Matching. Algorithms and Networks

Transcription

1 Matching Algorithms and Networks

2 This lecture Matching: problem statement and applications Bipartite matching (recap) Matching in arbitrary undirected graphs: Edmonds algorithm Diversion: generalized tic-tac-toe 2

3 1 Problem and applications

4 Matching Set of edges MM EE such that no vertex is endpoint of more than one edge. Maximal matching No ee EE with MM ee also a matching Maximum matching Matching with MM as large as possible Perfect matching MM = nn/2: each vertex endpoint of edge in MM. 4

5 Matching Not a maximal matching, Not a maximum matching Not a perfect matching 5

6 Maximum Matching A maximal matching, A maximum matching Not a perfect matching (Size 4) 6

7 Maximal Matching A maximal matching, Not a maximum matching Not a perfect matching (Size: 3) 7

8 Perfect Matching A maximal matching, A maximum matching A perfect matching (Size 4) 8

9 Cost versions Each edge has cost; look for perfect matching with minimum cost Also polynomial time solvable, but harder 9

10 Problems Given graph GG, find Maximal matching: easy (greedy algorithm) Maximum matching Polynomial time; not easy. Important easier case: bipartite graphs Perfect matching Special case of maximum matching A theorem for regular bipartite graphs and Schrijver s algorithm 10

11 Applications (1) Classroom assignment Scheduling Opponents selection for sport competitions In a chess (or tennis, or ) club, we have n players, that must be paired to play a match, each week. Some players have played already to each other, and we have rating differences. 11

12 Applications (2) Personnel assignment (bipartite graphs): We have n employees and n jobs. Each employee i has a proficiency index u(i,j) for job j. What assignment maximizes the total proficiency? Our company is international. We have a number of jobs abroad, and a number of employees. For each combination of a job and employee, we have: The information: can this employee do this job? The cost of moving the employee with his family to the new location 12

13 Application (3): matching Moving objects, seen at two successive time moments Which object came from where? moving objects 13

14 Application (3): matching moving objects Moving objects, seen at two successive time moments Which object came from where? Maybe 14

15 2 Bipartite matching (recap)

16 Bipartite graphs: using maximum flow algorithms Finding maximum matching in bipartite graphs: Model as flow problem, and solve it: make sure algorithm finds integral flow. Capacities 1 s t 16

17 Technique works for variants too Minimum cost perfect matching in bipartite graphs Model as min-cost flow problem b-matchings in bipartite graphs Function bb: VV N. Look for set of edges MM, with each vv endpoint of exactly bb(vv) edges in MM. 17

18 Steps by Ford-Fulkerson on the M-augmenting path: bipartite graph unmatched becomes in a flow augmentation step 18

19 A simple non-constructive proof of a well known theorem Theorem. Each regular bipartite graph has a perfect matching. Proof: Construct flow model of G. Set flow of edges from s, or to t to 1, and other edges flow to 1/d. This flow has value n/2, which is optimal. Ford-Fulkerson will find integral flow of value n/2; which corresponds to perfect matching. 19

20 3 Edmonds algorithm: matching in (possibly non-bipartite) undirected graphs

21 A theorem that also works when the graph is not bipartite Theorem. Let MM be a matching in graph GG. MM is a maximum matching, if and only if there is no MMaugmenting path. If there is an MM-augmenting path, then MM is not a maximum matching. Suppose MM is not a maximum matching. Let NN be a larger matching. Look at NN MM = NN MM (NN MM). Every node in NN MM has degree 0, 1, 2: collection of paths and cycles. All cycles alternatingly have edge from NN and from MM. There must be a path in NN MM with more edges from NN than from M: this is an augmenting path. 21

22 Algorithm of Edmonds Finds maximum matching in a graph in polynomial time 22

23 Jack Edmonds 23

24 Jack Edmonds 24

25 Definitions M-alternating walk: (Possibly not simple) path with edges alternating in M, and not M. M-flower M-alternating walk of odd length that starts in an unmatched vertex, and ends as an odd cycle: 25 M-blossom

26 Finding an M-augmenting walk Subroutine: given G and M, find an M- augmenting walk: Let X be the set of unmatched vertices. Let Y be the set of vertices adjacent to some vertex in X. Build digraph D = (V,A) with uu, vv there is an xx with uu, xx EE MM and xx, vv MM}. Find a shortest path P from a vertex in X to a vertex in Y in D. Transform P to a path P in G Take P : P, followed by an edge to X. P is M-alternating walk between two unmatched vertices. 26

