Complexity of partial covers of Theta graphs

1
Complexity of partial covers of Theta graphs

• Jirí Fiala, Jan Kratochvíl, Atilla Pór
• Charles University, Prague
• Renyi Insitute of Mathematics, Budapest

2

• Frequency assignment problem

3

• Frequency assignment problem
• Locally constrained graph homomorphism

4

• Frequency assignment problem
• Locally constrained graph homomorphism
• NP-hardness reduction based on edge-precoloring-ex
  tension of bipartite graphs

5

• Frequency assignment problem
• Locally constrained graph homomorphism
• NP-hardness reduction based on edge-precoloring-ex
  tension of bipartite graphs and
• Number theoretical conjecture

6

• Frequency assignment problem
• Locally constrained graph homomorphism
• NP-hardness reduction based on edge-precoloring-ex
  tension of bipartite graphs and
• Number theoretical conjecture
• Geometric reformulation

7

• Frequency assignment problem
• Locally constrained graph homomorphism
• NP-hardness reduction based on edge-precoloring-ex
  tension of bipartite graphs and
• Number theoretical conjecture
• Geometric reformulation
• Elementary geometric proof

8

• Frequency assignment problem
• Locally constrained graph homomorphism
• NP-hardness reduction based on edge-precoloring-ex
  tension of bipartite graphs and
• Number theoretical conjecture
• Geometric reformulation
• Elementary geometric proof
• NP-hardness result for the complexity of locally
  injective homomorphism into certain Theta graphs

9
L(2,1)-labelings of graphs

• f V(G) ? 0,1,2,,k
• uv ? E(G) ? f(u) f(v) ? 2
• dG(u,v) 2 ? f(u) ? f(v)

10
L(2,1)-labelings of graphs

• f V(G) ? 0,1,2,,k
• uv ? E(G) ? f(u) f(v) ? 2
• dG(u,v) 2 ? f(u) ? f(v)
• f(u) f(v) ? 1

11
L(2,1)-labelings of graphs

• f V(G) ? 0,1,2,,k
• uv ? E(G) ? f(u) f(v) ? 2
• dG(u,v) 2 ? f(u) ? f(v)
• f(u) f(v) ? 1
• L(2,1)(G) min such k

12
(No Transcript)
13
L(2,1)-labelings of graphs

• NP-complete for every fixed k ? 4
• Polynomial for graphs of bounded tree-width (when
  k fixed)

14
L(2,1)-labelings of graphs

• NP-complete for every fixed k ? 4
• Polynomial for graphs of bounded tree-width (when
  k fixed)
• Polynomial for trees (when k part of input)
• NP-complete for graphs of bounded tree-width
  (when k part of input, width ? 2)

15
H(2,1)-labelings of graphs

• f V(G) ? V(H)
• uv ? E(G) ? dH ( f(u), f(v)) ? 2
• dG(u,v) 2 ? f(u) ? f(v)

16
H(2,1)-labelings of graphs

• f V(G) ? V(H)
• uv ? E(G) ? dH ( f(u), f(v)) ? 2
• ? f(u)f(v) ? E(H)
• dG(u,v) 2 ? f(u) ? f(v)

17
H(2,1)-labelings of graphs

• f V(G) ? V(H)
• uv ? E(G) ? f(u)f(v) ? E(-H)
• dG(u,v) 2 ? f(u) ? f(v)

18
H(2,1)-labelings of graphs

• f V(G) ? V(H)
• uv ? E(G) ? f(u)f(v) ? E(-H)
• homomorphism from G to
  -H
• dG(u,v) 2 ? f(u) ? f(v)
• locally injective

19
(No Transcript)
20
H(2,1)-labelings of graphs locally injective
homomorphismsinto H
21
H(2,1)-labelings of graphs locally injective
homomorphismsinto Hpartial covers of H
22
locally bijective homomorphismsinto H

covers of H
23
(No Transcript)
24
cover
partial cover
25

• H-PARTIAL-COVER
• Input A graph G.
• Question ? locally injective homomorphism G? H?
• H-COVER
• Input A graph G.
• Question ? locally bijective homomorphism G? H?

26

• H-COLORING
• Input A graph G.
• Question ? homomorphism G? H?
• Thm (Hell, Neetril) H-COLORING is polynomial
  for H bipartite and NP-complete otherwise.

27

• H-ROLE-ASSIGNMENT
• Input A graph G.
• Question ? locally surjective homomorphism
  G? H?
• Conjecture (Kristianssen, JAT 2000)
  H-ROLE-ASSIGNMENT is polynomial for H with at
  most 3 vertices and NP-complete otherwise.
• Thm (Fiala, Paulusma 2002) True.

