Graph Sparsity. Patrice Ossona de Mendez. Shanghai Jiao Tong University November 6th, 2012

Transcription

1 Graph Sparsity Patrice Ossona de Mendez Centre d'analyse et de Mathématique Sociales CNRS/EHESS Paris, France Shanghai Jiao Tong University November 6th, 2012

2 Graph Sparsity (joint work with Jaroslav Ne²et il) Patrice Ossona de Mendez Centre d'analyse et de Mathématique Sociales CNRS/EHESS Paris, France Shanghai Jiao Tong University November 6th, 2012

3 What is dense? What is sparse?

4 Every kind of minors... Minor Topological minor Immersion δ > ct log t K t δ > ct 2 K t δ > ct K t Kostochka ('82) Komlós and Szemerédi ('94), DeVos et al. ('10) Thomason ('84) Bollobás and Thomason ('98)

5 Minors G m H: minor relation well quasi order (Robertson, Seymour; '04) Hadwiger's conjecture (proved for k 6 Robertson, Seymour, and Thomas '93; true for almost all graphs) G t H: topological minor relation not a well quasi order Hajós' conjecture (false for almost all graphs, but true if large girth) G i H: immersion relation well quasi order (Robertson, Seymour; '10) conjecture of Abu-Khzam and Langston (proved for k 7, DeVos, Kawarabayashi, Mohar, and Okamura '09)

6 Minors G m H: minor relation well quasi order (Robertson, Seymour; '04) Hadwiger's conjecture (proved for k 6 Robertson, Seymour, and Thomas '93; true for almost all graphs) G t H: topological minor relation not a well quasi order Hajós' conjecture (false for almost all graphs, but true if large girth) G i H: immersion relation well quasi order (Robertson, Seymour; '10) conjecture of Abu-Khzam and Langston (proved for k 7, DeVos, Kawarabayashi, Mohar, and Okamura '09)

7 Minors G m H: minor relation well quasi order (Robertson, Seymour; '04) Hadwiger's conjecture (proved for k 6 Robertson, Seymour, and Thomas '93; true for almost all graphs) G t H: topological minor relation not a well quasi order Hajós' conjecture (false for almost all graphs, but true if large girth) G i H: immersion relation well quasi order (Robertson, Seymour; '10) conjecture of Abu-Khzam and Langston (proved for k 7, DeVos, Kawarabayashi, Mohar, and Okamura '09)

8 Minors G m H: minor relation well quasi order (Robertson, Seymour; '04) Hadwiger's conjecture (proved for k 6 Robertson, Seymour, and Thomas '93; true for almost all graphs) G t H: topological minor relation not a well quasi order Hajós' conjecture (false for almost all graphs, but true if large girth) G i H: immersion relation well quasi order (Robertson, Seymour; '10) conjecture of Abu-Khzam and Langston (proved for k 7, DeVos, Kawarabayashi, Mohar, and Okamura '09)

9 Minors G m H: minor relation well quasi order (Robertson, Seymour; '04) Hadwiger's conjecture (proved for k 6 Robertson, Seymour, and Thomas '93; true for almost all graphs) G t H: topological minor relation not a well quasi order Hajós' conjecture (false for almost all graphs, but true if large girth) G i H: immersion relation well quasi order (Robertson, Seymour; '10) conjecture of Abu-Khzam and Langston (proved for k 7, DeVos, Kawarabayashi, Mohar, and Okamura '09)

10 Minors G m H: minor relation well quasi order (Robertson, Seymour; '04) Hadwiger's conjecture (proved for k 6 Robertson, Seymour, and Thomas '93; true for almost all graphs) G t H: topological minor relation not a well quasi order Hajós' conjecture (false for almost all graphs, but true if large girth) G i H: immersion relation well quasi order (Robertson, Seymour; '10) conjecture of Abu-Khzam and Langston (proved for k 7, DeVos, Kawarabayashi, Mohar, and Okamura '09)

