L(2, 1, 1)-Labeling Is NP-Complete for Trees

Transcription

1 L(2, 1, 1)-Labeling Is NP-Complete for Trees Petr A. Golovach 1, Bernard Lidický 2, and Daniël Paulusma 3 1 University of Bergen, Bergen, Norway 2 University of Illinois, Urbana, USA 3 University of Durham, Durham, UK SIAM DM 2012, Halifax June 19, 2012

2 Basic definitions Definition (L(p 1,..., p k )-labelings) For positive integers p 1,..., p k, p 1... p k, and λ, an L(p 1,..., p k )-labeling of a graph G with the span λ is a mapping f : V (G) {0, 1,..., λ} such that for any vertices u, v, f (u) f (v) p i if dist G (u, v) i, i {1,..., k}. The minimum span for which an L(p 1,..., p k )-labeling exists is denoted by λ p1,...,p k (G). L(1)-labeling is classical coloring.

3 Basic definitions Examples of L(2, 1) and L(2, 1, 1)-labelings

4 Basic definitions L(2, 1)-labeling of span

5 Basic definitions L(2, 1, 1)-labeling of span

6 Basic definitions Problem (L(p 1,..., p k )-labeling) Parameters: positive integers p 1,..., p k. Instance: a graph G and a positive integer λ. Question: does G have an L(p 1,..., p k )-labeling with span λ?

7 Known results General graphs: L(2, 1)-labeling is NP-complete (Griggs, Yeh; 92).

8 Known results General graphs: L(2, 1)-labeling is NP-complete (Griggs, Yeh; 92). L(2, 1)-labeling can be solved in polynomial time for λ < 4 and is NP-complete otherwise (Fiala, Kloks and Kratochvíl; 01).

9 Known results General graphs: L(2, 1)-labeling is NP-complete (Griggs, Yeh; 92). L(2, 1)-labeling can be solved in polynomial time for λ < 4 and is NP-complete otherwise (Fiala, Kloks and Kratochvíl; 01). L(2, 1, 1)-labeling can be solved in polynomial time for λ < 5 and is NP-complete otherwise (Fiala, Golovach and Kratochvíl; 04).

10 Known results General graphs: L(2, 1)-labeling is NP-complete (Griggs, Yeh; 92). L(2, 1)-labeling can be solved in polynomial time for λ < 4 and is NP-complete otherwise (Fiala, Kloks and Kratochvíl; 01). L(2, 1, 1)-labeling can be solved in polynomial time for λ < 5 and is NP-complete otherwise (Fiala, Golovach and Kratochvíl; 04). Exact algorithms for L(2, 1)-labeling of graphs (Král ; 05 and Havet, Klazar, Kratochvíl, Kratsch, Liedloff; 11).

11 Known results Graphs of bounded treewidth: For any fixed λ, L(p 1,..., p k )-labeling can be solved in polynomial (linear) time by the theorem of Courcelle.

12 Known results Graphs of bounded treewidth: For any fixed λ, L(p 1,..., p k )-labeling can be solved in polynomial (linear) time by the theorem of Courcelle. L(2, 1)-labeling is NP-complete for graphs tw 2 (Fiala, Golovach and Kratochvíl).

13 Known results Graphs of bounded treewidth: For any fixed λ, L(p 1,..., p k )-labeling can be solved in polynomial (linear) time by the theorem of Courcelle. L(2, 1)-labeling is NP-complete for graphs tw 2 (Fiala, Golovach and Kratochvíl). L(1, 1,..., 1)-labeling is solvable in polynomial time (Zhou, Kanari and Nishizeki; 00).

14 Known results Trees: L(2, 1)-labeling can be solved in polynomial time (Chang, Kuo; 96). in linear time (Hasunuma, Ishii, Ono, Uno; 09)

15 Known results Trees: L(2, 1)-labeling can be solved in polynomial time (Chang, Kuo; 96). in linear time (Hasunuma, Ishii, Ono, Uno; 09) L(p 1, 1)-labeling can be solved in polynomial time (Chang, Ke, Kuo, Liu, Yeh; 96).

16 Known results Trees: L(2, 1)-labeling can be solved in polynomial time (Chang, Kuo; 96). in linear time (Hasunuma, Ishii, Ono, Uno; 09) L(p 1, 1)-labeling can be solved in polynomial time (Chang, Ke, Kuo, Liu, Yeh; 96). L(p 1, p 2 )-labeling can be solved in polynomial time if p 2 divides p 1 and is NP-complete otherwise (Fiala, Golovach and Kratochvíl; 08).

