on the complexity of submap isomorphismon the complexity of submap isomorphism christine solnon1,...

On the Complexity of Submap Isomorphism

Christine Solnon1, Guillaume Damiand1, Colin de la Higuera2

and Jean-Christophe Janodet3

(1) LIRIS, INSA de Lyon / CNRS(2) LINA, Universite de Nantes / CNRS

(3) IBISC, Universite d’Evry

GbR 2013

1/18

Motivations

Plan

1 Motivations

2 Recalls on Generalized Maps

3 Recalls on Computational Complexity

4 Reduction of Planar-4 3-SAT to Submap Isomorphism

5 Conclusion

2/18

Motivations

Induced Subgraph Isomorphism

G’=(N’,E’)G=(N,E)

Is G a pattern of G′ ?

Search for an injective matching that preserves all edges betweenmatched vertices

NP-complete problem

3/18

Motivations

Partial Subgraph Isomorphism

G’=(N’,E’)G=(N,E)

Is G a partial pattern of G′ ?

Search for an injective matching that preserves pattern edges

NP-complete problem

4/18

Motivations

Plane Subgraph Isomorphism

Is the left-hand side graph included in the right-hand side one ?

Yes... but do they look similar ?

Graphs modelling images are embedded in planes Let us compare plane graphs Use generalized maps to model plane graphs

GbR 2009: Polynomial algorithm for submap iso.Precondition: the pattern map must be connected

Question: Polynomial algorithm when the pattern map is not connected ?

5/18

Motivations


Is the left-hand side graph included in the right-hand side one ?Yes... but do they look similar ?



Question: Polynomial algorithm when the pattern map is not connected ?

5/18

Motivations


Is the left-hand side graph included in the right-hand side one ?Yes... but do they look similar ?



Question: Polynomial algorithm when the pattern map is not connected ?5/18

Recalls on Generalized Maps

Plan

1 Motivations




5 Conclusion

6/18


From plane graphs to generalized mapsDefinition: nD generalized map M = (D, α0, . . . , αn)

D = finite set of darts∀i , αi is an involution on D∀i ,∀j ≥ i + 2, αi ◦ αj is an involution on D

Example:Graph:

2G-map:

Involutions:

a b c d e f g h i j k l m nα0 h c b e d g f a j i l k n m

α1 b a d c f e h g n k j m l iα2 a b c i j f g h d e k l m n

Notation:Given E ⊆ D, seamsM(E) = {(d , i , αi(d)),d ∈ E ,alphai(d) 6= d}Example: seamsM({a,b, c}) = {(a,1,b), (b,1,a), (b,0, c), (c,0,b)}

7/18




Example:Graph:

2G-map:

Involutions:

a b c d e f g h i j k l m nα0 h c b e d g f a j i l k n mα1 b a d c f e h g n k j m l i

α2 a b c i j f g h d e k l m n


7/18




Example: Graph: 2G-map:

Involutions:

a b c d e f g h i j k l m nα0 h c b e d g f a j i l k n mα1 b a d c f e h g n k j m l iα2 a b c i j f g h d e k l m n


7/18


Induced and Partial Submap Isomorphism

DefinitionsM = (D, α0, . . . , αn) and M ′ = (D′, α′

0, . . . , α′n) are isomorphic if ∃ a

bijection f : D → D′ such that ∀d ∈ D,∀i ∈ [0,n], f (αi(d)) = α′i (f (d)).

M ′ = (D′, α′0, . . . , α

′n) is an induced submap of M = (D, α0, . . . , αn) if

D′ ⊆ D and seamsM′(D′) = seamsM(D′).M ′ = (D′, α′

0, . . . , α′n) is a partial submap of M = (D, α0, . . . , αn) if D′ ⊆ D

and seamsM′(D′) ⊆ seamsM(D′).

Example:

b d

a c

M M ′

M ′′

M is isomorphic to an induced submap of M ′

but not M ′′

8/18






M ′ = (D′, α′0, . . . , α





Example:

b d

a c

M M ′ M ′′

M is isomorphic to an induced submap of M ′ but not M ′′

8/18






M ′ = (D′, α′0, . . . , α





Example:

b d

a c

M M ′ M ′′

M is isomorphic to a partial submap of M ′ and M ′′

8/18


Algorithm for Submap Isomorphism [GbR 2009]

Basic idea:Choose a dart d in the pattern map MFor each dart d ′ in the target map M ′:

Traverse M and M ′ in parallel and build a mappingIf the mapping is an isomorphism then answer Yes

Answer No

Complexity: O(|D| · |D′|)

Precondition: The pattern map M must be connected

M M ′9/18

Recalls on Computational Complexity

Plan

1 Motivations




5 Conclusion

10/18


Complexity classes

Class P∃ Polynomial time algorithm on a deterministic Turing machine

Class NPPolynomial time algorithm on a Non deterministic Turing machine There may be an exponential number of candidates to check

but ∃ polynomial algorithm for deciding if a given candidate is a solution

NP-complete problems

Hardest problems of NP: If one NP-complete problem is shown to belong to P then P = NP

How to prove that a problem p is NP-complete ?