11 Minors G m H: minor relation well quasi order (Robertson, Seymour; '04) Hadwiger's conjecture (proved for k 6 Robertson, Seymour, and Thomas '93; true for almost all graphs) G t H: topological minor relation not a well quasi order Hajós' conjecture (false for almost all graphs, but true if large girth) G i H: immersion relation well quasi order (Robertson, Seymour; '10) conjecture of Abu-Khzam and Langston (proved for k 7, DeVos, Kawarabayashi, Mohar, and Okamura '09)

12 Minors G m H: minor relation well quasi order (Robertson, Seymour; '04) Hadwiger's conjecture (proved for k 6 Robertson, Seymour, and Thomas '93; true for almost all graphs) G t H: topological minor relation not a well quasi order Hajós' conjecture (false for almost all graphs, but true if large girth) G i H: immersion relation well quasi order (Robertson, Seymour; '10) conjecture of Abu-Khzam and Langston (proved for k 7, DeVos, Kawarabayashi, Mohar, and Okamura '09)

13 Minors G m H: minor relation well quasi order (Robertson, Seymour; '04) Hadwiger's conjecture (proved for k 6 Robertson, Seymour, and Thomas '93; true for almost all graphs) G t H: topological minor relation not a well quasi order Hajós' conjecture (false for almost all graphs, but true if large girth) G i H: immersion relation well quasi order (Robertson, Seymour; '10) conjecture of Abu-Khzam and Langston (proved for k 7, DeVos, Kawarabayashi, Mohar, and Okamura '09)

14 Topological resolution of a class C Shallow topological minors at depth t: C t ={H : some 2t-subdivision of H is present in some G C}. 2t Example: C 0 is the monotone closure of C. Topological resolution in time: C C 0 C 1... C t... C time

15 Topological resolution of a class C Shallow topological minors at depth t: C t ={H : some 2t-subdivision of H is present in some G C}. 2t Example: C 0 is the monotone closure of C. Topological resolution in time: C C 0 C 1... C t... C time

16 Taxonomy of Classes A class C is somewhere dense if τ N : ω(c τ) = C is nowhere dense if t N : ω(c t) < C has bounded expansion if t N : d(c t) <

17 Taxonomy of Classes A class C is somewhere dense if τ N : ω(c τ) = C is nowhere dense if t N : ω(c t) < C has bounded expansion if t N : t N : d(c t) < χ(c t) <

18 Examples Graphs with maximum degree 100 Planar graphs Class of G with G > G 1+ɛ

19 Examples Graphs with maximum degree 100 Bounded expansion: d(c t) < 100 Planar graphs Class of G with G > G 1+ɛ

20 Examples Graphs with maximum degree 100 Bounded expansion: d(c t) < 100 Planar graphs Bounded expansion: d(c t) < 6 (Euler) Class of G with G > G 1+ɛ

21 Examples Graphs with maximum degree 100 Bounded expansion: d(c t) < 100 Planar graphs Bounded expansion: d(c t) < 6 (Euler) Class of G with G > G 1+ɛ Somewhere dense: 10/ɛ-subdivisions of K n (Jiang '10)

22 More Examples Class of G without cycles of length Class of G such that (G) f (girth(g)) Random graphs G(n, d/n)

23 More Examples Class of G without cycles of length Somewhere dense: subdivisions of K n Class of G such that (G) f (girth(g)) Random graphs G(n, d/n)

24 More Examples Class of G without cycles of length Somewhere dense: subdivisions of K n Class of G such that (G) f (girth(g)) Nowhere dense: ω(g t) f (6t) Random graphs G(n, d/n)

25 More Examples Class of G without cycles of length Somewhere dense: subdivisions of K n Class of G such that (G) f (girth(g)) Nowhere dense: ω(g t) f (6t) Random graphs G(n, d/n) bounded expansion class R d s.t. G(n, d/n) R d a.a.s.

