Efficient Solutions for the -coloring Problem on Classes of Graphs

Transcription

1 Efficient Solutions for the -coloring Problem on Classes of Graphs Daniel Posner (PESC - UFRJ) PhD student - posner@cos.ufrj.br Advisor: Márcia Cerioli LIPN Université Paris-Nord 29th november 2011

2 distance d(u, v) = distance between u and v. diameter = max{d(u, v) u, v V(G)} Ex: d(u, v) = 3; diameter of the graph is 3.

3 coloring coloring of a graph G = (V, E) f: V ℕ*, such that if uv E, then f(u) f(v)

4 -coloring -coloring of a graph G = (V, E) f: V ℕ if uv E,, such that then f(u) f(v) 2, if dist(u, v) = 2, then f(u) f(v)

5 Motivation

6 Motivation

7 Motivation (G) = 4

8 Motivation (G) = 9

9 Decision version L(2,1)-coloring Problem Instance: G = (V, E), k ℕ Question: Is there an λ-coloring f of G with f : V {0, 1,..., k}? the minimum span is denoted λ

10 Examples λ +1

11 Known results Class trees p-quasi trees Comp. P P Class diameter 2 k fixed proper interval Comp. NP-c NP-c Open bipartite chain P permutation bipartite planar NP-c Open regular grids Open P cographs P P P bipartite permutation split NP-c P4-tidy split permutation P graphs (q, q-4), q fixed

12 Known results

13 Trees λ = + 1 or + 2

14 Trees [Griggs e Yeh 92] conjectured λ-col. of trees was NP-complete. complete [Chang e Kuo 96] showed an O(n 4.5) algorithm. [Hasunuma et al. 09] gave a linear time algorithm. It is still open a structural characterization of trees.

15 pv(g) pv(g) = minimum number of disjoint paths pv(g) = 3

16 (G) and pv(g) [Griggs e Yeh 92] λ(g λ( ^ K1) = n G is hamiltonian c [Georges et al. 94] pv(gc) 2 λ = n + pv(gc) 2 pv(gc) = 1 λ n - 1

17 (q, q-4) graphs A graph G is (q, q-4) if each set of q vertices induces at most q - 4 P4's. [Babel e Olariu] Ex.: q = 4 a.k.a. cografos (G is cograph P4-free) q = 5 a.k.a. P4-sparse q = 7 superclass of P4-lite (P4-tidy and perfect)

18 (q, q-4)separável p-componente Teo (Jamison e Olariu): If G is (q, q-4), then: (i) union of two (q, q-4) graphs or; (ii) join of two (q, q-4) graphs or; (iii) spider where the head is a (q, q-4) graph or; (iv) it has a separaple p-component H = (H1, H2), H q, G[V \ H2] = G[V \ H] ^ G[H1], G[V \ H1] = G[V \ H] U G[H2].

19 Union and Join

20 Spider graph If G = (V, E) is a spider, then V=S K R. S is a stable set. K is a clique, S = K G[R K] = G[R] ^ G[K] bijective function f : S K (a) edges: thin spider (b) no edges: thick spider

21 Thin spider and Thick spider

22 Separaple p-component

23 λ-coloring of union and join λ-coloring of G H λ = max{ λ(g), λ(h) } λ-coloring G ^ H λ = λ'(g) + λ'(h) + 2

24 λ-coloring of thin spider Teo If G is thin spider with K > 3. then λ = max{ R -1, λ(g[r]) } + 2 K K -1 em {0,..., 2 K - 2}

25 λ-coloring of thick spider Teo If G is thick spider with K 3 λ= { λ(g[r]) + 2 K K n K if λ(g[r]) R otherwise

26 pv(g) in separable p-component Teo If G has a separable p-component H = (H1, H2), then pv(g) = min{ max{ pv(g \ H) - B1( ), CH B3( ), 1} + B2( ) } 2

27 FPT FPT(fixed parameter tractable) in q(g) q(g) = smallest q for which G is (q, q-4) graph Algorithm FPT in q(g) Linear algorithms for (q, q-4) graphs with q fixed Ex: O(2q n) or O(qq n)

28 λ-coloring of separable p-component Teo If G is (q, q-4) graph, q fixed, with a separable p-component then λ can be obtained in linear time. proof. Gc is (q, q-4) graph and Hc is a separable p-component. If G has less than 2q vertices, one can obtain λ in O(2q4q). Otherwise, pv(gc) can be obtained in O (n), as CH qq

29 λ-coloring of separable p-component Teo If G is (q, q-4) graph, q fixed, with a separable p-component then λ can be obtained in linear time. proof. If d(u,v) 3, then u, v H1 U H2.

