11/21/02CSE Max Flow CSE Algorithms Max Flow Problems

CSE Max Flow2 Max Flow problem Input: a directed graph with a non-negative weight (capacity) on each edge, and a designated source node s and sink node t. A flow assigns to each edge a number (from 0 to its capacity) such that for each node except s and t, the sum of the flow into the node equals sum of the flow out. [The fine print: this is slightly different from the book’s definition.] The value of a flow is the sum of the flow out of the source (which equals total flow into the sink). Maximum-Flow problem: find a flow with the maximum possible value. Note: A max flow can assign 0 to each edge going into the source, and to each edge going out of the sink.

CSE Max Flow3 Example b a d s t c source sink b a d s t c A flow of value 18 This is wasteful! The text’s formulation doesn’t even let it happen.

CSE Max Flow4 Leftover capacity b a d s t c source sink b a d s t c Residual network New backedge showing we can reduce forward flow by 10

CSE Max Flow5 Augmenting Path b a d s t c New residual network We can push 5 more through network on indicated path (New flow has value 23.) b a d s t c Now we’re stuck! We can’t get any more across cut.

CSE Max Flow6 Duality: Max Flow = Min Cut Original problem had a cut of value 23. We can’t possibly get more from source to sink. Theorem: Max flow = min cut Proof: Obviously, any flow  min cut. But if max flow < min cut, there would be an augmenting path from source to sink, leading to a higher-valued flow. This is a contradiction. Thus, max flow = min cut. QED. b a d s t c

CSE Max Flow7 Ford-Fulkerson Methods initialize flow to 0; while (there’s an augmenting path){ update flow; compute new residual graph; } If there are several augmenting paths, does it matter which we pick?? b a s t

CSE Max Flow8 Edmonds-Karp Algorithm Always choose an augmenting path with as few edges as possible (say it has L edges). –This might create a new backedge, e.g (v,u). –This new edge can’t be in a new path of L edges.. (Handle multiple new backedges by induction proof.) –Meanwhile, at least one original edge has been eliminated. –Thus, there are at most E iterations using L long paths. u v s j long path t k long path u v s m long t n long j+1+k is a minimal length path (L) m  j+1 and n  k+1 so m+n > j+k+1 = L

CSE Max Flow9 Edmonds-Karp Algorithm Always choose an augmenting path with as few edges as possible. –At most E iterations use L-edge augmenting path –Longest augmenting path has V-1 edges. –Thus, there are at most O(VE) augmentations. –How long does it take to find and process a shortest augmenting path??

CSE Max Flow10 Max Flow Algorithms Edmonds-Karp is O(VE 2 ) Some faster algorithms based on “Push-Relabel”. –E.g. Goldberg’s algorithm is O(V 2 E) Start source at height V, all others at 0 Add in flow from higher nodes to lower ones. If a non-sink node can’t push all its incoming flow out, increase it’s height. When done, return excess back to source. –Carefully choosing order gives O(V 3 ) algorithm. Fastest known algorithm is O( min(V 2/3, E 1/2 ) E lg(1+V 2 /E) lg(max capacity) ).

CSE Max Flow11 An Application Maximum matching problem: –Given an (undirected) graph (V,E) find a maximum-sized set of disjoint edges. Not the same as a maximal set of disjoint edges. –Amazingly, not NP-complete (but it’s not easy) Bipartite graph: Graph such that you can partition the nodes V = V 1 U V 2, and every edge goes between a node of V 1 and one of V 2. Maximum matching for a bipartite graph can be reduced to a max flow problem.

CSE Max Flow12 Glossary (in case symbols are weird)            subset  element of  infinity  empty set  for all  there exists  intersection  union  big theta  big omega  summation  >=  <=  about equal  not equal  natural numbers(N)  reals(R)  rationals(Q)  integers(Z)