorem Basic Ford-Fulkerson Algorithm Ford-Fulkerson Example Analysis of Ford-Fulkerson Edmonds-Karp Algorithm Maximum 11 / 35 Bipartite

  12. Flow Augmentation CSCE423/823 G f is like a flow network (except that it can have an edge and its reversal); so we can find a flow within it Introduction If f is a flow in G and f 0 is a flow in G f , can define the Flow Networks augmentation of f by f 0 as Ford-Fulkerson Method ⇢ f ( u, v ) + f 0 ( u, v ) � f 0 ( v, u ) Residual if ( u, v ) 2 E Networks ( f " f 0 )( u, v ) = Flow 0 otherwise Augmentation Augmenting Path Max-Flow Lemma: f " f 0 is a flow in G with value | f " f 0 | = | f | + | f 0 | Min-Cut Theorem Basic Proof: Not di ffi cult to show that f " f 0 satisfies capacity constraint Ford-Fulkerson Algorithm Ford-Fulkerson and and flow conservation; then show that | f " f 0 | = | f | + | f 0 | Example Analysis of (pp. 718–719) Ford-Fulkerson Edmonds-Karp Result: If we can find a flow f 0 in G f , we can increase flow in G Algorithm Maximum 12 / 35 Bipartite

  13. Augmenting Path CSCE423/823 By definition of residual network, an edge ( u, v ) 2 E f with Introduction c f ( u, v ) > 0 can handle additional flow Flow Networks Ford-Fulkerson Since edges in E f all have positive residual capacity, it follows that if Method there is a simple path p from s to t in G f , then we can increase flow Residual Networks Flow along each edge in p , thus increasing total flow Augmentation Augmenting Path We call p an augmenting path Max-Flow Min-Cut Theorem The amount of flow we can put on p is p ’s residual capacity: Basic Ford-Fulkerson Algorithm Ford-Fulkerson c f ( p ) = min { c f ( u, v ) : ( u, v ) is on p } Example Analysis of Ford-Fulkerson Edmonds-Karp Algorithm Maximum 13 / 35 Bipartite

  14. Augmenting Path (2) CSCE423/823 Introduction Flow Networks Ford-Fulkerson Method Residual Networks Flow Augmentation Augmenting Path Max-Flow Min-Cut Theorem Basic Ford-Fulkerson Algorithm p is shaded; what is c f ( p ) ? Ford-Fulkerson Example Analysis of Ford-Fulkerson Edmonds-Karp Algorithm Maximum 14 / 35 Bipartite

  15. Augmenting Path (3) CSCE423/823 Lemma: Let G = ( V, E ) be a flow network, f be a flow in G , and p be an augmenting path in G f . Define f p : V ⇥ V ! R as Introduction ⇢ c f ( p ) Flow Networks if ( u, v ) 2 p Ford-Fulkerson f p ( u, v ) = Method 0 otherwise Residual Networks Flow Then f p is a flow in G f with value | f p | = c f ( p ) > 0 Augmentation Augmenting Path Corollary: Let G , f , p , and f p be as above. Then f " f p is a flow in Max-Flow Min-Cut Theorem G with value | f " f p | = | f | + | f p | > | f | Basic Ford-Fulkerson Algorithm Thus, every augmenting path increases flow in G Ford-Fulkerson Example When do we stop? Will we have a maximum flow if there is no Analysis of Ford-Fulkerson augmenting path? Edmonds-Karp Algorithm Maximum 15 / 35 Bipartite

  16. Max-Flow Min-Cut Theorem CSCE423/823 Used to prove that once we run out of augmenting paths, we have a maximum flow Introduction A cut ( S, T ) of a flow network G = ( V, E ) is a partition of V into Flow Networks S ✓ V and T = V \ S such that s 2 S and t 2 T Ford-Fulkerson Net flow across the cut ( S, T ) is Method Residual Networks Flow X X X X f ( S, T ) = f ( u, v ) � f ( v, u ) Augmentation Augmenting Path u 2 S v 2 T u 2 S v 2 T Max-Flow Min-Cut Theorem Capacity of cut ( S, T ) is Basic Ford-Fulkerson Algorithm Ford-Fulkerson X X Example c ( S, T ) = c ( u, v ) Analysis of Ford-Fulkerson u 2 S v 2 T Edmonds-Karp Algorithm A minimum cut is one whose capacity is smallest over all cuts Maximum 16 / 35 Bipartite

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