Graph Algorithms Maximum Flow Applications

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1 Chapter 5 Graph Algorithms Maximum Flow Applications Algorithm Theory WS 202/3 Fabian Kuhn

2 Maximum Flow Applications Maximum flow has many applications Reducing a problem to a max flow problem can even be seen as an important algorithmic technique Examples: related network flow problems computation of small cuts computation of matchings computing disjoint paths scheduling problems assignment problems with some side constraints Algorithm Theory, WS 202/3 Fabian Kuhn 2

3 Undirected Edges and Vertex Capacities Undirected Edges: Undirected edge, : add edges, and, to network Vertex Capacities: Not only edge, but also (or only) nodes have capacities Capacity of node,: Replace node by edge, : Algorithm Theory, WS 202/3 Fabian Kuhn 3

4 Minimum Cut Given: undirected graph,, nodes, cut: Partition, of such that, Size of cut, : number of edges crossing the cut Objective: find cut of minimum size Algorithm Theory, WS 202/3 Fabian Kuhn 4

5 Edge Connectivity Definition: A graph, is edge connected for an integer if the graph, is connected for every edge set,. Goal: Compute edge connectivity of (and edge set of size that divides into 2parts) minimum set is a minimum cut for some, Actually for all, in different components of, Possible algorithm: fix and find min cut for all Algorithm Theory, WS 202/3 Fabian Kuhn 5

6 Minimum Vertex Cut Given: undirected graph,, nodes, vertex cut: Set such that, and and are in different components of the sub graph induced by Size of vertex cut: Objective: find vertex cut of minimum size Replace undirected edge, by, and, Compute max flow for edge capacities and node capacities for, Replace each node by and : Min edge cut corresponds to min vertex cut in Algorithm Theory, WS 202/3 Fabian Kuhn 6

7 Vertex Connectivity Definition: A graph, is vertex connected for an integer if the sub graph induced by is connected for every edge set,. Goal: Compute vertex connectivity of (and node set of size that divides into 2parts) Compute minimum vertex cut for fixed and all Algorithm Theory, WS 202/3 Fabian Kuhn 7

8 Edge Disjoint Paths Given: Graph,with nodes, Goal: Find as many edge disjoint paths as possible Solution: Find max flow in with edge capacities for all Flow induces edge disjoint paths: Integral capacities can compute integral max flow Get edge disjoint paths by greedily picking them Correctness follows from flow conservation Algorithm Theory, WS 202/3 Fabian Kuhn 8

9 Edge Disjoint Paths Flow induces edge disjoint paths: Get edge disjoint paths by greedily picking them Algorithm Theory, WS 202/3 Fabian Kuhn 9

10 Edge Disjoint Paths Flow induces edge disjoint paths: Get edge disjoint paths by greedily picking them Algorithm Theory, WS 202/3 Fabian Kuhn 0

11 Edge Disjoint Paths Flow induces edge disjoint paths: Decompose into one path and a flow of value Algorithm Theory, WS 202/3 Fabian Kuhn

12 Vertex Disjoint Paths Given: Graph,with nodes, Goal: Find as many internally vertex disjoint paths as possible Solution: Find max flow in with node capacities for all Flow induces vertex disjoint paths: Integral capacities can compute integral max flow Get vertex disjoint paths by greedily picking them Correctness follows from flow conservation Algorithm Theory, WS 202/3 Fabian Kuhn 2

13 Menger s Theorem Theorem: (Menger s theorem, edge version) For every graph,with nodes,, the minimum number of edges that need to be removed to disconnect and equals the maximum number of pairwise edge disjoint paths from to. Theorem: (Menger s theorem, node version) For every graph,with nodes,, the minimum number of nodes,that need to be removed to disconnect and equals the maximum number of pairwise internally vertexdisjoint paths from to Both versions can be seen as a special case of the max flow min cut theorem Algorithm Theory, WS 202/3 Fabian Kuhn 3

14 Baseball Elimination Team Wins Losses To Play Against = l NY Balt. T. Bay Tor. Bost. New York Baltimore Tampa Bay Toronto Boston Only wins/losses possible (no ties), winner: team with most wins Which teams can still win (as least as many wins as top team)? Boston is eliminated (cannot win): Boston can get at most 79 wins, New York already has 8 wins If for some, : team is eliminated Sufficient condition, but not a necessary one! Algorithm Theory, WS 202/3 Fabian Kuhn 4

15 Baseball Elimination Team Wins Losses To Play Against = l NY Balt. T. Bay Tor. Bost. New York Baltimore Tampa Bay Toronto Boston Can Toronto still finish first? Toronto can get 82 8 wins, but: NY and Tampa have to play 4 more times against each other if NY wins one, it gets 82 wins, otherwise, Tampa has 82 wins Hence: Toronto cannot finish first How about the others? How can we solve this in general? Algorithm Theory, WS 202/3 Fabian Kuhn 5

16 Max Flow Formulation Can team 3 finish with most wins? Remaining number of games between the 2 teams game nodes 5 team nodes Number of wins team can have to not beat team Team 3 can finish first iff all source game edges are saturated Algorithm Theory, WS 202/3 Fabian Kuhn 6

