Max Flow: Algorithms and Applications

Transcription

1 Max Flow: Algorithms and Applications

2 Outline for today Quick review of the Max Flow problem Applications of Max Flow The Ford-Fulkerson (FF) algorithm for Max Flow Analysis of FF, and the Max Flow Min Cut Theorem

3 Review: Flow Network A (directed) flow network G=(V,E) has the following properties: A source node s with no incoming edges A sink node t with no outgoing edges A nonnegative integer capacity c(e) for each edge e. Assume that G has no isolated nodes. Then, if V =n and E = m, n m. Also assume that if (u,v) is in G then (v,u) is not in G. (If both are in G, replace (v,u) by (v,z), (z,u) where z is a new node, and let (v,z), (z,u) have the capacity c(v,u).

4 Review: (s,t)-flow An (s-t)-flow f that maps edges of G to nonnegative real numbers, with the following properties: Capacity condition: 0 f(e) c(e) Conservation condition: For any internal (i.e., not source or sink) node v, flow into v = flow out of v: e into v f(e) = e out of v f(e). The value of f is v(f) = e out of s f(e)

5 The Max Flow Problem Given a flow network, find a flow of maximum value. Today we ll look at an efficient algorithm for Max Flow, that finds an integral max flow. That is, the flow along any edge is an integer.

6 The Max Flow Problem

7 The Max Flow Problem Map of rail network linking Russia to Eastern Europe from the mid-1950 s. Weights represent the rate at which material could be shipped from one region to the next. Created by Theodore E. Harris and Frank S. Ross. From a survey of Alexander Schriver, and also in Erickson s notes (online).

8 An Application of Max Flow An efficient algorithm for Max Flow that produces an integral flow can also be used to find a maximum bipartite matching of a bipartite graph. Bipartite matching and its generalizations (when nodes and/or edges have weights) have applications in resource allocation: scheduling jobs to machines, cell phones to cell towers, etc. Exercise: How to use a Max Flow algorithm to find maximum bipartite matchings?

9 Reducing Maximum Bipartite Matching to Max Flow G = (U,V,E) à (G' = ({s,t} U V, E'), c()) where E' contains edges (s,u) for each u in U (v,t) for each v in V (u,v) for each {u,v} in E and the capacity c(e) = 1 for all e in E'.

10 Reducing Maximum Bipartite Matching to Max Flow From Jeff Erickson s notes

11 Towards an Algorithm for Max Flow

12 Towards an Algorithm for Max Flow We need to define: Residual graphs Augmenting flows

13 Residual Graphs The residual graph G f has node set V and for each edge e = (u,v) of E, If f(e) < c(e), then e = (u,v) is a "forwards" edge of G f, with residual capacity c f (e) = c(e)-f(e). If f(e) > 0, then e' = (v,u) is a "backwards" edge of G f, with residual capacity c f (e') = f(e). Note that all residual capacities are positive.

14 Augmenting Flows Let P be a simple s-t path in G f. Let bottleneck(p, f) be the minimum residual capacity on the edges of P. augment(f, P) // yields a new flow f' in G b := bottleneck(p, f) for each edge e = (u,v) in P IF (u,v) is a forwards edge THEN increase f(e) in G by b ELSE // e is a backwards edge decrease f(v,u) in G by b

15 Properties of f = augment(f, P) f' satisfies the capacity constraints on all edges of G f' satisfies the conservation constraints on all internal nodes of G f' is a flow of G, and v(f') > v(f) augment(f, P) can be implemented in O(m) time

16 The Ford-Fulkerson Algorithm (circa 1956) Exercise: With the augment procedure in hand, can you design an algorithm for Max Flow?