26 Every kind of shallow minors... but one taxonomy! Shallow Minor Shallow Topological minor Shallow Immersion t 2t 2t s + 1

27 General diagram bounded degree ultra sparse minor closed Bounded expansion τ, d(g τ) < τ, χ(g τ) < Ω(n 1+ɛ ) edges Ω(n 2 ) edges Nowhere dense τ, ω(g τ) < Somewhere dense τ, ω(g τ) =

28 More precise diagram Bounded degree Planar Bounded genus Locally bounded tree-width Excluded minor Locally excluded minor Excluded topological minor Bounded expansion Locally bounded expansion Almost wide Nowhere dense Somewhere dense

29 Application to Counting

30 The Subgraph Isomorphism Problem Problem Input: a graph G. Question: does G contain a copy of F? n γt in general (Ne²et il and Poljak '85, using Coppersmith and Winograd '90, γ 0.792); d F α(f ) n α(f ) for d-degenerate graphs ( Chrobak and Eppstein '91).

31 The Subgraph Isomorphism Problem Problem Input: a graph G. Question: does G contain a copy of F? n γt in general (Ne²et il and Poljak '85, using Coppersmith and Winograd '90, γ 0.792); d F α(f ) n α(f ) for d-degenerate graphs ( Chrobak and Eppstein '91).

32 The Subgraph Isomorphism Problem Problem Input: a graph G. Question: does G contain a copy of F? n γt in general (Ne²et il and Poljak '85, using Coppersmith and Winograd '90, γ 0.792); d F α(f ) n α(f ) for d-degenerate graphs ( Chrobak and Eppstein '91).

33 Finding F in a d-degenerate graph Compute an acyclic orientation with indegree at most d. Consider an acyclic orientation of F. For each choice of sinks, backtrack to form a copy of F.

34 The Subgraph Isomorphism Problem for xed F Tractable classes Some easy classes of graphs G : bounded tree-width (Courcelle '90; Eppstein '99); planar graphs (Eppstein '95); bounded genus (Eppstein '00). Key notion: classes with bounded local tree-width: tw(g) f (diam(g)) Theorem (Eppstein '00) A minor closed class C has bounded local tree-width if and only if C excludes an apex graph.

35 The Subgraph Isomorphism Problem for xed F Tractable classes Some easy classes of graphs G : bounded tree-width (Courcelle '90; Eppstein '99); planar graphs (Eppstein '95); bounded genus (Eppstein '00). Key notion: classes with bounded local tree-width: tw(g) f (diam(g)) Theorem (Eppstein '00) A minor closed class C has bounded local tree-width if and only if C excludes an apex graph.

36 The Subgraph Isomorphism Problem for xed F Tractable classes Some easy classes of graphs G : bounded tree-width (Courcelle '90; Eppstein '99); planar graphs (Eppstein '95); bounded genus (Eppstein '00). Key notion: classes with bounded local tree-width: tw(g) f (diam(g)) Theorem (Eppstein '00) A minor closed class C has bounded local tree-width if and only if C excludes an apex graph.

37 Minor Closed Excluding an Apex Split the vertex set by distances to the root modulo F + 1. The union of the subsets i mod F + 1 induces a graph G i with bounded tree-width, and every copy of F is included in such a G i. Solve the problem in the graphs G i (using Courcelle or Eppstein). More generally, we can count the number of copies of F using Arnborg, Lagergren, and Seese '91 and exclusion/inclusion.

38 Minor Closed Excluding an Apex Split the vertex set by distances to the root modulo F + 1. The union of the subsets i mod F + 1 induces a graph G i with bounded tree-width, and every copy of F is included in such a G i. Solve the problem in the graphs G i (using Courcelle or Eppstein). More generally, we can count the number of copies of F using Arnborg, Lagergren, and Seese '91 and exclusion/inclusion.

39 Minor Closed Excluding an Apex Split the vertex set by distances to the root modulo F + 1. The union of the subsets i mod F + 1 induces a graph G i with bounded tree-width, and every copy of F is included in such a G i. Solve the problem in the graphs G i (using Courcelle or Eppstein). More generally, we can count the number of copies of F using Arnborg, Lagergren, and Seese '91 and exclusion/inclusion.