17 Reason for Elimination AL East: Aug 30, 996 Team Wins Losses To Play Against = l NY Balt. Bost. Tor. Detr. New York Baltimore Boston Toronto Detroit Detroit could finish with wins Consider NY, Bal, Bos, Tor Have together already won 278 games Must together win at least 27more games On average, teams in win games Algorithm Theory, WS 202/3 Fabian Kuhn 7

18 Reason for Elimination Certificate of elimination:,,,, #wins of nodes in #remaining games among nodes in Team is eliminated by if. Algorithm Theory, WS 202/3 Fabian Kuhn 8

19 Reason for Elimination Theorem: Team is eliminated if and only if there exists a subset of the teams such that is eliminated by. Proof Idea: Minimum cut gives a certificate If is eliminated, max flow solution does not saturate all outgoing edges of the source. Team nodes of unsaturated source game edges are saturated Source side of min cut contains all teams of saturated team dest. edges of unsaturated source game edges Set of team nodes in source side of min cut give a certificate Algorithm Theory, WS 202/3 Fabian Kuhn 9

20 Circulations with Demands Given: Directed network with positive edge capacities Sources & Sinks: Instead of one source and one destination, several sources that generate flow and several sinks that absorb flow. Supply & Demand: sources have supply values, sinks demand values Goal: Compute a flow such that source supplies and sink demands are exactly satisfied The circulation problem is a feasibility rather than a maximization problem Algorithm Theory, WS 202/3 Fabian Kuhn 20

21 Circulations with Demands: Formally Given: Directed network, with Edge capacities 0for all Node demands for all 0: node needs flow and therefore is a sink 0: node has a supply of and is therefore a source 0: node is neither a source nor a sink Flow: Function : satisfying Capacity Conditions: : 0 Demand Conditions: : Objective: Does a flow satisfying all conditions exist? If yes, find such a flow. Algorithm Theory, WS 202/3 Fabian Kuhn 2

22 Example Algorithm Theory, WS 202/3 Fabian Kuhn 22

23 Condition on Demands Claim: If there exists a feasible circulation with demands for, then Proof: 0. of each edge appears twice in the above sum with different signs overall sum is 0 Total supply = total demand: Define : : Algorithm Theory, WS 202/3 Fabian Kuhn 23

24 Reduction to Maximum Flow Add super source and super sink to network 0 0 supplies sources with flow siphons flow out of sinks Algorithm Theory, WS 202/3 Fabian Kuhn 24

25 Example Algorithm Theory, WS 202/3 Fabian Kuhn 25

26 Formally Reduction: Get graph from graph as follows Node set of is, Edge set is and edges,for all with 0, capacity of edge is, for all with 0, capacity of edge is Observations: Capacity of min cut is at least (e.g., the cut, A feasible circulation on can be turned into a feasible flow of value of by saturating all,and, edges. Any flow of of value induces a feasible circulation on, and, edges are saturated By removing these edges, we get exactly the demand constraints Algorithm Theory, WS 202/3 Fabian Kuhn 26

27 Circulation with Demands Theorem: There is a feasible circulation with demands, on graph if and only if there is a flow of value on. If all capacities and demands are integers, there is an integer circulation The max flow min cut theorem also implies the following: Theorem: The graph has a feasible circulation with demands, if and only if for all cuts,,,. Algorithm Theory, WS 202/3 Fabian Kuhn 27

28 Circulation: Demands and Lower Bounds Given: Directed network, with Edge capacities 0and lower bounds l for Node demands for all 0: node needs flow and therefore is a sink 0: node has a supply of and is therefore a source 0: node is neither a source nor a sink Flow: Function : satisfying Capacity Conditions: : l Demand Conditions: : Objective: Does a flow satisfying all conditions exist? If yes, find such a flow. Algorithm Theory, WS 202/3 Fabian Kuhn 28

29 Solution Idea Define initial circulation l Satisfies capacity constraints: : l Define l l If 0, demand condition is satisfied at by, otherwise, we need to superimpose another circulation such that Remaining capacity of edge : l We get a circulation problem with new demands, new capacities, and no lower bounds Algorithm Theory, WS 202/3 Fabian Kuhn 29

30 Eliminating a Lower Bound: Example Lower bound of Algorithm Theory, WS 202/3 Fabian Kuhn 30

31 Reduce to Problem Without Lower Bounds Graph, : Capacity: For each edge : l Demand: For each node : Model lower bounds with supplies & demands: Flow: l Create Network (without lower bounds): For each edge : l For each node : Algorithm Theory, WS 202/3 Fabian Kuhn 3

32 Circulation: Demands and Lower Bounds Theorem: There is a feasible circulation in (with lower bounds) if and only if there is feasible circulation in (without lower bounds). Given circulation in, l is circulation in The capacity constraints are satisfied because l Demand conditions: l l Given circulation in, l is circulation in The capacity constraints are satisfied because l Demand conditions: l l Algorithm Theory, WS 202/3 Fabian Kuhn 32

33 Integrality Theorem: Consider a circulation problem with integral capacities, flow lower bounds, and node demands. If the problem is feasible, then it also has an integral solution. Proof: Graph has only integral capacities and demands Thus, the flow network used in the reduction to solve circulation with demands and no lower bounds has only integral capacities The theorem now follows because a max flow problem with integral capacities also has an optimal integral solution It also follows that with the max flow algorithms we studied, we get an integral feasible circulation solution. Algorithm Theory, WS 202/3 Fabian Kuhn 33

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