Graph Algorithms Edge Coloring Graph Algorithms 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 Matchings ⋆ A matching, M ⊆ E, is a set of edges such that any 2 edges from the set do not intersect. − ∀(u,v)6=(u′ ,v′)∈M (u 6= u′, u 6= v ′, v 6= u′, v 6= v ′). ⋆ A perfect matching, M ⊆ E, is a matching that covers all the vertices. − ∀u∈V ∃{(u, v) ∈ M }. Graph Algorithms 2 Example: Matching Graph Algorithms 3 Example: Maximal Matching Graph Algorithms 4 Example: Maximum Size Matching Graph Algorithms 5 Example: Perfect Matching Graph Algorithms 6 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 Example: Edge Covering Graph Algorithms 8 Example: Minimal Edge Covering Graph Algorithms 9 Example: Minimum Size Edge Covering Graph Algorithms 10 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 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 EC + M ≥ n EC = M ′ + (n − 2M ′) = n − M′ ≥ n−M (∗ since M ≥ M ′ ∗) ⇒ EC + M ≥ n . Graph Algorithms 13 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 EC + M ≤ n EC ′ = M + (n − 2M ) = n−M ⇒ EC ≤ n − M (∗ since EC ′ ≥ EC ∗) ⇒ EC + M ≤ n . Graph Algorithms 15 Edge Coloring Definition I: ⋆ A disjoint collection of matchings that cover all the edges in the graph. ⋆ A partition E = M1 ∪ M2 ∪ · · · ∪ Mψ such that Mj 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 6= w and (u, v), (u, w) ∈ E then c(u, v) 6= c(u, w). Observation: Both definitions are equivalent. Graph Algorithms 16 Example: Edge Coloring Graph Algorithms 17 Example: Edge Coloring with Minimum Number of Colors Graph Algorithms 18 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 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 More Bounds on the Chromatic Index Notation: M (G) – size of the maximum matching in G. ⋆ M (G) ≤ ⌊n/2⌋. l m Observation: ψ(G) ≥ Mm (G) . ⋆ A pigeon hole argument: the size of each color-set is at most M (G). m l m Corollary: ψ(G) ≥ ⌊n/2⌋ . Graph Algorithms 21 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 Coloring the Edges the Star Graph Sn ⋆ In a star graph ∆ = n − 1. ⋆ Each edge must be colored with a different color ⇒ ψ(Sn) = ∆. Graph Algorithms 23 Coloring the Edges of the Cycle Cn ⋆ In any cycle ∆ = 2. ⋆ For even n = 2k, edges alternate colors and 2 colors are enough ⇒ ψ(C2k ) = ∆. ⋆ For odd n = 2k + 1, at least 1 edge is colored with a third color ⇒ ψ(C2k+1) = ∆ + 1. Graph Algorithms 24 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 Ka,b: There are a vertices in A, b vertices in B, and all possible a · b edges exist. Graph Algorithms 25 Coloring the Edges of K3,4 Graph Algorithms 26 Coloring the Edges of K3,4 Graph Algorithms 27 Coloring the Edges of K3,4 Graph Algorithms 28 Coloring the Edges of K3,4 Graph Algorithms 29 Coloring the Edges of Complete Bipartite Graphs ⋆ ∆ = max {a, b} in the complete bipartite graph Ka,b. ⋆ Let the vertices be v0, . . . , va−1 ∈ A and u0, . . . , ub−1 ∈ B. ⋆ Assume a ≤ b. ⋆ Color the edges in b = ∆ rounds with the colors 0, . . . , ∆−1. ⋆ In round 0 ≤ i ≤ ∆ − 1, color edges (vj , u(j+i) mod b) with color i for 0 ≤ j ≤ a. Graph Algorithms 30 Complete Graphs – Even n ⋆ ψ(Kn) = 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 Complete Graphs – n = 2k power of 2 ⋆ If k = 1 color the only edge with 1 = ∆ color. ⋆ If k > 1, partition the vertices into two Kn/2 cliques A and B each with 2k−1 vertices. ⋆ Color the complete bipartite Kn/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 = ∆(Kn/2) colors. ⋆ All together, n − 1 = ∆ colors were used. Graph Algorithms 32 Coloring the edges of K8 Graph Algorithms 33 Coloring the edges of K8 Graph Algorithms 34 Coloring the edges of K8 Graph Algorithms 35 Coloring the edges of K8 Graph Algorithms 36 Complete Graphs – Odd n ⋆ ψ(Kn) = n = ∆ + 1 for an odd n. − ψ(Kn) ≤ ψ(Kn+1) = n because n + 1 is even. n(n−1) − ψ(Kn) ≥ (n(n−1)/2) = n because there are edges ((n−1)/2) 2 in Kn and the size of the maximum