Outline. 1 The matching problem. 2 The Chinese Postman Problem

Transcription

1 Outline The matching problem Maximum-cardinality matchings in bipartite graphs Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm 2

2 Let G = (V, E) be an undirected graph. Definition: and M E is a Matching in G if the degree of each node in (V, M) is, at the most,. (no node of G is incident with more than one edge in M) M is a Perfect Matching in G, if the degree of each node in (V, M) is exactly. (each node of G is incident with exactly one edge in M)

3 G = (V, E) Matching Not a Matching

4 The Matching Problem Given an undirected graph G = (V, E) with weights on the edges, to find a matching in G of maximal (or minimal) weight. Maximum-cardinality matching (all weights are ) Maximum(minimum)-weight matching (general weights) Note: Generalization to b-matching (uncapacitated or capacitated).

5 G = (V, E) Maximum cardinality matching Maximum weight matching

6 Maximum-cardinality matchings in bipartite graphs Maximum-cardinality matchings in bipartite graphs G = (V, E) is bipartite if V = V V 2 with E = δ(v ). Alternatively: G bipartite G does not contain odd cycles. Definition: Let M E a matching and P E a path in G. v V is exposed (wrt M) if it is not incident with any matching edge. P is an alternating path (wrt M) if matching and nonmatching edges alternate in P. P is augmenting (wrt M) if P alternating and its two endnodes are exposed. When an alternating path P exists, a larger matching can be obtained by replacing the matching edges of P with the nonmatching edges of P.

7 Maximum-cardinality matchings in bipartite graphs Maximum-cardinality matchings in bipartite graphs Alternating path Augmenting path Larger matching

8 Maximum-cardinality matchings in bipartite graphs Maximum-cardinality matchings in bipartite graphs Proposition: A matching M is optimal if and only if there does not exist any augmenting path (wrt M).

9 Maximum-cardinality matchings in bipartite graphs Maximum-cardinality matchings in bipartite graphs M optimal there exists no augmenting path (wrt M) Proof: = : If an alternating path exists, a matching M, M = M + can be obtained. =: Suppose M not maximal. Let M be a matching with M = M +. Consider the symmetric difference F = M M = (M M ) \ (M M ). F = M + M 2 M M = 2 M + 2 M M F is odd. F contains an augmenting path. There are 3 possible cases for a node v in (V, F ): v has degree 0: v is exposed in M and in M (otherwise v is incident to some edge in M M ). v has degree : v is incident to an edge of M \ M or to a matching edge of M \ M. v has degree 2: v is incident to an matching edge of M \ M and to a matching edge of M \ M.

10 Maximum-cardinality matchings in bipartite graphs Maximum-cardinality matchings in bipartite graphs Proof: = (Cont): Therefore, (V, F ) contains: Isolated nodes. Even cycles (where edges of M and edges of M alternate). Paths where edges of M and edges of M alternate. Since F is odd, an odd alternating path must exist. Since M > M, one such alternating path must be augmenting wrt M.

11 Maximum-cardinality matchings in bipartite graphs Maximum-cardinality matchings in bipartite graphs M M (V, F) Alternating path

12 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm The augmenting path algorithm Idea of the algorithm For a given pair u, v V with u exposed. The alternating paths that might exist from u to v are either all even or all odd. Grow tree of alternating paths rooted at an exposed node. Scanned nodes are classified in labeled and unlabeled Labels have two components: First component: Even (E) or Odd (O). E label means that there exists an even alternating path from the root of the tree to the node. O label means that there exists an even alternating path from the root of the tree to the node. Second component: predecessor of node in alternating path.

13 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm MaximumBipartiteMatching(G) () Let M be a Matching, all nodes are unlabeled and not scanned. (2) Optimality test. If all exposed nodes are labeled, STOP (The current Matching M is maximal) Otherwise, chose an unlabeled exposed node r. Label r with (E, ). (3) Grow an alternating tree. Chose a labeled unscanned node i. (3.) If i is even, label all unlabeled neighbors of i (in G) with (O, i). Declare i as scanned. Go to (4). (3.2) If i is odd and not exposed, label the node connected with i in the matching with (E, i). Declare i as scanned. Go to (4). (3.3) If i is odd and exposed, go to (5). (4) If a labeled unscanned node exists, go to (3), otherwise go to (2). (5) Augmenting path identification. An augmenting path from r to i has been found (it can be used to find a larger matching). Use the second component of the labels to trace the augmenting path. Let M denote the new Matching. Delete all labels, declare all nodes as unscanned, go to (2).

