Graph Algorithms Matching

Transcription

1 Chapter 5 Graph Algorithms Matching Algorithm Theory WS 2012/13 Fabian Kuhn

2 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 2012/13 Fabian Kuhn 2

3 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 2012/13 Fabian Kuhn 3

4 Matrix Rounding Given: matrix, of real numbers row sum:,, column sum:, Goal: Round each,, as well as and up or down to the next integer so that the sum of rounded elements in each row (column) equals the rounded row (column) sum Original application: publishing census data Example: original data possible rounding Algorithm Theory, WS 2012/13 Fabian Kuhn 4

5 Matrix Rounding Theorem: For any matrix, there exists a feasible rounding. Remark: Just rounding to the nearest integer doesn t work original data rounding to nearest integer feasible rounding Algorithm Theory, WS 2012/13 Fabian Kuhn 5

6 Reduction to Circulation Matrix elements and row/column sums give a feasible circulation that satisfies all lower bound, capacity, and demand constraints rows: 3,4 columns: 12,13 2,3 10,11 all demands 0 Algorithm Theory, WS 2012/13 Fabian Kuhn 6

7 Matrix Rounding Theorem: For any matrix, there exists a feasible rounding. Proof: The matrix entries, and the row and column sums and give a feasible circulation for the constructed network Every feasible circulation gives matrix entries with corresponding row and column sums (follows from demand constraints) Because all demands, capacities, and flow lower bounds are integral, there is an integral solution to the circulation problem gives a feasible rounding! Algorithm Theory, WS 2012/13 Fabian Kuhn 7

8 Matching Algorithm Theory, WS 2012/13 Fabian Kuhn 8

9 Gifts Children Graph Which child likes which gift can be represented by a graph Algorithm Theory, WS 2012/13 Fabian Kuhn 9

10 Matching Matching: Set of pairwise non incident edges Maximal Matching: A matching s.t. no more edges can be added Maximum Matching: A matching of maximum possible size Perfect Matching: Matching of size (every node is matched) Algorithm Theory, WS 2012/13 Fabian Kuhn 10

11 Bipartite Graph Definition: A graph, is called bipartite iff its node set can be partitioned into two parts such that for each edge u, v,, 1. Thus, edges are only between the two parts Algorithm Theory, WS 2012/13 Fabian Kuhn 11

12 Santa s Problem Maximum Matching in Bipartite Graphs: Every child can get a gift iff there is a matching of size #children Clearly, every matching is at most as big If #children #gifts, there is a solution iff there is a perfect matching Algorithm Theory, WS 2012/13 Fabian Kuhn 12

13 Reducing to Maximum Flow Like edge disjoint paths all capacities are Algorithm Theory, WS 2012/13 Fabian Kuhn 13

14 Reducing to Maximum Flow Theorem: Every integer solution to the max flow problem on the constructed graph induces a maximum bipartite matching of. Proof: 1. An integer flow of value induces a matching of size Left nodes (gifts) have incoming capacity 1 Right nodes (children) have outgoing capacity 1 Left and right nodes are incident to 1edge of with 1 2. A matching of size implies a flow of value For each edge, of the matching:,,, 1 All other flow values are 0 Algorithm Theory, WS 2012/13 Fabian Kuhn 14

15 Running Time of Max. Bipartite Matching Theorem: A maximum matching of a bipartite graph can be computed in time Algorithm Theory, WS 2012/13 Fabian Kuhn 15

16 Perfect Matching? There can only be a perfect matching if both sides of the partition have size. There is no perfect matching, iff there is an cut of size in the flow network. 2 2 Algorithm Theory, WS 2012/13 Fabian Kuhn 16

17 Cuts Partition, of node set such that and If : edge,is in cut, If : edge, is in cut, Otherwise (if, ), all edges from to some are in cut, Algorithm Theory, WS 2012/13 Fabian Kuhn 17

18 Hall s Marriage Theorem Theorem: A bipartite graph,for which has a perfect matching if and only if :, where is the set of neighbors of nodes in. Proof: No perfect matching some cut has capacity 1. Assume there is for which U : Algorithm Theory, WS 2012/13 Fabian Kuhn 18