matching is n−1 2 . Graph Algorithms 37 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 Coloring the Edges of K7 Graph Algorithms 39 Coloring the Edges of K7 Graph Algorithms 40 Coloring the Edges of K7 Graph Algorithms 41 Coloring the Edges of K7 Graph Algorithms 42 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(K2k+1) = 2k + 1 = ∆ + 1. Graph Algorithms 43 Coloring the Edges of Even Complete Graphs – Algorithm I ⋆ Let the vertices be 0, . . . , n − 1. ⋆ Color the edges of Kn−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 Coloring the Edges of K8 Graph Algorithms 45 Coloring the Edges of K8 Graph Algorithms 46 Coloring the Edges of K8 Graph Algorithms 47 Coloring the Edges of Even Complete Graphs – Algorithm I Correctness: The coloring is legal: ⋆ The coloring of Kn−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(K2k ) = 2k − 1 = ∆. Graph Algorithms 48 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−( n2 −1)) mod (n−1), ((x+( n2 −1)) mod (n−1))). Correctness and number of colors: The same as Algorithm I since both algorithms are equivalent! Graph Algorithms 49 Coloring the Edges of Even Complete Graphs – Algorithm II 6 5 4 0 7 1 3 2 Graph Algorithms 50 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 Coloring the Edges of Even Complete Graphs – Algorithm III 6 5 4 0 7 1 3 2 Graph Algorithms 52 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 Complexity ⋆ m edges to color. ⋆ O(∆) to find the available color for any edge. ⋆ Overall, O(∆m) complexity. Graph Algorithms 54 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 Coloring the Vertices of the Line Graph L(G) Graph Algorithms 56 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 Gw (x, y) be the connected component of G(x, y) that contains w. Graph Algorithms 57 Observation ⋆ G(x, y) is a collection of even size cycles and paths. Graph Algorithms 58 Exchanging Colors Tool ⋆ Exchanging between the colors x and y in the connected component Gw (x, y) of G(x, y) results with another legal coloring. w Graph Algorithms w 59 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 cu is missing at u and one color cv is missing at v. ⋆ If cu = cv then color the edge (u, v) with the color cu = cv . u(c u) v( c v) Graph Algorithms 60 Coloring the Edges of Bipartite Graphs with ∆ Colors ⋆ Assume cu 6= cv . ⋆ Let Gu(cu, cv ) be the connected component of the subgraph of G containing only edges colored with cu and cv . ⋆ Gu(cu, cv ) is a path starting at u with a cv colored edge. u(c u) v( c v) Graph Algorithms 61 Coloring the Edges of Bipartite Graphs with ∆ Colors ⋆ The colors in Gu(cu, cv ) alternate between cv and cu. ⋆ The first edge in the path starting at u is colored cv ⇒ any edge in the path that starts at the side of u must be colored with cv . ⋆ v does not belong to Gu(cu, cv ) because cv is missing at v. u(c u) v( c v) Graph Algorithms 62 Coloring the Edges of Bipartite Graphs with ∆ Colors ⋆ Exchange between the colors cu and cv in Gu(cu, cv ). ⋆ cv is missing at both u and v: color the edge (u, v) with the color cv . v( c v) u(c u) Graph Algorithms v( c v) u(c v) 63 Complexity ⋆ m edges to color. ⋆ O(∆) to find the missing colors at u and v. ⋆ O(n) to change the colors in Gu(cu, cv ). ⋆ Overall, O(nm) complexity. Graph Algorithms 64 Coloring the Edges of Any Graph with ∆ + 1 Colors ⋆ Color the edges with the colors {1, 2, . . . , ∆ + 1} following an arbitrary order. Let (v0, v1) be the next edge to color. ⋆ At most ∆ − 1 edges containing v0 or v1 are colored ⇒ one color c0 is missing at v0 and one color c1 is missing at v1. v1(c 1) Graph Algorithms v0(c 0) 65 Coloring the edges of Any Graph with ∆ + 1 Colors ⋆ If c0 = c1 then color the edge (v0, v1) with the color c0 = c1. v1(c 1) Graph Algorithms v0(c 0) 66 Coloring the Edges Any Graph with ∆ + 1 Colors ⋆ Assume c0 6= c1. ⋆ Construct a sequence of distinct colors c0, c1, c2, . . . , cj−1, cj and a