14 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm Runtime of the augmenting path algorithm The runtime of the algorithm is O(VA). With more elaborate techniques it can be implemented in O( VA).

15 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm The augmenting path algorithm: Example Consider the following graph and matching of cardinality The tree of alternating paths after the first iteration of the algorithm is (E, -) 5 (O, 5) 7 (O, 5) 8 (E, 7) 2 (E, 8) 3 (O, 3) 6 (O, 5) 0 (E, 0) 4 9 (O, 4) The augmenting path (5, 8, 3, 6) is found, which allows to obtain the following cardinality 4 matching:

16 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm

17 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm The algorithm does not find any augmenting path algorithm: 9 (E, -) (E, -) (O, ) (E, 7) 7 2 (O, 2) 8 3 (O, 9) 4 (E, 0) 0 (O, 5) 5 (E, 8) 6 (E, 3) Thus, the current matching is optimal

18 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm Correctness of the augmenting path algorithm The algorithm terminates with a matching M 0 and set of labeled trees T i = (V i, E i ), i =, s and a set of unlabeled nodes V s. (V s, E s ), contains a perfect matching. (No nodes in V s are exposed). (Vs, Vs 2 ) bipartition of V s with E s = δ(vs ). Vi 0 V i odd nodes of T i, i =,, s. W = s i= V i 0 Vs. Each e s i= E i is incident to a node in W. In (V i, E i ) there cannot be an edge joining two even nodes. (there would be an odd cycle) There cannot be an edge joining even nodes in different trees. (one of these nodes would have been labeled from the other) There cannot be an edge joining an even node and an unlabeled node. (the unlabeled node would have been labeled from the even node)

19 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm Correctness of the augmenting path algorithm V O O E T= (V, E) E O E V s O E V O 2 O Gs= (Vs, Es) E T2= (V2, E2) E E O s W = E i M 0 = M 0 i= Each element of W is incident to exactly one edge of M 0. Each edge of E is incident to exactly one element of W.

20 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm Correctness of the augmenting path algorithm LP formulation of matching problem without integrality constraints. max s.t. e E x e x e for all v V min s.t. v V π v π v for all e E e δ(v) 0 x e for all e E v:e δ(v) 0 π v for all v V For general graphs, the associated polytope may have non-integer vertices. For bipartite graphs, all the vertices of the associated polytope are integer. (In the bipartite case the above model is an exact formulation)

21 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm Correctness of the augmenting path algorithm LP formulation of matching problem without integrality constraints. max s.t. e E x e x e for all v V min s.t. v V π v π v for all e E e δ(v) 0 x e for all e E v:e δ(v) 0 π v for all v V For general graphs, the associated polytope may have non-integer vertices. For bipartite graphs, all the vertices of the associated polytope are integer. (In the bipartite case the above model is an exact formulation)

22 Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm Correctness of the augmenting path algorithm Define the pair of solutions to LP and the dual problem, respectively: x e = {, e M 0 0, e / M 0, π v = {, v W 0, i / W, By construction these solutions are feasible for their respective problems. Further they take the same objective function value.

23 2 3 4 (O, ) 2 (O, ) 2 (E, -) (E, -) 3 (E, 2) 4 (O, 3) 3 (O, ) 4 When G = (V, E) is not bipartite... The labels assigned to the nodes are no longer unique. The augmenting path algorithm may fail to identify an augmenting path even if it exists.

24 max s.t. e E x e e δ(v) x e for all v V 0 x e for all e E When G = (V, E) is not bipartite the LP formulation does not necessarily give an integer solution Consider G = K 3. The optimal solution to the LP formulation is ( 2, 2, 2 ). The polytope is not the convex hull of incidence vectors associated with matchings. The LP formulation is not enough. The inequality x 2 + x 3 + x 23 is needed (at most one of the tree edges can be used).

