Graph Algorithms. Edge Coloring. Graph Algorithms

Transcription

1 Graph Algorithms Edge Coloring Graph Algorithms

2 The Input Graph A simple and undirected graph G = (V,E) with n vertices in V, m edges in E, and maximum degree. Graph Algorithms 1

3 Matchings A matching, M E, is a set of edges such that any 2 edges from the set do not intersect. (u,v) (u,v ) M (u u,u v,v u,v v ). A perfect matching, M E, is a matching that covers all the vertices. u V {(u,v) M}. Graph Algorithms 2

4 Example: Matching Graph Algorithms 3

5 Example: Maximal Matching Graph Algorithms 4

6 Example: Maximum Size Matching Graph Algorithms 5

7 Example: Perfect Matching Graph Algorithms 6

8 Edge Covering An edge covering, EC E, is a set of edges such that any vertex in V belongs to at least one of the edges in EC. v V e EC (e = (u,v)). Graph Algorithms 7

9 Example: Edge Covering Graph Algorithms 8

10 Example: Minimal Edge Covering Graph Algorithms 9

11 Example: Minimum Size Edge Covering Graph Algorithms 10

12 Matching and Edge Covering Definition An isolated vertex is a vertex with no neighbors. Proposition: EC + M = n for G with no isolated vertices. Proof outline: Show that EC + M n and EC + M n which imply EC + M = n. Graph Algorithms 11

13 EC + M n Construct a matching M. Consider the edges of EC in any order. Add an edge to M if it does not intersect any edge that is already in M. The rest of the edges in EC connect a vertex from M to a vertex that is not in M. Therefore, the number of edges in EC is the number of edges in M plus n 2M additional edges. Graph Algorithms 12

14 EC + M n EC = M + (n 2M ) = n M n M ( since M M ) EC + M n. Graph Algorithms 13

15 EC + M n Construct an edge covering EC. EC contains all the edges of the maximum matching M. For each vertex that is not covered by M, add an edge that contains it to EC. Due to the maximality of M, there is no edge that covers 2 vertices that are not covered by M. Therefore, the number of edges in EC is the number of edges in M plus n 2M additional edges. Graph Algorithms 14

16 EC + M n EC = M + (n 2M) = n M EC n M ( since EC EC ) EC + M n. Graph Algorithms 15

17 Edge Coloring Definition I: A disjoint collection of matchings that cover all the edges in the graph. A partition E = M 1 M 2 M ψ such that M j is a matching for all 1 j ψ. Definition II: An assignment of colors to the edges such that two intersecting edges are assigned different colors. A function c : E {1,...,ψ} such that if v w and (u,v),(u,w) E then c(u,v) c(u,w). Observation: Both definitions are equivalent. Graph Algorithms 16

18 Example: Edge Coloring Graph Algorithms 17

19 Example: Edge Coloring with Minimum Number of Colors Graph Algorithms 18

20 The Edge Coloring Problem The optimization problem: Find an edge coloring with minimum number of colors. Notation: ψ(g) the chromatic index of G the minimum number of colors required to color all the edges of G. Hardness: A Hard problem to solve. It is NP-Hard to decide if ψ(g) = or ψ(g) = + 1 where is the maximum degree in G. Graph Algorithms 19

21 Bounds on the Chromatic Index Let be the maximum degree in G. Any edge coloring must use at least colors. ψ(g). A greedy first-fit algorithm colors the edges of any graph with at most 2 1 colors. ψ(g) 2 1. There exists a polynomial time algorithm that colors any graph with at most + 1 colors. ψ(g) + 1. Graph Algorithms 20

22 More Bounds on the Chromatic Index Notation: M(G) size of the maximum matching in G. M(G) n/2. Observation: ψ(g) m M(G). A pigeon hole argument: the size of each color-set is at most M(G). Corollary: ψ(g) m n/2. Graph Algorithms 21

23 Vertex Coloring vs. Edge Coloring Matching is an easy problem while Independent Set is a hard problem. Edge Coloring is a hard problem while Vertex Coloring is a very hard problem. Graph Algorithms 22

24 Coloring the Edges the Star Graph S n In a star graph = n 1. Each edge must be colored with a different color ψ(s n ) =. Graph Algorithms 23

25 In any cycle = 2. Coloring the Edges of the Cycle C n For even n = 2k, edges alternate colors and 2 colors are enough ψ(c 2k ) =. For odd n = 2k + 1, at least 1 edge is colored with a third color ψ(c 2k+1 ) = + 1. Graph Algorithms 24