28

• H-PARTIAL-COVER
• Input A graph G.
• Question ? locally injective homomorphism G? H?
• H-COVER
• Input A graph G.
• Question ? locally bijective homomorphism G? H?
• Thm H-COVER ? H-PARTIAL-COVER

29
Complexity of H-COVER

• Bodlaender 1989
• Abello, Fellows, Stilwell 1991
• JK, Proskurowski, Telle 1994, 1996, 1997
• JF 2000
• Complexity of H-PARTIAL-COVER
• JF, JK 2001

30
Complexity of H-COVER

• NP-complete for k-regular graphs H (k?3)

31
Complexity of H-COVER

• NP-complete for k-regular graphs H (k?3)
• Polynomial for graphs with at most 2 vertices in
  each block of the degree partition

32
Complexity of H-COVER

• NP-complete for k-regular graphs H (k?3)
• Polynomial for graphs with at most 2 vertices in
  each block of the degree partition
• Polynomial for graphs arising from affine
  mappings

33
Complexity of H-COVER

• NP-complete for k-regular graphs H (k?3)
• Polynomial for graphs with at most 2 vertices in
  each block of the degree partition
• Polynomial graphs arising from affine mappings
• Polynomial for Theta graphs (based on König-Hall
  thm)

34
(No Transcript)
35
(No Transcript)
36

• Thm ?(ak,bm)-PARTIAL-COVER is
• - polynomial if a,b are odd
• - NP-complete if a-b is odd

37

• Thm ?(ak,bm)-PARTIAL-COVER is
• - polynomial if a,b are odd
• - NP-complete if a-b is odd
• Thm ?(a,b,c)-PARTIAL-COVER is
• - NP-complete if abc

38
?(1,2,3)
39
(No Transcript)
40
-?(1,2,3) P5
41
Eq

• ?(1,2,3)-PARTIAL-COVER
• P5(2,1)-labeling
• L(2,1)(G) ? 4

42
Eq

• ?(1,2,3)-PARTIAL-COVER
• P5(2,1)-labeling
• L(2,1)(G) ? 4
• And hence all NP-complete.

43

• Thm ?(ak,bm)-PARTIAL-COVER is
• - polynomial if a,b are odd
• - NP-complete if a-b is odd
• Thm ?(a,b,c)-PARTIAL-COVER is
• - NP-complete if abc
• Question ?(1,3,5)-PARTIAL-COVER ?

44

• Thm ?(ak,bm)-PARTIAL-COVER is
• - polynomial if a,b are odd
• - NP-complete if a-b is odd
• Thm ?(a,b,c)-PARTIAL-COVER is
• - NP-complete if abc
• Thm ?(a,b,c)-PARTIAL-COVER is
• - NP-complete if a,b,c are odd and distinct

45
Theorem

• Let a? b ? c be odd integers. If there exists an
  m such that the equation
• xaybzc m
• I) has no solution satisfying xy ? z, xz ?
  y, yz ? x
• II) has a solution satisfying x yz1
• III) has a solution satisfying y xz1
• IV) has a solution satisfying z xy1,
• then ?(a,b,c)-PARTIAL-COVER is NP-complete.

46
Proof

• Given cubic bipartite graph G with some edges
  precolored by two colors, it is NP-complete to
  decide if the precoloring can be extended to a
  proper 3-edge-coloring. (JF 2003)

47
(No Transcript)
48
(No Transcript)
49
(No Transcript)
50
Proof

• Given cubic bipartite graph G with some edges
  precolored by two colors, it is NP-complete to
  decide if the precoloring can be extended to a
  proper 3-edge-coloring.
• Construct G by replacing edges of color 1 by
  paths of length a, edges of color 2 by paths of
  length b and uncolored edges by paths of length
  m.

51
(No Transcript)
52
Proof

• Given cubic bipartite graph G with some edges
  precolored by two colors, it is NP-complete to
  decide if the precoloring can be extended to a
  proper 3-edge-coloring.
• Construct G by replacing edges of color 1 by
  paths of length a, edges of color 2 by paths of
  length b and uncolored edges by paths of length
  m.
• Then G partially covers ?(a,b,c) iff G allows a
  precoloring extension.

