on tractable parameterizations of graph isomorphism
DESCRIPTION
On Tractable Parameterizations of Graph Isomorphism. Adam Bouland, Anuj Dawar and Eryk Kopczyński. G. H. Is ?. G 1 G 2. What is the parameterized complexity of Graph Isomorphism?. Size of smallest excluded minor. Tree-Width. Genus. Crossing Number. Path-Width. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/1.jpg)
On Tractable Parameterizations of Graph Isomorphism
Adam Bouland, Anuj Dawar and Eryk Kopczyński
![Page 2: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/2.jpg)
G H
G1 G2Is ?
![Page 3: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/3.jpg)
What is the parameterized complexity of Graph
Isomorphism?
![Page 4: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/4.jpg)
Tree-Width
Path-Width
Tree-Depth Max Leaf Number
Vertex Cover Number
Size of smallest excluded minor
Genus
Crossing Number
![Page 5: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/5.jpg)
Tree-Width
Path-Width
Tree-Depth Max Leaf Number
Vertex Cover Number
Size of smallest excluded minor
Genus
Crossing Number
XPnf(k)
![Page 6: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/6.jpg)
Tree-Width
Path-Width
Tree-Depth Max Leaf Number
Vertex Cover Number
Size of smallest excluded minor
Genus
Crossing Number
FPT?
? ?
? ?
?
+ Others
f(k)nO(1)
![Page 7: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/7.jpg)
Tree-Width
Path-Width
Tree-Depth Max Leaf Number
Vertex Cover Number
Size of smallest excluded minor
Genus
Crossing Number
FPT?
? ?
? ?
![Page 8: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/8.jpg)
Tree-Width
Path-Width
Tree-Depth Max Leaf Number
Vertex Cover Number
Size of smallest excluded minor
Genus
Crossing Number
FPT?
? ?
? ?
Generalized Tree-Depth
![Page 9: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/9.jpg)
Why tree-depth?
Theorem [Elberfeld Grohe Tantau 2012]:
FO=MSO on a class of graphs C iff C has bounded tree-depth
Game definition – similar to path-width
Matrix factorization
![Page 10: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/10.jpg)
Tree-Depth: 2 definitions
“Closure” of ForestRooted Forest
G has td(G)<=d iff G is a subgraph of the closure of a forest of depth d.
![Page 11: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/11.jpg)
Proof Outline
• Decomposition
• Modify tree isomorphism algorithm• Bound # vertices which can serve as root
of decomposition
![Page 12: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/12.jpg)
Proof Outline
• Decomposition
• Bound # vertices which can serve as root of decomposition
• Modify tree isomorphism algorithm
![Page 13: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/13.jpg)
Tree-Depth: 2 definitions
d cops 1 robber
Cop player wins if a cop lands on the robber
![Page 14: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/14.jpg)
Tree-Depth: 2 definitions
d cops 1 robber
![Page 15: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/15.jpg)
Tree-Depth: 2 definitions
d cops 1 robber
![Page 16: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/16.jpg)
Tree-Depth: 2 definitions
d cops 1 robber
![Page 17: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/17.jpg)
Tree-Depth: 2 definitions
d cops 1 robber
![Page 18: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/18.jpg)
Tree-Depth: 2 definitions
d cops 1 robber
![Page 19: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/19.jpg)
Tree-Depth: 2 definitions
d cops 1 robber
![Page 20: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/20.jpg)
Tree-Depth: 2 definitions
d cops 1 robber
Cop player wins if a cop lands on the robber
![Page 21: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/21.jpg)
Tree-Depth: 2 definitions
Fact: A graph has tree-depth d iff the Cop player has a winning strategy in the game using d cops
![Page 22: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/22.jpg)
Tree-Depth: 2 definitions
![Page 23: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/23.jpg)
Tree-Depth: 2 definitions
![Page 24: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/24.jpg)
Tree-Depth: 2 definitions
![Page 25: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/25.jpg)
Tree-Depth: 2 definitions
Cop Wins
![Page 26: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/26.jpg)
Bounding the Number of Roots
Thm [Dvorak, Giannopolou and Thilikos 12]: The class C={G:td(G)≤d} is characterized by a finite set of forbidden subgraphs, each of size at most 2^2^(d-1)
Cor: Number of roots of a graph of tree-depth d is at most 2^2^(d-1)
![Page 27: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/27.jpg)
Bounding the Number of Roots
H is forbidden subgraph for tree-depth <=d-1, and H has tree-depth d
Cor: Number of roots of a graph of tree-depth d is at most 2^2^(d-1)
G H
![Page 28: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/28.jpg)
Bounding the Number of Roots
SkS1 …S2
B
![Page 29: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/29.jpg)
Bounding the Number of Roots
SkS1 …S2
BSi ≈Sj iff there is an isomorphism from Si U B to Sj U B which also preserves edges
to B
![Page 30: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/30.jpg)
Bounding the Number of Roots
SkS1 …S2
BThm: Deleting more than d
copies of same component does not affect set of roots of the tree-
depth
![Page 31: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/31.jpg)
Bounding the Number of Roots
SkS1 …S2
BThm: Deleting more than d
copies of same component does not affect set of roots of the tree-
depth
Idea: Never play cops in more than d copies
Can “mirror” strategies using only d copies
![Page 32: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/32.jpg)
Bounding the Number of Roots
S1 S2 Sk…
G’
B
S1S1 SkS1 SkS1 S2
WLOG G is minimal
#Vertices in component containing robber (and
hence #Roots) bounded by reverse induction
![Page 33: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/33.jpg)
Bounding the Number of Roots
S1 S2 Sk…
G’
B
S1S1 SkS1 SkS1 S2
WLOG G is minimal
#Vertices in component containing robber (and
hence #Roots) bounded by reverse induction
![Page 34: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/34.jpg)
Isomorphism Algorithm
s
Define S<T if
1. |S|<|T|
2. |S|=|T| and #s <#t
3. |S|=|T|, #s=#t. and
(S1…S#s)<(T1…T#t)
where S_i and T_i are inductively ordered components of S and T
![Page 35: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/35.jpg)
Isomorphism AlgorithmDefine S<T if
1. |S|<|T|
2. |S|=|T| and #s <#t
3. |S|=|T|, #s=#t and
(E(s,r1)..E(s,rk))< (E(t,r1)..E(t,rk))
4. Above equal and
(S1…S#s)<(T1…T#t)
s
r1
Theorem 1: Graph Isomorphism is FPT in tree-depth
![Page 36: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/36.jpg)
Extension: Subdivisions
Defn: A graph has generalized tree-depth d iff it is a subdivision of a graph of tree-depth d
Theorem 2: Graph Isomorphism is FPT in the generalized tree-depth
![Page 37: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/37.jpg)
Tree-Width
Path-Width
Tree-Depth Max Leaf Number
Vertex Cover Number
Size of smallest excluded minor
Genus
Crossing Number
FPT?
? ?
? ?
Generalized Tree-Depth
![Page 38: On Tractable Parameterizations of Graph Isomorphism](https://reader035.vdocuments.net/reader035/viewer/2022062305/56814e44550346895dbbbb0e/html5/thumbnails/38.jpg)
Questions
?