17 The Ford-Fulkerson Algorithm (circa 1956) FF(G) // given a flow network G, return its max (s,t)-flow initialize f(e)=0 for all e in G WHILE there is a simple s-t path P in in G f. f = augment(f, P) ENDWHILE return f

18 Termination and Running Time of FF Exercise: Can you explain why FF terminates? Can bound the running time of FF?

19 Termination and Running Time of FF Let C be the sum of the capacities of edges out of s. Claim: FF terminates within C iterations of the while loop, and thus runs in O(m C) time. Proof: On each iteration of FF, the flow f increases, and all flow values f(e) are integers. Moreover the max flow is bounded by C. Thus, the while loop executes at most C times. Each iteration of the loop can be implemented in O(m) time: constructing G f, finding P, augmenting the flow.

20 FF Produces a Max Flow To show this, we ll take one more detour to introduce (s,t)-cuts.

21 (s,t)-cuts and Min Cuts An (s,t)-cut is a pair (S, T) that partitions G's vertices, with s in S and t in T. (Note that S T = V, and S T =.) The capacity of (S,T) is c(s, T) = v in S w in T c(v,w). A min cut is an (s,t)-cut for which c(s, T) is minimum.

22 Relating Flows and Cuts Claim: The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) this is the flow into s, which is 0

23 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s)

24 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) )

25 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) ) this is just the previous line, plus the sum of flows in and out of internal nodes which is 0 by conservation of flow

26 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) )

27 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) ) = v in S ( w in T f(v,w) - u in T f(u,v) )

28 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) ) = v in S ( w in T f(v,w) - u in T f(u,v) ) we throw out all the terms that cancel in the previous line

29 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) ) = v in S ( w in T f(v,w) - u in T f(u,v) )

30 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) ) = v in S ( w in T f(v,w) - u in T f(u,v) ) v in S w in T f(v,w)

31 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) ) = v in S ( w in T f(v,w) - u in T f(u,v) ) v in S w in T f(v,w) since flow is always positive

32 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) ) = v in S ( w in T f(v,w) - u in T f(u,v) ) v in S w in T f(v,w)

33 Relating Flows and Cuts Claim (cut bound on flow): The value of any flow f is at most the capacity of any cut (S, T). Proof: v(f) = w f(s,w) = w f(s,w) - u f(u,s) = v in S ( w f(v,w) - u f(u,v) ) = v in S ( w in T f(v,w) - u in T f(u,v) ) v in S w in T f(v,w) v in S w in T c(v,w) = c(s, T)

34 Relating Flows and Cuts A corollary of the cut bound on flow claim is that v(f) = c(s,t) if and only if f saturates every edge from S to T and avoids every edge from T to S. Moreover, if we have a flow f and a cut (S, T ) that satisfies this equality condition, f must be a maximum flow, and (S, T ) must be a minimum cut.

35 Relating Flows and Cuts A corollary of the cut bound on flow claim is that v(f) = c(s,t) if and only if f saturates every edge from S to T and avoids every edge from T to S. Moreover, if we have a flow f and a cut (S, T ) that satisfies this equality condition, f must be a maximum flow, and (S, T ) must be a minimum cut. (Why? If f is not a max flow, then there is some flow f' with v(f') > c(s,t), contradicting the previous claim. Similarly, if (S,T) is not a min cut, then there is some smaller cut (S',T') with v(f) > c(s,t), again contradicting the previous claim.)

36 The Max Flow Min Cut Theorem Theorem (max flow min cut): In any flow network the value of the max flow is equal to the capacity of the min cut. Proof: Let f be a max flow. We will find a cut (S,T) such that f saturates all edges from S to T, and avoids all edges from T to S. By the previous corollary, v(f) = c(s,t), and (S,T) must be a min cut.

37 Summary: Max Flow The Max Flow problem has many applications in transportation of goods, matching, resource allocation and more. The Ford-Fulkerson (FF) algorithm for Max Flow uses an augmenting paths approach

38 Summary: Max Flow Correctness of FF relies on the Max Flow Min Cut Theorem, the max flow min cut theorem is an example of duality, where two seemingly different optimization problems one a maximization problem and the other a minimization problem have the same solution Next time: Application of Min Cut, and a faster variant of FF