Find a reduction of a known NP-complete problem to p

11/18


The Boolean Satisfiability Problem (SAT)

Definition of the SAT problem:Input: a Boolean expression EQuestion: Is there a truth assignment of the variables that satisfies E?

First known NP-complete problem [Cook 1971]

Example:Input: E = (¬x ∨ y ∨ u)

∧ (¬x ∨ y ∨ ¬z)∧ (¬y ∨ z ∨ u)∧ (¬z ∨ u ∨ ¬w)∧ (x ∨ w ∨ ¬u)

Solution = {¬x , y ,¬z,u,w}

Special case of SAT which is still NP-complete: Planar-4 3-SATAll clauses have 3 literalsEach variable occurs in at least 2 and at most 4 different clausesThe bipartite graph associated with E is planar

12/18


The Boolean Satisfiability Problem (SAT)

Definition of the SAT problem:Input: a Boolean expression EQuestion: Is there a truth assignment of the variables that satisfies E?

First known NP-complete problem [Cook 1971]

Example:Input: E = (¬x ∨ y ∨ u)

∧ (¬x ∨ y ∨ ¬z)∧ (¬y ∨ z ∨ u)∧ (¬z ∨ u ∨ ¬w)∧ (x ∨ w ∨ ¬u)

Solution = {¬x , y ,¬z,u,w}

C4

C2

C3

C5

w

C1

y

x

u

z

Special case of SAT which is still NP-complete: Planar-4 3-SATAll clauses have 3 literalsEach variable occurs in at least 2 and at most 4 different clausesThe bipartite graph associated with E is planar

12/18

Reduction of Planar-4 3-SAT to Submap Isomorphism

Plan

1 Motivations




5 Conclusion

13/18



Goal of the reduction:

Let E be a boolean formula with v variables and c clauses

Give a polynomial time algorithm for building two maps M and M ′

such that there exists a solution to E iff M is a submap of M ′

Basic idea: Associate patterns with variables and clauses

The pattern map M is composed of v + c connected components

The target map M ′ is derived from the plane graph associated with E :

Replace vertices by the corresponding patterns2-sew patterns associated with adjacent vertices

Submap isomorphism⇔ Selection of 1 satisfied litteral for each clause

14/18


Example for induced submap isomorphism

Instance of Planar-4 3-SAT:(¬x ∨ y ∨ u) ∧ (¬x ∨ y ∨ ¬z)

∧ (¬y ∨ z ∨ u) ∧ (¬z ∨ u ∨ ¬w)∧ (x ∨ w ∨ ¬u)

C4

C2

C3

C5

w

C1

y

x

u

z

Pattern map M:

w:

x: y: z:

u:

C3:

C4:

C5:C2:

C1:

Target map M ′:

15/18


Example for induced submap isomorphism


∧ (¬y ∨ z ∨ u) ∧ (¬z ∨ u ∨ ¬w)∧ (x ∨ w ∨ ¬u)

C4

C2

C3

C5

w

C1

y

x

u

z

Pattern map M:

w:

x: y: z:

u:

C3:

C4:

C5:C2:

C1:

Target map M ′:

w

x

x

x

x

x

x

u u

u

u

uu

u

x

x

x

u

u

u

C1

C2

C3 C4

C5

y

z

y

y

y

yy

y

y

y

z

z zz

z

z

z

z

z

u

u

w

w

w

w

w

15/18


Example for partial submap isomorphism


∧ (¬y ∨ z ∨ u) ∧ (¬z ∨ u ∨ ¬w)∧ (x ∨ w ∨ ¬u)

C4

C2

C3

C5

w

C1

y

x

u

z

Pattern map M: Target map M ′:

w

x

x

x

x

x

x

u u

u

u

uu

u

x

x

x

u

u

u

C1

C2

C3 C4

C5

y

z

y

y

y

yy

y

y

y

z

z zz

z

z

z

z

z

u

u

w

w

w

w

w

16/18

Conclusion

Plan

1 Motivations




5 Conclusion

17/18

Conclusion

Conclusion

Tractability of deciding if M is isomorphic to a submap of M ′ ?

Let k be the number of connected components in M:

We can search for all occurrences of each component of M inpolynomial time

Let m be the maximum number of occurrences of a connectedcomponent of M

Number of candidate solutions to explore bounded by mk

The problem is tractable if k is small enough

Further workComplexity of the maximum common submap problem:

NP-hard in the general case

Open question if the submap must be connected

18/18

on the complexity of submap isomorphismon the complexity of submap isomorphism christine solnon1,...

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