26 Complete Bipartite Graphs Bipartite graphs: V = A B and each edge is incident to one vertex from A and one vertex from B. Complete bipartite graphs K a,b : There are a vertices in A, b vertices in B, and all possible a b edges exist. Graph Algorithms 25

27 Coloring the Edges of K 3,4 Graph Algorithms 26

28 Coloring the Edges of K 3,4 Graph Algorithms 27

29 Coloring the Edges of K 3,4 Graph Algorithms 28

30 Coloring the Edges of K 3,4 Graph Algorithms 29

31 Coloring the Edges of Complete Bipartite Graphs = max {a,b} in the complete bipartite graph K a,b. Let the vertices be v 0,...,v a 1 A and u 0,...,u b 1 B. Assume a b. Color the edges in b = rounds with the colors 0,..., 1. In round 0 i 1, color edges (v j,u (j+i) mod b ) with color i for 0 j a. Graph Algorithms 30

32 Complete Graphs Even n ψ(k n ) = n 1 = for an even n. n 1 disjoint perfect matchings each of size n/2. m = (n 1)(n/2) in a complete graph with n vertices. Graph Algorithms 31

33 Complete Graphs n = 2 k power of 2 If k = 1 color the only edge with 1 = color. If k > 1, partition the vertices into two K n/2 cliques A and B each with 2 k 1 vertices. Color the complete bipartite K n/2,n/2 implied by the partition V = A B with n/2 colors. Recursively and in parallel, color both cliques A and B with n/2 1 = (K n/2 ) colors. All together, n 1 = colors were used. Graph Algorithms 32

34 Coloring the edges of K 8 Graph Algorithms 33

35 Coloring the edges of K 8 Graph Algorithms 34

36 Coloring the edges of K 8 Graph Algorithms 35

37 Coloring the edges of K 8 Graph Algorithms 36

38 Complete Graphs Odd n ψ(k n ) = n = + 1 for an odd n. ψ(k n ) ψ(k n+1 ) = n because n + 1 is even. ψ(k n ) (n(n 1)/2) n(n 1) ((n 1)/2) = n because there are 2 edges in K n and the size of the maximum matching is n 1 2. Graph Algorithms 37

39 Coloring the Edges of Odd Complete Graphs Arrange the vertices as a regular n-polygon. Color the n edges on the perimeter of the polygon using n colors. Color an inside edge with the color of its parallel edge on the perimeter of the polygon. Graph Algorithms 38

40 Coloring the Edges of K 7 Graph Algorithms 39

41 Coloring the Edges of K 7 Graph Algorithms 40

42 Coloring the Edges of K 7 Graph Algorithms 41

43 Coloring the Edges of K 7 Graph Algorithms 42

44 Coloring the Edges of Odd Complete Graphs Correctness: The coloring is legal: Parallel edges do not intersect. Each edge is parallel to exactly one perimeter edge. Number of colors: ψ(k 2k+1 ) = 2k + 1 = + 1. Graph Algorithms 43

45 Coloring the Edges of Even Complete Graphs Algorithm I Let the vertices be 0,...,n 1. Color the edges of K n 1 on the vertices 0,...,n 2 using the polygon algorithm. For 0 x n 2, color the edge (n 1,x) with the only color missing at x. Graph Algorithms 44

46 Coloring the Edges of K 8 Graph Algorithms 45

47 Coloring the Edges of K 8 Graph Algorithms 46

48 Coloring the Edges of K 8 Graph Algorithms 47

49 Coloring the Edges of Even Complete Graphs Algorithm I Correctness: The coloring is legal: The coloring of K n 1 implies a legal coloring for all the edges except those incident to vertex n 1. Exactly 1 color is missing at a vertex x {0,...,n 2}. This is the color of the perimeter edge opposite to it. The n 1 perimeter edges are colored with different colors. Number of colors: ψ(k 2k ) = 2k 1 =. Graph Algorithms 48

50 Coloring the Edges of Even Complete Graphs Algorithm II Let the vertices be 0,...,n 1. In round 0 x n 2, color the following edges with the color x: (n 1,x). (((x 1) mod (n 1)),((x + 1) mod (n 1))). (((x 2) mod (n 1)),((x + 2) mod (n 1)))... (((x ( n 2 1)) mod (n 1),((x+(n 2 1)) mod (n 1))). Correctness and number of colors: The same as Algorithm I since both algorithms are equivalent! Graph Algorithms 49

51 Coloring the Edges of Even Complete Graphs Algorithm II Graph Algorithms 50

52 Coloring the Edges of Even Complete Graphs Algorithm III Let 0 i < j n 1 be 2 vertices. Case j = n 1: color the edge (i,j) with the color i. Case i + j is even: color the edge (i,j) with the color i+j 2. Case i + j is odd and i + j < n 1: color the edge (i,j) with the color i+j+(n 1) 2. Case i + j is odd and i + j n 1: color the edge (i,j) with the color i+j (n 1) 2. Correctness and number of edges: The same as Algorithm I and Algorithm II since all three algorithms are equivalent! Graph Algorithms 51