53
(No Transcript)
54
(No Transcript)
55

• m a b a c a b a
• b a b a b
• c a c
• but never
• m a b,

56
Conjecture (1998-2003)

• Let a? b ? c be odd integers. Then there exists
  an m such that the equation
• xaybzc m
• I) has no solution satisfying xy ? z, xz ?
  y, yz ? x
• II) has a solution satisfying x yz1
• III) has a solution satisfying y xz1
• IV) has a solution satisfying z xy1.

57
Examples

• (a,b,c) (1,3,5) m 11
• 11 13151 353 515
• (a,b,c) (1,3,7) m 17
• (a,b,c) (1,5,7) m 23
• (a,b,c) (39,47,49) m 993
• (a,b,c) (45,47,49) m 2255

58
Geometric formulation

• Integer points in the triangle cut on the plane
• xaybzc m
• by the coordinate planes

59
Reformulation

• Set
• abc, bac, cab
• x(-xyz1)/2, y(x-yz1)/2,
  z(xy-z1)/2
• Mabcm.

60
Reformulation

• Set
• abc, bac, cab
• x(-xyz1)/2, y(x-yz1)/2,
  z(xy-z1)/2
• Mabcm.
• Then
• xaybzcM ? xaybzc
  m

61
Reformulation

• Set
• abc, bac, cab
• x(-xyz1)/2, y(x-yz1)/2,
  z(xy-z1)/2
• Mabcm.
• Then
• xaybzcM ? xaybzc
  m
• (bc) (-xyz1)/2 (ac) (x-yz1)/2
  (ab)(xy-z1)/2

62
Reformulation

• Set
• abc, bac, cab
• x(-xyz1)/2, y(x-yz1)/2,
  z(xy-z1)/2
• Mabcm.
• Then
• xaybzcM ? xaybzc
  m
• (bc) (-xyz1)/2 (ac) (x-yz1)/2
  (ab)(xy-z1)/2
• axabybczc abcm M

63
Reformulation

• Set
• abc, bac, cab
• x(-xyz1)/2, y(x-yz1)/2,
  z(xy-z1)/2
• Mabcm.
• Then
• xaybzcM ? xaybzc
  m
• and
• xy ? z ? z ? 1, i.e.
• I) ? i) there is no solution satifying
  x,y,z ? 1.

64
Reformulation

• Set
• abc, bac, cab
• x(-xyz1)/2, y(x-yz1)/2,
  z(xy-z1)/2
• Mabcm.
• Then
• xaybzcM ? xaybzc
  m
• similarly
• xy z-1 ? z 0, i.e.
• II) ? ii) there is a solution satifying
  x,y? 1, z0.

65
Reformulation

• And the Conjecture is equivalent to
• For distinct even integers a, b, c , there
  exists M such that the equation
• xaybzcM
• i) has no integer solution satifying x,y,z ?
  1,
• ii) has a solution satifying x,y? 1, z0,
• iii) has a solution satifying x,z? 1, y0,
• iv) has a solution satifying y,z? 1, x0.

66
Geometric interpretation

• Integer solutions to the equation
• xaybzcM

67
Solution

• Theorem
• For integers a, b, c , there exists M such
  that the equation
• xaybzcM
• i) has no integer solution satifying x,y,z ?
  1,
• ii) has a solution satifying x,y? 1, z0,
• iii) has a solution satifying x,z? 1, y0,
• iv) has a solution satifying y,z? 1, x0.

68
Proof

• Consider the plane ?
• xaybzc 0
• Which intersects the integer lattice in a
  2-dimensional lattice L and each translate ?M
• xaybzc M
• intersects the integer lattice either in a
  translate of L or in emptyset.
• All triangles cut in ?M by the coordinate planes
  have the same directions of sides, say lx , ly ,
  lz.

69
(No Transcript)
70
(No Transcript)
71
In ?
lx
lz
0
72
(No Transcript)
73
(No Transcript)
74
(No Transcript)
75
(No Transcript)
76
(No Transcript)
77
ly
78
(No Transcript)
79
(No Transcript)
80
(No Transcript)
81
(No Transcript)
82
(No Transcript)
83
(No Transcript)
84
(No Transcript)
85
(No Transcript)
86
(No Transcript)
87
?
88
?M
89
Questions

• More than 3 paths - ?(a,b,c,d,)
• Multiple lengths - ?(an,bm,ck)

90
Questions

• More than 3 paths - ?(a,b,c,d,)
• Multiple lengths - ?(an,bm,ck)
• Beyond Theta graphs
• H-PARTIAL-COVER is conjectured
• NP-complete for H containing a
• subdivision of K4

91

• Thank you