25 max s.t. e E x e e δ(v) e E(U) x e x e U 2 v V 0 x e, e E U V, U 3, U odd, min s.t. π v + v V v:e δ(v) U V U 3, U odd π v + U:e E(U) U y U 2 y U e E 0 π v v V 0 y U U V, U 3, U odd

26 max s.t. e E x e e δ(v) e E(U) x e x e U 2 v V 0 x e, e E U V, U 3, U odd, min s.t. π v + v V v:e δ(v) U V U 3, U odd π v + U:e E(U) U y U 2 y U e E 0 π v v V 0 y U U V, U 3, U odd

27 Detecting odd circuits A nonmatching edge connects to even nodes. The two branches of the tree meet at an even node. In principle each node of the odd circuit could be labeled as even (because there are two possible alternating paths, one even and one odd). E O O E E O O O E E Relative to M, M U = U 2. We shrink the odd cycle into a pseudonode and we mark it as even E

28 Detecting odd circuits A nonmatching edge connects to even nodes. The two branches of the tree meet at an even node. In principle each node of the odd circuit could be labeled as even (because there are two possible alternating paths, one even and one odd). E O O E E O O O E E Relative to M, M U = U 2. We shrink the odd cycle into a pseudonode and we mark it as even E

29 Let U V, U 3 U odd with M U = U 2 The subgraph (U, E(U)) is called blossom relative to M. The common node, b(u) is denoted base of the blossom. It is either the root of the tree. or adjacent to a matching edge in the tree. Then we shrink the odd cycle into a pseudonode and we mark it as even E O E O E

30 Shrinking a blossom The blossom (U, E(U)) is replaced by the new pseudonode B(U). If v V \ U is connected with a node of U, then connect v with B(U). Matching and nonmatching edges retain their status. B(U) inherits the label of b(u) Each node v V \ U which was labeled from a node of U, receives B(U) as the second component of his label. The resulting pseudonode will be treated as a regular node (it can be part of a new blossom).

31 Reconstruction of an augmenting path. Suppose we find in the reduced graph an augmenting path P that contains B(U). To reconstruct the path P in the original graph we do the following: Let v be node after B(U) in P. Identify in G the alternating path of even length that connects b(u) and v.

32 Maximum matchings in general graphs: Example G, M (E, -) (O, ) (E, 2) 2 3 (O, 3) (E, 4) (O, 3) (E, 5) U={3, 4, 5, 6, 7}; b(u)=3 9 2

33 Maximum matchings in general graphs: Example U={3, 4, 5, 6, 7} 9 2 Reduced graph, G, and matching, M, after shrinking U 2 B(U)

34 Maximum matchings in general graphs: Example Reduced graph, G, and matching, M, after shrinking U 2 B(U) (E, -) (O, ) (E, 2) (O, B(U)) (E, 8) 2 B(U) (O, B(U)) 2 (E, 9) U ={B(U)}, 8, 9, 0, }; b(u )=B(U)

35 Maximum matchings in general graphs: Example 2 B(U) 8 0 U ={B(U), 8, 9, 0, } 9 2 Reduced graph, G, and matching M, after shrinking U 2 B(U ) 2

36 Maximum matchings in general graphs: Example Reduced graph, G, and matching M, after shrinking U 2 B(U ) 2 (E, -) (O,) 2 (E,2) B(U ) Augmenting path, P, in G : (, 2, B(U ), 2) 2 (O, B(U )

37 Maximum matchings in general graphs: Example Reconstructing alternating path in U (E, -) (O, ) (E, 2) (O, B(U)) (E, 8) 2 B(U) (O, B(U)) 2 (E, 9) (E, -) (O, ) (E, 2) 2 B(U) 8 0 Augmenting path, P, in G : (, 2, B(U), 9,, 2) 9 (O, B(U)) 2 (E, 9)

38 Maximum matchings in general graphs: Example Reconstructing alternating path in U (E, -) (O,) (E, 2) 2 3 (O, 3) (E, 4) (E, 5) 9 2 (E, -) (O,) (E, 2) (E, 5) Augmenting path, P, in G: (, 2, B(U), 9,, 2) 9 2

39 Observation Let G denote the graph obtained after shrinking a blossom B(U). If G contains an augmenting path, then G also contains an augmenting path. We will prove that the reverse is also true.

