L(2,1,1)-Labeling Is NP-Complete for Treesorion.math.iastate.edu › lidicky › slides › 2012-siam-l211.pdf · L(2;1;1)-Labeling Is NP-Complete for Trees Petr A. Golovach 1, Bernard

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

Petr A. Golovach 1, Bernard Lidicky 2, and Daniel Paulusma 3

1University of Bergen, Bergen, Norway

2University of Illinois, Urbana, USA

3University of Durham, Durham, UK

SIAM DM 2012, HalifaxJune 19, 2012

Basic definitions

Definition (L(p1, . . . , pk)-labelings)

For positive integers p1, . . . , pk , p1 ≥ . . . ≥ pk , and λ, anL(p1, . . . , pk)-labeling of a graph G with the span λ is a mappingf : V (G )→ {0, 1, . . . , λ} such that for any vertices u, v ,|f (u)− f (v)| ≥ pi if distG (u, v) ≤ i , i ∈ {1, . . . , k}.The minimum span for which an L(p1, . . . , pk)-labeling exists isdenoted by λp1,...,pk (G ).

L(1)-labeling is classical coloring.

Basic definitions

Examples of L(2, 1) and L(2, 1, 1)-labelings

3

4

5 5

0

4

1

0 2

Basic definitions

L(2, 1)-labeling of span 5

3

4

5 5

0

4

1

0 2

Basic definitions

L(2, 1, 1)-labeling of span 7

3

4

5 5

0

4

1

0 2

3

4

5 7

3

6

1

0 2

Basic definitions

Problem (L(p1, . . . , pk)-labeling)

Parameters: positive integers p1, . . . , pk .

Instance: a graph G and a positive integer λ.

Question: does G have an L(p1, . . . , pk)-labeling with span λ?

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 < 4and is NP-complete otherwise (Fiala, Kloks and Kratochvıl;’01).

• L(2, 1, 1)-labeling can be solved in polynomial time for λ < 5and is NP-complete otherwise (Fiala, Golovach andKratochvıl; ’04).

• Exact algorithms for L(2, 1)-labeling of graphs (Kral’; ’05 andHavet, Klazar, Kratochvıl, Kratsch, Liedloff; ’11).

Known results

General graphs:





Known results

General graphs:





Known results

General graphs:





Known results

Graphs of bounded treewidth:

• For any fixed λ, L(p1, . . . , pk)-labeling can be solved inpolynomial (linear) time by the theorem of Courcelle.

• L(2, 1)-labeling is NP-complete for graphs tw ≤ 2 (Fiala,Golovach and Kratochvıl).

• L(1, 1, . . . , 1)-labeling is solvable in polynomial time (Zhou,Kanari and Nishizeki; ’00).

Known results





Known results





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(p1, 1)-labeling can be solved in polynomial time (Chang,Ke, Kuo, Liu, Yeh; ’96).

• L(p1, p2)-labeling can be solved in polynomial time if p2

divides p1 and is NP-complete otherwise (Fiala, Golovach andKratochvıl; ’08).

• L(p1, 1)-labeling is NP-complete if p1 is part of the input(Golovach; ’06).

Known results

Trees:






Known results

Trees:






Known results

Trees:






Known results

TheoremEvery tree T satisfies

∆(T ) + 1 ≤ λ2,1(T ) ≤ ∆(T ) + 2.

Theorem (King, Ras, Zhou; ’10 and indep. Fiala, Golovach,Kratochvıl; ’04)

Every tree T satisfies

ω(T 3)− 1 ≤ λ2,1,1(T ) ≤ ω(T 3).

Main result

Theorem (Golovach, L., Paulusma)

The L(2, 1, 1)-labeling problem is NP-complete for the class oftrees.

Sketch of the proof

Problem (3-Satisfiability)

Instance: variables x1, . . . , xx and clauses C1, . . . ,Cm.

Question: can φ = C1 ∧ . . . ∧ Cm be satisfied?

Sketch of the proof - Idea of the reduction

......

xi{4i, λ− 4i, 4i+ 2,

λ− (4i+ 2)}

xi ∨ xj ∨ xs{4i, λ− 4i, 4j + 2,λ− (4j + 2), 4s, λ− 4s}


distance 2 distance 4

......

xi{4i, λ− 4i, 4i+ 2,

λ− (4i+ 2)}




......

xi{4i, λ− 4i, 4i+ 2,

λ− (4i+ 2)}


4i 4i+ 2 λ− (4i+ 2) λ− 4i

xi =

{true, if 4i or λ− 4i is not used,

false, if 4i + 2 or λ− (4i + 2) is not used.



......

xi{4i, λ− 4i, 4i+ 2,

λ− (4i+ 2)}


4i λ− 4i λ− (4i+ 2) ? not xi

xi =

{true, if 4i or λ− 4i is not used,

false, if 4i + 2 or λ− (4i + 2) is not used.

Sketch of the proof - Forcing lists

......

xi{4i, λ− 4i, 4i+ 2,

λ− (4i+ 2)}



T (k){2, 4, . . . , 2k}∪∪{λ− 2k , λ− 2k − 2, . . . , λ− 2}

{0, λ} {1, λ− 1}


T (k) {2k , λ− 2k}

2k − 2 copies of T (k − 1)

Trees F (k)


T (k) S

F (i)F (i)

s.t. 2i /∈ SFor each i ∈ {1, . . . , k},

Forcing of a listS ⊆ {2, 4, . . . , 2k} ∪ {λ− 2k , λ− 2k − 2, . . . , λ− 2} s.t. ∀x ∈ S ,

λ− x ∈ S .

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 (p1, . . . , pk)-labelings)

For positive integers p1, . . . , pk , p1 ≥ . . . ≥ pk , and λ, anC (p1, . . . , pk)-labeling of a graph G with the span λ is a mappingf : V (G )→ {0, 1, . . . , λ} such that for any vertices u, v ,|f (u)− f (v)|c ≥ pi if distG (u, v) ≤ i , i ∈ {1, . . . , k}.

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 (p1, . . . , pk)-labelings)

For positive integers p1, . . . , pk , p1 ≥ . . . ≥ pk , and λ, anC (p1, . . . , pk)-labeling of a graph G with the span λ is a mappingf : V (G )→ {0, 1, . . . , λ} such that for any vertices u, v ,|f (u)− f (v)|c ≥ pi if distG (u, v) ≤ i , i ∈ {1, . . . , k}.

Cyclic labelings

C (2, 1)-labeling of span 6

0 2

4

0

1

5

4

3

5

Cyclic labelings

What is the computational complexity of C (2, 1, 1)-Labeling ontrees?

Thank you for your attention!

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L(2,1,1)-Labeling Is NP-Complete for Treesorion.math.iastate.edu › lidicky › slides › 2012-siam-l211.pdf · L(2;1;1)-Labeling Is NP-Complete for Trees Petr A. Golovach 1, Bernard