30 λ-coloring of separable p-component Teo If G is (q, q-4) graph, q fixed, with a separable p-component then λ can be obtained in linear time. proof. Let G' be obtained from G merging vertices in the same class. If pv(g'c) > 1, then λ(g') = n' + pv(g'c) 2 (Georges et al.) If pv(g'c) = 1, then λ(g') = n' -1 (use hamiltonian path.)

31 λ-coloring of separable p-component Teo If G is (q, q-4) graph, q fixed, with a separable p-component then λ can be obtained in linear time. proof. Assign the same color to the merged vertices For each O(q q) possible G' one can obtain λ'. λ will be the minimum among all these λ' Complexity: O(n 2q5q)

32 Example

33 Example

34 Exemplo Example

35 Exemplo Example

36 Exemplo Example

37 Exemplo Example

38 Exemplo Example

39 Exemplo Example

40 Exemplo Example

41 Exemplo Example

42 Exemplo Example

43 Exemplo Example

44 Exemplo Example

45 Exemplo Example

46 interval graph Interval graph: G = Ω(I) I

47 comparability graph Comparability graph: transitive orientation of the edges of the graph. xy, yz xz Cocomparability graph: Gc is a comparability graph.

48 permutation graph Permutation graph: G = Ω(S) S

49 split graph Split graph: G = (V, E), V = S K. K is a clique S is a stable set

50 split permutation graphs Split permutation graph: G is split and permutation. [Brandstädt, Bang Le and Spinrad-99]

51 split permutation graphs 4n There are θ ( ) split permutation graphs. n [Guruswami-99] Split permutation clique Helly [ISGCI] Extended triangle G is clique-helly every extended triangle of G has an universal vertex [Szwarcfiter-97]

52 split permutation graphs

53 split permutation graphs Our work: For a split permutation graph G, (G) = max{ (GR), (GL) } GL = G \ SR GR = G \ SL (G) can be computed in linear time. O(n2) algorithm that obtain an -coloring with this span

54 split permutation graphs

55 split permutation graphs Chain ordering a1 < a2 < < a L + M such that: N(a 1 ) N(a 2 )... N(a L + M) Asteroidal triple (AT) (AT-free)

56 split permutation graphs c SM KL N(c) or KR N(c).

57 split permutation graphs For a split permutation graph G, (G) max{ (GR), (GL) } diameter of GL is 2 (GL) = nl + pv(glc) - 2 GL subgraph of G (G) (GL).

58 split permutation graphs P(R)? (GR) P(R) (G) = max{ (GR), (GL) }

59 split permutation graphs c G

60 split permutation graphs Interval model has to be modified: (a) (b) (c)

61 split permutation graphs A linear-time algorithm to find (GR) and (GL) For split permutation graphs (G) = max{ (GR), (GL)}. This proof also gives a O(n2) algorithm to find an optimum -coloring of graphs on this class.

62 Open Problems

63 Open Problems Griggs and Yeh Conjecture: λ 2 only proved for a few classes of graphs, and it is still open for bipartite graphs. λ [Gonçalves 06]

64 Upper bounds in λ Class Upper bound diameter 2 2 [Griggs and Yeh] regular grids +2 [Calamoneri et al.] cocomparability 4-1 [Calamoneri et al.] cograph n + pv(gc) -2 [Chang e Kuo], 2 [C. and P.] planar [van den Heuvel and McGuinness] bipartite permutation weakly chordal wb(g)+1 [Araki] 2 [C. and P.] split [C. and P.] interval 2 [Calamoneri et al.]