sequence of edges (v0, v1), (v0, v2), . . . , (v0, vj ). − Color ci is missing at vi for 0 ≤ i ≤ j. − ci is the color of the edge (v0, vi+1) for 1 ≤ i ≤ j. v3(c 3) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 67 Constructing the Sequence v1 (c 1) Graph Algorithms v0(c 0) 68 Constructing the Sequence v2 (c 2) v1 (c 1) Graph Algorithms v0(c 0) 69 Constructing the Sequence v3(c 3) v2 (c 2) v1 (c 1) Graph Algorithms v0(c 0) 70 Constructing the Sequence v3(c 3) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 71 Constructing the Sequence ⋆ The edge (v0, v1) and the colors c0, c1 are initially defined. ⋆ Assume the colors c0, c1, . . . , cj−1 and the edges (v0, v1), (v0, v2), . . . , (v0, vj ) are defined: − ci is missing at vi for 0 ≤ i ≤ j − 1. − ci is the color of the edge (v0, vi+1) for 1 ≤ i ≤ j − 1. ⋆ Let cj be a color missing at vj . ⋆ If there exists an edge (v0, vj+1) colored with cj , where vj+1 ∈ / {v1, . . . , vj }, then continue constructing the sequence with the defined cj and vj+1. Graph Algorithms 72 The Process Terminates ⋆ Since v0 has only ∆ neighbors, the construction process stops with one of the following cases: Case I: There is no edge (v0, v) colored with cj . Case II: For some 2 ≤ k < j: cj = ck−1 ⇒ the edge (v0, vk ) is colored with cj . Graph Algorithms 73 Case I ⋆ The color cj is missing at v0. v3(c 3) v2 (c 2) vj-1(c j-1) vj (c j ) v1 (c 1) v0(c 0, c j) Graph Algorithms 74 Case I ⋆ Shift colors: Color (v0, vi) with ci for 1 ≤ i ≤ j − 1. v3 vj-1 v2 v1 Graph Algorithms v0(c 0, c j) vj (c j ) 75 Case I ⋆ cj is missing at both v0 and vj : color (v0, vj ) with cj . v3 vj-1 v2 v1 Graph Algorithms v0(c 0, c j) vj (c j ) 76 Case II ⋆ For some 2 ≤ k < j: cj = ck−1 ⇒ (v0, vk ) is colored with cj . vk (c k) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 77 Case II ⋆ Shift colors: color (v0, vi) with ci for 1 ≤ i ≤ k − 1. ⋆ The edge (v0, vk ) is not colored. ⋆ cj is missing at both vk and vj . v k( c j ) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 78 Case II ⋆ Consider the sub-graph G(c0, cj ) of G containing only the edges colored with c0 and cj . ⋆ G(c0, cj ) is a collection of paths and cycles; v0, vk , vj are end-vertices of paths in G(c0, cj ). ⋆ Not all of the 3 vertices v0, vk , vj are in the same connected component of G(c0, cj ). Case II.I v0 and vk are in different connected components of G(c0, cj ). Case II.II v0 and vj are in different connected components of G(c0, cj ). Graph Algorithms 79 Case II.I ⋆ v0, vk are in different connected components of G(c0, cj ). v k( c j ) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 80 Case II.I ⋆ Exchange between c0 and cj in the vk -path in G(c0, cj ). v k(c 0) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 81 Case II.I ⋆ c0 is missing at both v0 and vk : color (v0, vk ) with c0. v k(c 0) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 82 Case II.II ⋆ v0, vj are in different connected components of G(c0, cj ). v k( c j ) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 83 Case II.II ⋆ Shift colors: Color (v0, vi) with ci for k ≤ i ≤ j − 1. ⋆ The edge (v0, vj ) is not colored. v k( c j ) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 84 Case II.II ⋆ The shift process does not involve c0 and cj ⇒ v0 and vj are still in different connected components of G(c0, cj ). ⋆ cj is missing at vj and c0 is missing in v0. v k( c j ) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) vj (c j ) v0(c 0) 85 Case II.II ⋆ Exchange between c0 and cj in the vj -path in G(c0, cj ). v k( c j ) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) v j (c 0) v0(c 0) 86 Case II.II ⋆ c0 is missing at both v0 and vj : color (v0, vj ) with c0. v k( c j ) v2 (c 2) v1 (c 1) Graph Algorithms vj-1(c j-1) v j (c 0) v0(c 0) 87 Complexity ⋆ m edges to color. ⋆ O(∆2) to find v1, . . . , vj . ⋆ O(n) to exchange colors. ⋆ Overall, O(nm + ∆2m) complexity. Graph Algorithms 88