19 Hall s Marriage Theorem Theorem: A bipartite graph,for which has a perfect matching if and only if :, where is the set of neighbors of nodes in. Proof: No perfect matching some cut has capacity 2. Assume that there is a cut, of capacity Algorithm Theory, WS 2012/13 Fabian Kuhn 19

20 Hall s Marriage Theorem Theorem: A bipartite graph,for which has a perfect matching if and only if :, where is the set of neighbors of nodes in. Proof: No perfect matching some cut has capacity 2. Assume that there is a cut, of capacity Algorithm Theory, WS 2012/13 Fabian Kuhn 20

21 What About General Graphs Can we efficiently compute a maximum matching if is not bipartitie? How good is a maximal matching? A matching that cannot be extended Vertex Cover: set of nodes such that,,,. A vertex cover covers all edges by incident nodes Algorithm Theory, WS 2012/13 Fabian Kuhn 21

22 Vertex Cover vs Matching Consider a matching and a vertex cover Claim: Proof: At least one node of every edge, is in Needs to be a different node for different edges from Algorithm Theory, WS 2012/13 Fabian Kuhn 22

23 Vertex Cover vs Matching Consider a matching and a vertex cover Claim: If is maximal and is minimum, 2 Proof: is maximal: for every edge,, either or (or both) are matched Every edge is covered by at least one matching edge Thus, the set of the nodes of all matching edges gives a vertex cover of size 2. Algorithm Theory, WS 2012/13 Fabian Kuhn 23

24 Maximal Matching Approximation Theorem: For any maximal matching and any maximum matching, it holds that Proof: 2. Theorem: The set of all matched nodes of a maximal matching is a vertex cover of size at most twice the size of a min. vertex cover. Algorithm Theory, WS 2012/13 Fabian Kuhn 24

25 Augmenting Paths Consider a matching of a graph,: A node is called free iff it is not matched Augmenting Path: A (odd length) path that starts and ends at a free node and visits edges in and edges in alternatingly. free nodes alternating path Matching can be improved using an augmenting path by switching the role of each edge along the path Algorithm Theory, WS 2012/13 Fabian Kuhn 25

26 Augmenting Paths Theorem: A matching of,is maximum if and only if there is no augmenting path. Proof: Consider non max. matching and max. matching and define, Note that and Each node is incident to at most one edge in both and induces even cycles and paths Algorithm Theory, WS 2012/13 Fabian Kuhn 26

27 Finding Augmenting Paths free nodes augmenting path odd cycle Algorithm Theory, WS 2012/13 Fabian Kuhn 27

28 Blossoms If we find an odd cycle free node Graph Matching stem root contract blossom contracted blossom Graph blossom Matching, is a matching of. Algorithm Theory, WS 2012/13 Fabian Kuhn 28

29 Contracting Blossoms Lemma: Graph has an augmenting path w.r.t. matching iff has an augmenting path w.r.t. matching Note: If stem has length 0, root of blossom if free and thus also the node is free in. Also: The matching can be computed efficiently from. Algorithm Theory, WS 2012/13 Fabian Kuhn 29

30 Edmond s Blossom Algorithm Algorithm Sketch: 1. Build a tree for each free node 2. Starting from an explored node at even distance from a free node in the tree of, explore some unexplored edge, : 1. If is an unexplored node, is matched to some neighbor : add to the tree ( is now explored) 2. If is explored and in the same tree: at odd distance from root ignore and move on at even distance from root blossom found 3. If is explored and in another tree at odd distance from root ignore and move on at even distance from root augmenting path found Algorithm Theory, WS 2012/13 Fabian Kuhn 30

31 Running Time Finding a Blossom: Repeat on smaller graph Finding an Augmenting Path: Improve matching Theorem: The algorithm can be implemented in time. Algorithm Theory, WS 2012/13 Fabian Kuhn 31

32 Matching Algorithms We have seen: time alg. to compute a max. matching in bipartite graphs time alg. to compute a max. matching in general graphs Better algorithms: Best known running time (bipartite and general gr.): Weighted matching: Edges have weight, find a matching of maximum total weight Bipartite graphs: flow reduction works in the same way General graphs: can also be solved in polynomial time (Edmond s algorithms is used as blackbox) Algorithm Theory, WS 2012/13 Fabian Kuhn 32

33 Happy Holidays! Algorithm Theory, WS 2012/13 Fabian Kuhn 33