40 MaximumMatching(G) Notation: M(v): Neighbor of v connected with a matching edge. (otherwise, not defined) p(v): Predecessor of v in the tree.

41 MaximumMatching(G) () Find a Matching M. (2) Label all exposed nodes v as even; set p(v) = NULL. All other nodes are unlabeled. (3) Let (v, w) be an edge, such that v is an even node and (v, w) has not been observed from v. If no such edge exists, Stop ( Matching is maximal ). If w is not labeled, label w as odd and M(w) as even. Set p(w) = v and p(m(w)) = w. If w is even, and v and w are in different trees, an augmenting path has been found. Expand the blossoms and augment the matching. Go to (2). If w is even and v and W are in the same tree, a blossom (U, E(U)) has been found. Let b(u) be the nearest common predecessor of v and w (base of the blossom). Shrink the nodes of the blossom B(U). Set p(b(u)) = p(b(u)) and p(v) = B(U) for each node with p(v) in U. Label B(U) as even and go to (3). If w is odd, go to (3).

42 Example M(2) = 3, M(3) = 2, M(4) = 7, M(7) = 4, M(5) = 6, M(6) = 5, M(9) = 0, M(0) = 9.

43 Example (E, -) (O, 8) (E, 4) (O, 7) U = {5, 6, 7}; b(u)=7; p(u) = 4. 5 (E, 6)

44 Example (E, -) (O, 8) 8 4 (E, 4) (O, B(U)) 2 3 (E, 2) U = {U, 2, 3}; b(u )=U; p(u ) = 4 U ={U, 2, 3} b(u )=U

45 Example (E, -) (O, 8) 8 4 B(U ) (E, 4) v = B(U ), w = ; They belong to different trees. Both are labeled even. Augmenting path found: (8, 4, B(U ), ). (E, -) (O, ) (E, 9) 9 0

46 Correctness of MaximumMatching algorithm max s.t. e E x e e δ(v) e E(U) x e x e U 2 v V 0 x e, e E U V, U 3, U odd, min s.t. π v + v V v:e δ(v) U V U 3, U odd π v + U:e E(U) U 3, U odd U y U 2 y U 0 π v v V e E 0 y U U V, U 3, U odd

47 Correctness of MaximumMatching algorithm max s.t. e E x e e δ(v) e E(U) x e x e U 2 v V 0 x e, e E U V, U 3, U odd, min s.t. π v + v V v:e δ(v) U V U 3, U odd π v + U:e E(U) U 3, U odd U y U 2 y U 0 π v v V e E 0 y U U V, U 3, U odd

48 Correctness of MaximumMatching algorithm The algorithm terminates with: A matching M of G (and a corresponding matching of G, M 0 ) A set of labeled trees T i = (Ṽi, Ẽi), i =, s A set of unlabeled nodes Ṽs. Gs = (Ṽs, E(Ṽs)) G s, contains a perfect matching. (No nodes in Ṽs are exposed). The edges of M 0 are: edges of some tree, internal edges in some pseudonode, or edges of G s. The solution associated with M 0 is feasible to the LP primal. We build a feasible solution to the LP dual problem of value M 0. By the Theorem of weak duality the result will follow.

49 Correctness of MaximumMatching algorithm O E q ~ T = ~ ~ ( V, E ) O O B(U) E E O E ~ Ts = ( V ~ ~ s, E( Vs ) Let B(U) be a pseudonode. R(U) = {u V u U or v in blossom nested in B(U)} R(U) odd. ( U odd; by replacing pseudonodes by odd sets R(U)) ~ T 2 = ~ ( V ~ 2, E2 ) E O E For edges of E(Ṽ i ), i =,, s, dual constraint satisfied by setting π v = for each odd node in Ṽ i, i =,, s. For edges in (R(U), E(R(U))), dual constraint satisfied by setting y U =. For edges of G s (assuming that Ṽ s ). Select q Ṽ s and set π q =. If Ṽs = 2, the dual constraint associated with the only edge of E(Ṽs) is satisfied. Otherwise, set yṽs \{q} =. Dual constraints associated with edges of E(Ṽs \ {q}) are satisfied. Note that Ṽs \ {q} is an odd set with (Ṽs 2)/2 matching edges.