53 Coloring the Edges of Even Complete Graphs Algorithm III Graph Algorithms 52

54 Greedy First Fit Edge Coloring Algorithm: Color the edges of G with 2 1 colors. Consider the edges in an arbitrary order. Color an edge with the first available color among {1,2,...,2 1}. Correctness: At the time of coloring, at most 2 2 colors are not available. By the pigeon hole argument there exists an available color. Graph Algorithms 53

55 Complexity m edges to color. O( ) to find the available color for any edge. Overall, O( m) complexity. Graph Algorithms 54

56 Coloring the Vertices of the Line Graph L(G) Coloring the edges of G is equivalent to coloring the vertices of the line graph L(G) of G. If the maximum degree in G is then the maximum degree in L(G) is (L(G)) = 2 2. The greedy coloring of the edges of G with 2 1 colors is equivalent to the greedy coloring of the vertices of L(G) with (L(G)) + 1 = 2 1 colors. Graph Algorithms 55

57 Coloring the Vertices of the Line Graph L(G) Graph Algorithms 56

58 Subgraphs Definitions For colors x and y, let G(x,y) be the subgraph of G containing all the vertices of G and only the edges whose colors are x or y. For a vertex w, let G w (x,y) be the connected component of G(x,y) that contains w. Graph Algorithms 57

59 Observation G(x,y) is a collection of even size cycles and paths. Graph Algorithms 58

60 Exchanging Colors Tool Exchanging between the colors x and y in the connected component G w (x,y) of G(x,y) results with another legal coloring. w w Graph Algorithms 59

61 Coloring the edges of Bipartite Graphs with Colors Color the edges with the colors {1,2,..., } following an arbitrary order. Let (u,v) be the next edge to color. At most 1 edges containing u or v are colored one color c u is missing at u and one color c v is missing at v. If c u = c v then color the edge (u,v) with the color c u = c v. u( c u ) v( c v ) Graph Algorithms 60

62 Coloring the Edges of Bipartite Graphs with Colors Assume c u c v. Let G u (c u,c v ) be the connected component of the subgraph of G containing only edges colored with c u and c v. G u (c u,c v ) is a path starting at u with a c v colored edge. u( c u ) v( c v ) Graph Algorithms 61

63 Coloring the Edges of Bipartite Graphs with Colors The colors in G u (c u,c v ) alternate between c v and c u. The first edge in the path starting at u is colored c v any edge in the path that starts at the side of u must be colored with c v. v does not belong to G u (c u,c v ) because c v is missing at v. u( c u ) v( c v ) Graph Algorithms 62

64 Coloring the Edges of Bipartite Graphs with Colors Exchange between the colors c u and c v in G u (c u,c v ). c v is missing at both u and v: color the edge (u,v) with the color c v. u( c u ) v( c v ) u( c v ) v( c v ) Graph Algorithms 63

65 Complexity m edges to color. O( ) to find the missing colors at u and v. O(n) to change the colors in G u (c u,c v ). Overall, O(nm) complexity. Graph Algorithms 64

66 Coloring the Edges of Any Graph with + 1 Colors Color the edges with the colors {1,2,..., + 1} following an arbitrary order. Let (v 0,v 1 ) be the next edge to color. At most 1 edges containing v 0 or v 1 are colored one color c 0 is missing at v 0 and one color c 1 is missing at v 1. v ( c ) 1 1 v ( c ) 0 0 Graph Algorithms 65

67 Coloring the edges of Any Graph with + 1 Colors If c 0 = c 1 then color the edge (v 0,v 1 ) with the color c 0 = c 1. v ( c ) 1 1 v ( c ) 0 0 Graph Algorithms 66

68 Coloring the Edges Any Graph with + 1 Colors Assume c 0 c 1. Construct a sequence of distinct colors c 0,c 1,c 2,...,c j 1,c j and a sequence of edges (v 0,v 1 ),(v 0,v 2 ),...,(v 0,v j ). Color c i is missing at v i for 0 i j. c i is the color of the edge (v 0,v i+1 ) for 1 i j. v 3( c 3) v 2( c 2 ) v j-1( cj-1) v 1( c 1 ) v 0( c 0 ) v ( c ) j j Graph Algorithms 67