27 Finding M-augmenting path or M-flower Look at the path P we just found. Two cases: P is a simple path: it is an M-augmenting path P is not simple. Look to start of P until the first time a vertex is visited for the second time. This is an M-flower: Cycle-part of walk cannot be of even size, as it then can be removed and we have a shorter walk in D. 27

28 Algorithmic idea Start with some matching M, and find either M-augmenting path or M-blossom. If we find an M-augmenting path: Augment M, and obtain matching of one larger size; repeat. If we find an M-blossom, we shrink it, and obtain an equivalent smaller problem; recurs. 28

29 Shrinking M-blossoms Let B be a set of vertices in G. G/B is the graph, obtained from G by contracting B to a single vertex. Contracted Blossom 29

30 Theorem Theorem: Let B be an M-blossom. Then M has an augmenting path iff M/B has an augmenting path in G/B. Suppose G/B has an M/B-augmenting path P. P does not traverse the vertex representing B: P also M-augmenting path in G. P traverses B: lift to M-augmenting path in G (case analysis). 30

31 Case 1a: P passes through B M/B-Augmenting Path in G/B M-Augmenting Path in G 31

32 Case 1b: P passes through B M/B-Augmenting Path in G/B Q: does path always pass through stem? 32

33 Case 2a: P has B as endpoint Q: when does this case occur? 33

34 Case 2b: P has B as endpoint 34

35 Converse Flip stem so that B is unmatched to obtain M. Suppose G has M -augmenting path P. If P does not intersect B then P also M /B-augmenting path in G/B. Otherwise, assume P does not start in B, if not reverse P. Since B is free, the part of P until B is an M /Baugmenting path. 35

36 Subroutine Given: Graph G, matching M Question: Find M-augmenting path if it exists. Let X be the vertices not endpoint of edge in M. Build D, and test if there is an M-alternating walk P from X to X of positive length. (Using Y, etc.) If no such walk exists: M is maximum matching. If P is a path: output P. If P is not a path: Find M-blossom B on P. Shrink B, and recurse on G/B and M/B. If G/B has no M/B-augmenting path, then M is maximum matching. Otherwise, expand M/B-augmenting path to an M-augmenting path. 36

37 Edmonds algorithm A maximum matching can be found in O(n 2 m) time. Start with empty (or any) matching, and repeat improving it with M-augmenting paths until this stops. O(n) iterations. Recursion depth is O(n); work per recursive call O(m). A perfect matching in a graph can be found in O(n 2 m) time, if it exists. 37

38 Improvements Better analysis and data structures gives O(n 3 ) algorithm. Faster is possible: O(n 1/2 m) time. Minimum cost matchings with more complicated structural ideas. 38

39 5 Diversion: multidimensional tic-tactoe

40 Trivial drawing strategies in multidimensional tic-tac-toe Tic-tac-toe Generalizations More dimensions Larger board size Who has a winning strategy? Either first player has winning strategy, or second player has drawing strategy 40

41 Trivial drawing strategy If lines are long enough: pairing of squares such that each line has a pair If player 1 plays in a pair, then player 2 plays to other square in pair v i a a f j b h u b c i g c d u h d f j e e g v 41

42 Trivial drawing strategies and generalized matchings Bipartite graph: line-vertices and square-vertices; edge when square is part of line Look for set of edges M, such that: Each line-vertex is incident to two edges in M Each square-vertex is incident to at most one edge in M There exists such a set of edges M, if and only if there is a trivial drawing strategy (of the described type). 42

43 Consequences Testing if trivial drawing strategy exists and finding one if so can be done efficiently (flow algorithm). n by n by by n tic-tac-toe (d-dimensional) has a trivial drawing strategy if n is at least 3 dd 1 A square belongs to at most 3 dd 1 lines. So, if n is at least 3 dd 1 then line-square graph has an edge coloring with n colors. Let M be the set of edges with colors 1 and 2. 43

44 6 Conclusions

45 Conclusion Many applications of matching! Often bipartite Algorithms for finding matchings: Bipartite: flow models Bipartite, regular: Schrijver (See course Algorithms for decision support) General: with M-augmenting paths and blossomshrinking Minimum cost matching can also be solved in polynomial time: more complex algorithm Min cost matching on bipartite graphs is solved using min cost flow 45