50 Correctness of MaximumMatching algorithm Define the pair of feasible solutions to LP and the dual problem: x e = {, e M 0 0, e / M 0., if v odd node in Ṽ i, i =,, s, π v =, if v = q arbitrarily selected in Ṽ s, 0, otherwise. y R(U) =, if B(U) is a pseudonode, yṽs \{q} =, if Ṽs 2, y U = 0, otherwise.

51 Correctness of MaximumMatching algorithm, if v odd node in Ṽi, i =,, s, π v =, if v = q arbitrarily selected in Ṽ s, 0, otherwise. y R(U) =, if B(U) is a pseudonode, yṽs \{q} =, if Ṽs 2, y U = 0, otherwise. Each odd node is incident with one matching edge. Each pseudonode B(U) contains at most R(U) matching edges. 2 E(Ṽs \ {q}) contains (Ṽs 2)/2 matching edges. Thus, π v + v V U V U 3, U odd U y U M 0. 2

52 Complexity of MaximumMatching algorithm The complexity of the MaximumMatching algorithm is O(AV ) An efficient implementation of the algorithm may result in an overall O( AV ) complexity. Keep pseudonodes after an augmenting path is found. Grow new alternating forest in G. Update bases of the pseudonodes after an augmenting path is found. Expand pseudonodes if thew receive odd labels in the new alternating forest.

53 Outline The matching problem Maximum-cardinality matchings in bipartite graphs Maximum-cardinality matchings in bipartite graphs: The augmenting path algorithm 2

54 Classic example application for the matching problem. Stems from Chinese mathematician Mei-Ko. Given a connected graph and a cost function on the edges, to find a minimum cost closed walk that traverses at least once all the edges. Special case: The graph is Eulerian. Each Eulerian tour is an optimal solution. In all other cases, one or more edges are traversed more than once. Introduce parallel edges of minimum total cost, so that the resulting graph is Eulerian.

55 Classic example application for the matching problem. Stems from Chinese mathematician Mei-Ko. Given a connected graph and a cost function on the edges, to find a minimum cost closed walk that traverses at least once all the edges. Special case: The graph is Eulerian. Each Eulerian tour is an optimal solution. In all other cases, one or more edges are traversed more than once. Introduce parallel edges of minimum total cost, so that the resulting graph is Eulerian.

56 Let G = (V, E) the edge graph with edge costs c e 0, e E. We look for a subgraph G such that G G is Eulerian, and the sum of the edges in G is minimal. Since c e 0, e E, G contains no cycle. Look for a disjoint collection of (shortest) paths connecting odd degree nodes.

57 MinimumChinesePostmanTour(G, c) () Let W denote the set of odd degree nodes in G. (2) For each pair of nodes u, v W find a shortest (u, v)-path in G. (3) Compute a perfect matching, M in teh complete graph with nodes set W and edge lengths given by shortest distances of (2). (4) Add to G parallel edges corresponding to the paths represented by M. (5) Determine an Euler tour in the extended graph. This solves the Chinese postman problem.

58 Example El CPP en un grafo no dirigido G = (V, E) G = (V, E) Compute shortest paths in G

59 Example El CPP en un grafo no dirigido G = (V, E) Identify odd nodes in G Compute minimum weight perfect matching, M in graph induced by odd degree nodes

60 Example Eulerian graph (V, E M)

61 on a directed graph D = (V, A) strongly connected with non-negative arc costs c a 0, a A, can be formulated as a minimum cost network flow problem. min c uv x uv s.t. (u,v) A x uv x vu = 0 u V (u,v) A x uv (v,u) A (u, v) A

62 The Chinese Postman problem on a general graph is NP-hard.