40 Minor Closed Excluding an Apex Split the vertex set by distances to the root modulo F + 1. The union of the subsets i mod F + 1 induces a graph G i with bounded tree-width, and every copy of F is included in such a G i. Solve the problem in the graphs G i (using Courcelle or Eppstein). More generally, we can count the number of copies of F using Arnborg, Lagergren, and Seese '91 and exclusion/inclusion.

41 A Tool for Sparse Graphs: Sections

42 Principle Color the vertices of G by N colors, consider the subgraphs G I induced by subsets I of p colors.

43 Principle Color the vertices of G by N colors, consider the subgraphs G I induced by subsets I of p colors.

44 Low Tree-Width decompositions Theorem (Devos, Oporowski, Sanders, Reed, Seymour, Vertigan; '04) For every proper minor closed class C and integer p 1, there is an integer N, such that every graph G C has a vertex partition into N graphs such that any j p parts form a graph with tree-width at most p 1. Remark This theorem relies on Robertson-Seymour structure theorem.

45 Low Tree-Width decompositions Theorem (Devos, Oporowski, Sanders, Reed, Seymour, Vertigan; '04) For every proper minor closed class C and integer p 1, there is an integer N, such that every graph G C has a vertex partition into N graphs such that any j p parts form a graph with tree-width at most p 1. Remark This theorem relies on Robertson-Seymour structure theorem.

46 Tree-depth Denition The tree-depth td(g) of a graph G is the minimum height of a rooted forest Y s.t. G Closure(Y ). td(p n ) = log 2 (n + 1)

47 Tree-depth Denition The tree-depth td(g) of a graph G is the minimum height of a rooted forest Y s.t. G Closure(Y ). td(p n ) = log 2 (n + 1)

48 Properties max H td(h), (H connected component of G) td(g) = 1 + min v td(g v), (G connected, v vertex of G) 0, if G is empty the tree-depth is minor-monotone: H minor of G = td(h) td(g). for every graph G it holds tw(g) td(g) (tw(g) + 1) log 2 G.

49 Low Tree-Depth Decompositions Chromatic numbers χ p (G) χ p (G) is the minimum of colors such that any subset I of p colors induce a subgraph G I so that td(g I ) I. χ(g) = χ 1 (G) χ 2 (G)... χ p (G) χ G (G) = td(g).

50 Low tree-depth decompositions Let C be an innite class of graphs. Theorem (Ne²et il and POM, '06) sup χ p (G) < C has bounded expansion. G C Theorem (Ne²et il and POM, '10) p, lim sup G C log χ p (G) log G = 0 C is nowhere dense.

51 Low tree-depth decompositions Let C be an innite class of graphs. Theorem (Ne²et il and POM, '06) sup χ p (G) < C has bounded expansion. G C Theorem (Ne²et il and POM, '10) p, lim sup G C log χ p (G) log G = 0 C is nowhere dense.

52 Algorithmic version of LTDD theorem Procedure A for k = 1 to 2 p do Compute a fraternal augmentation. end for Compute depth p transitivity Greedily color vertices according to the augmented graph Theorem (Ne²et il, POM; '08, '10) Procedure A computes a χ p -coloring of G with N p (G) colors in time O(N p (G) G ), where N p (G) = O(1) if G C with bounded expansion; N p (G) = G o(1) if G C nowhere dense;

53 Algorithmic Consequences Theorem (Ne²et il, POM) For every xed F, counting the number of copies of F in G may de done in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a nowhere dense class. Extension Let p N and φ[x] be a Boolean query with x = p. Counting the vectors v such that G φ[v] may be achieved in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a nowhere dense class.

54 Algorithmic Consequences Theorem (Ne²et il, POM) For every xed F, counting the number of copies of F in G may de done in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a nowhere dense class. Extension Let p N and φ[x] be a Boolean query with x = p. Counting the vectors v such that G φ[v] may be achieved in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a nowhere dense class.