17 Known results Trees: L(2, 1)-labeling can be solved in polynomial time (Chang, Kuo; 96). in linear time (Hasunuma, Ishii, Ono, Uno; 09) L(p 1, 1)-labeling can be solved in polynomial time (Chang, Ke, Kuo, Liu, Yeh; 96). L(p 1, p 2 )-labeling can be solved in polynomial time if p 2 divides p 1 and is NP-complete otherwise (Fiala, Golovach and Kratochvíl; 08). L(p 1, 1)-labeling is NP-complete if p 1 is part of the input (Golovach; 06).

18 Known results Theorem Every tree T satisfies (T ) + 1 λ 2,1 (T ) (T ) + 2. Theorem (King, Ras, Zhou; 10 and indep. Fiala, Golovach, Kratochvíl; 04) Every tree T satisfies ω(t 3 ) 1 λ 2,1,1 (T ) ω(t 3 ).

19 Main result Theorem (Golovach, L., Paulusma) The L(2, 1, 1)-labeling problem is NP-complete for the class of trees.

20 Sketch of the proof Problem (3-Satisfiability) Instance: variables x 1,..., x x and clauses C 1,..., C m. Question: can φ = C 1... C m be satisfied?

21 Sketch of the proof - Idea of the reduction x i {4i, λ 4i, 4i + 2, λ (4i + 2)}.. x i x j x s {4i, λ 4i, 4j + 2, λ (4j + 2), 4s, λ 4s}

22 Sketch of the proof - Idea of the reduction x i {4i, λ 4i, 4i + 2, λ (4i + 2)}.. x i x j x s {4i, λ 4i, 4j + 2, λ (4j + 2), 4s, λ 4s} distance 2 distance 4

23 Sketch of the proof - Idea of the reduction x i {4i, λ 4i, 4i + 2, λ (4i + 2)} 4i 4i + 2 λ (4i + 2).. λ 4i x i x j x s {4i, λ 4i, 4j + 2, λ (4j + 2), 4s, λ 4s} distance 2 distance 4 x i = { true, false, if 4i or λ 4i is not used, if 4i + 2 or λ (4i + 2) is not used.

24 Sketch of the proof - Idea of the reduction x i {4i, λ 4i, 4i + 2, λ (4i + 2)} 4i λ 4i λ (4i + 2)..? not x i x i x j x s {4i, λ 4i, 4j + 2, λ (4j + 2), 4s, λ 4s} distance 2 distance 4 x i = { true, false, if 4i or λ 4i is not used, if 4i + 2 or λ (4i + 2) is not used.

25 Sketch of the proof - Forcing lists x i {4i, λ 4i, 4i + 2, λ (4i + 2)}.. x i x j x s {4i, λ 4i, 4j + 2, λ (4j + 2), 4s, λ 4s}

26 Sketch of the proof - Forcing lists T (k) {2, 4,..., 2k} {λ 2k, λ 2k 2,..., λ 2} {0, λ} {1, λ 1}

27 Sketch of the proof - Forcing lists T (k) {2k, λ 2k} 2k 2 copies of T (k 1) Trees F (k)

28 Sketch of the proof - Forcing lists T (k) S F (i) F (i) For each i {1,..., k}, s.t. 2i / S Forcing of a list S {2, 4,..., 2k} {λ 2k, λ 2k 2,..., λ 2} s.t. x S, λ x S.

29 Cyclic labelings Definition (Cyclic metric (modulo λ + 1)) For positive integers a, b {0,..., λ}, a b c = min{ a b, λ + 1 a b }.

30 Cyclic labelings Definition (Cyclic metric (modulo λ + 1)) For positive integers a, b {0,..., λ}, a b c = min{ a b, λ + 1 a b }. Definition (C(p 1,..., p k )-labelings) For positive integers p 1,..., p k, p 1... p k, and λ, an C(p 1,..., p k )-labeling of a graph G with the span λ is a mapping f : V (G) {0, 1,..., λ} such that for any vertices u, v, f (u) f (v) c p i if dist G (u, v) i, i {1,..., k}.

31 Cyclic labelings C(2, 1)-labeling of span

32 Cyclic labelings What is the computational complexity of C(2, 1, 1)-Labeling on trees?

33 Thank you for your attention!