69 Constructing the Sequence v ( c ) 1 1 v ( c ) 0 0 Graph Algorithms 68

70 Constructing the Sequence v ( c ) 2 2 v ( c ) 1 1 v ( c ) 0 0 Graph Algorithms 69

71 Constructing the Sequence v ( c ) 2 2 v ( c ) 3 3 v ( c ) 1 1 v ( c ) 0 0 Graph Algorithms 70

72 Constructing the Sequence v ( c ) 3 3 v ( c ) 2 2 v ( c ) j-1 j-1 v ( c ) 1 1 v ( c ) j j v ( c ) 0 0 Graph Algorithms 71

73 Constructing the Sequence The edge (v 0,v 1 ) and the colors c 0,c 1 are initially defined. Assume the colors c 0,c 1,...,c j 1 and the edges (v 0,v 1 ),(v 0,v 2 ),...,(v 0,v j ) are defined: c i is missing at v i for 0 i j 1. c i is the color of the edge (v 0,v i+1 ) for 1 i j 1. Let c j be a color missing at v j. If there exists an edge (v 0,v j+1 ) colored with c j, where v j+1 / {v 1,...,v j }, then continue constructing the sequence with the defined c j and v j+1. Graph Algorithms 72

74 The Process Terminates Since v 0 has only neighbors, the construction process stops with one of the following cases: Case I: There is no edge (v 0,v) colored with c j. Case II: For some 2 k < j: c j = c k 1 the edge (v 0,v k ) is colored with c j. Graph Algorithms 73

75 Case I The color c j is missing at v 0. v ( c ) 2 2 v ( c ) 3 3 v ( c ) j-1 j-1 v ( c ) 1 1 v (, ) 0 c 0 c j v ( c ) j j Graph Algorithms 74

76 Case I Shift colors: Color (v 0,v i ) with c i for 1 i j 1. v 3 v 2 v j-1 v 1 v (, ) 0 c 0 c j v ( c ) j j Graph Algorithms 75

77 Case I c j is missing at both v 0 and v j : color (v 0,v j ) with c j. v 3 v 2 v j-1 v 1 v (, ) 0 c 0 c j v ( c ) j j Graph Algorithms 76

78 Case II For some 2 k < j: c j = c k 1 (v 0,v k ) is colored with c j. v k( c k ) v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v ( c ) j j Graph Algorithms 77

79 Case II Shift colors: color (v 0,v i ) with c i for 1 i k 1. The edge (v 0,v k ) is not colored. c j is missing at both v k and v j. v ( c ) k j v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v ( c ) j j Graph Algorithms 78

80 Case II Consider the sub-graph G(c 0,c j ) of G containing only the edges colored with c 0 and c j. G(c 0,c j ) is a collection of paths and cycles; v 0,v k,v j are end-vertices of paths in G(c 0,c j ). Not all of the 3 vertices v 0,v k,v j are in the same connected component of G(c 0,c j ). Case II.I v 0 and v k are in different connected components of G(c 0,c j ). Case II.II v 0 and v j are in different connected components of G(c 0,c j ). Graph Algorithms 79

81 Case II.I v 0,v k are in different connected components of G(c 0,c j ). v ( c ) k j v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v ( c ) j j Graph Algorithms 80

82 Case II.I Exchange between c 0 and c j in the v k -path in G(c 0,c j ). v k( c 0) v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v ( c ) j j Graph Algorithms 81

83 Case II.I c 0 is missing at both v 0 and v k : color (v 0,v k ) with c 0. v k( c 0) v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v ( c ) j j Graph Algorithms 82

84 Case II.II v 0,v j are in different connected components of G(c 0,c j ). v ( c ) k j v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v ( c ) j j Graph Algorithms 83

85 Case II.II Shift colors: Color (v 0,v i ) with c i for k i j 1. The edge (v 0,v j ) is not colored. v ( c ) k j v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v ( c ) j j Graph Algorithms 84

86 Case II.II The shift process does not involve c 0 and c j v 0 and v j are still in different connected components of G(c 0,c j ). c j is missing at v j and c 0 is missing in v 0. v ( c ) k j v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v ( c ) j j Graph Algorithms 85

87 Case II.II Exchange between c 0 and c j in the v j -path in G(c 0,c j ). v ( c ) k j v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v j( c 0) Graph Algorithms 86

88 Case II.II c 0 is missing at both v 0 and v j : color (v 0,v j ) with c 0. v ( c ) k j v 2( c 2 ) v 1( c 1 ) v 0( c 0 ) v j-1( cj-1) v j( c 0) Graph Algorithms 87

89 Complexity m edges to color. O( 2 ) to find v 1,...,v j. O(n) to exchange colors. Overall, O(nm + 2 m) complexity. Graph Algorithms 88