55 Testing First Order Properties

56 Checking rst-order properties Theorem (Ne²et il, POM) Existential rst-order properties may be checked in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a nowhere dense class. Theorem (Dvo ák, Krá, Thomas; '10) First-order properties may be checked in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a class with locally bounded expansion. Problem Can rst-order properties be checked in n 1+o(1) time for G in a nowhere dense class?

57 Checking rst-order properties Theorem (Ne²et il, POM) Existential rst-order properties may be checked in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a nowhere dense class. Theorem (Dvo ák, Krá, Thomas; '10) First-order properties may be checked in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a class with locally bounded expansion. Problem Can rst-order properties be checked in n 1+o(1) time for G in a nowhere dense class?

58 Checking rst-order properties Theorem (Ne²et il, POM) Existential rst-order properties may be checked in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a nowhere dense class. Theorem (Dvo ák, Krá, Thomas; '10) First-order properties may be checked in O(n) time for G in a class with bounded expansion, n 1+o(1) time for G in a class with locally bounded expansion. Problem Can rst-order properties be checked in n 1+o(1) time for G in a nowhere dense class?

59 Other Applications

60 Degrees of freedom Looking for the exponent / the degree of freedom log(#f G) How is log G bounded when G is restricted to a class C? How much vertices can be chosen independently when looking for a copy of F?

61 Degrees of freedom Looking for the exponent / the degree of freedom log(#f G) How is log G bounded when G is restricted to a class C? How much vertices can be chosen independently when looking for a copy of F? Theorem (Ne²et il, POM, 2011) For every innite class of graphs C and every graph F log(#f G) lim lim sup {, 0, 1,..., α(f ), F }, i G C i log G where α(f ) is the stability number of F.

62 Degrees of freedom Looking for the exponent / the degree of freedom log(#f G) How is log G bounded when G is restricted to a class C? How much vertices can be chosen independently when looking for a copy of F? Theorem (Ne²et il, POM, 2011) For every innite class of graphs C and every graph F log(#f G) lim lim sup i G C i log G where α(f ) is the stability number of F. nowhere dense {}}{ {, 0, 1,..., α(f ), F },

63 Chromatic number of the p-distance graph G [ p] : two vertices x and y are adjacent in G [ p] if the distance between x and y in G is exactly p. Theorem (Ne²et il, POM) Let p be an odd-integer. Then χ(g [ p] χp(g) p 2p ) 2 Corollary For every bounded expansion class C and every odd integer p it holds sup χ(g [ p] ) <. G C

64 Chromatic number of the p-distance graph G [ p] : two vertices x and y are adjacent in G [ p] if the distance between x and y in G is exactly p. Theorem (Ne²et il, POM) Let p be an odd-integer. Then χ(g [ p] χp(g) p 2p ) 2 Corollary For every bounded expansion class C and every odd integer p it holds sup χ(g [ p] ) <. G C

65 Chromatic number of the p-distance graph G [ p] : two vertices x and y are adjacent in G [ p] if the distance between x and y in G is exactly p. Theorem (Ne²et il, POM) Let p be an odd-integer. Then χ(g [ p] χp(g) p 2p ) 2 Corollary For every bounded expansion class C and every odd integer p it holds sup χ(g [ p] ) <. G C

66 Problems Van den Heuvel and Naserasr Does there exist a constant C 6 such that for every odd-integer p and any planar graph G it holds χ(g [ p] ) C?

67 Problems Van den Heuvel and Naserasr Does there exist a constant C 6 such that for every odd-integer p and any planar graph G it holds χ(g [ p] ) C? Thomassé For a graph G, let G odd be the graph where two vertices are adjacent if they are at odd distance in G. Does there exist a function f such that for every planar graph G it holds χ(g odd ) f (ω(g odd ))?

68 Conclusion

69 Thank you!