Identifying codes and VC-dimension

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Identifying codes and VC-dimension

Aur´ elie Lagoutte

LIP, ENS Lyon Joint work with:

N. Bousquet, Z. Li, A. Parreau and S. Tomass´e BGW - November 19, 2014

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Part I

Identifying codes

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Modelization with a graph

Identifying codeC = subset of vertices which is

• dominating : ∀u ∈V,N[u]∩C 6=∅,

• separating : ∀u,v ∈V,N[u]∩C 6=N[v]∩C.

5 1

2

6 3 b 4

a c d

5

1 6 4

V\ C a b c d

1 • • - -

2 - • - -

3 - • • -

4 - - • •

5 • • • -

6 - • • •

Given a graph G, what is the size γ^ID(G) of minimum identifying code ?

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Modelization with a graph

5 1

2

6 3 b 4

a c d

5

1 6 4

V\ C a b c d

1 • • - -

2 - • - -

3 - • • -

4 - - • •

5 • • • -

6 - • • •

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Modelization with a graph

5 1

2

6 3 b 4

a c d

5

1 6 4

V\ C a b c d

1 • • - -

2 - • - -

3 - • • -

4 - - • •

5 • • • -

6 - • • •

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Modelization with a graph

5 1

2

6 3 b 4

a c d

5

1 6 4

V\ C a b c d

1 • • - -

2 - • - -

3 - • • -

4 - - • •

5 • • • -

6 - • • •

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Some facts about identifying codes

• Introduced in 1998 by Karpvosky, Chakrabarty and Levitin

• Exists iff there is no twins u v

Twins: N[u] =N[v]

• NP-complete (Charon, Hudry, Lobstein, 2001)

• Hard to approximate: best approximation factor log(|V|)

• Lower bound:

→A vertex is identified by a nonempty subset ofC ⇒ |V| ≤2^γ^ID^(G⁾−1

γ^ID(G)≥log(|V|+ 1) Tight example:

b c

a

bc ac ab abc

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Some facts about identifying codes

• Exists iff there is no twins

• NP-complete(Charon, Hudry, Lobstein, 2001)

• Lower bound:

b c

a

bc ac ab abc

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Some facts about identifying codes

• Lower bound:

b c

a

bc ac ab abc

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Some facts about identifying codes

• Lower bound:

→A vertex is identified by a nonempty subset ofC ⇒ |V| ≤2^γ^ID^(G)−1

b c

a

bc ac ab abc

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Some facts about identifying codes

• Lower bound:

→A vertex is identified by a nonempty subset ofC ⇒ |V| ≤2^γ^ID^(G)−1

b c

a

bc ac ab abc

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In restricted classes of graphs?

Example 1 : Class of interval graphs

I1 I4

I2 I5

I3

1 2 3

4 5

IfG is an interval graph, γ^ID(G)≥p 2|V|.

Proposition Foucaud, Naserasr, Parreau, Valicov, 2012+

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In restricted classes of graphs?

Example 1 : Class of interval graphs

I1 I4

I2 I5

I3

1 2 3

4 5

IfG is an interval graph, γ^ID(G)≥p 2|V|.

Proposition Foucaud, Naserasr, Parreau, Valicov, 2012+

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In restricted classes of graphs?

Example 2: Class of split graphs

Clique Stable set

For infinitely many split graphs G,γ^ID(G) = log(|V|+ 1). Proposition

Min-Id-Codeis log-APX-hard for split graphs. Proposition Foucaud, 2013

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In restricted classes of graphs?

Example 2: Class of split graphs

Clique Stable set

For infinitely many split graphs G,γ^ID(G) = log(|V|+ 1).

Proposition

Min-Id-Codeis log-APX-hard for split graphs.

Proposition Foucaud, 2013

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Part II

VC-dimension

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Shattered set

• H= (V,E) an hypergraph

• A set X ⊆V is shattered if for allY ⊆X, there exists e ∈ E, s.te∩X =Y.

A 2-shattered set A 3-shattered set

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Vapnik Chervonenkis (VC) dimension of an hypergraph

• A set X is shatteredif ∀Y ⊆X,∃e ∈ E, s.te∩X =Y.

• VC-dimension of H: largest size of a shattered set.

No 3-shattered set⇒ VC-dim≤2 A 2-shattered set ⇒VC-dim = 2

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Vapnik Chervonenkis (VC) dimension of an hypergraph

No 3-shattered set⇒ VC-dim≤2

A 2-shattered set ⇒VC-dim = 2

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Vapnik Chervonenkis (VC) dimension of an hypergraph

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Vapnik Chervonenkis (VC) dimension of an hypergraph

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Vapnik Chervonenkis (VC) dimension of an hypergraph

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Vapnik Chervonenkis (VC) dimension of an hypergraph

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Vapnik Chervonenkis (VC) dimension of an hypergraph

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VC dimension of a graph / of a class of graph

• VC-dimension of G: VC-dim of the hypergraph of closed neighborhoods

⇒

VC-dim(G) = 2

• VC-dimension of a class C: maximal VC-dimension overC

I Class ofinterval graphshas VC-dimension 2.

I Class ofsplit graphshas infinite VC-dimension.

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VC dimension of a graph / of a class of graph

• VC-dimension of G: VC-dim of the hypergraph of closed neighborhoods

⇒

VC-dim(G) = 2

• VC-dimension of a class C: maximal VC-dimension overC

I Class ofinterval graphshas VC-dimension 2.

I Class ofsplit graphshas infinite VC-dimension.

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Split graphs have infinite VC-dimension

For anyk, there is a split graph with VC-dimension k.

Clique of sizek 1

2 3 4 Stable set

of size 2^k

{1,2,4}

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Split graphs have infinite VC-dimension

For anyk, there is a split graph with VC-dimension k.

Clique of sizek 1

2 3 4 Stable set

of size 2^k

{1,2,4}

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Intervals have finite VC-dimension

There is no interval graph with VC-dimension 3.

Assume there is a shattered set{1,2,3}.

Shattered set

1 3 2

Intervals 1 2 3 1

2 3 1

2 3

Objection {1,3} {1,3} {1,3}

1 3 2

{1,3} {1,2}

{2,3}

Forbidden for intervals !

Interval graphs have VC-dimension at most 2.

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Intervals have finite VC-dimension

Shattered set

1 3 2

Intervals 1 2 3 1

2 3 1

2 3

Objection {1,3} {1,3} {1,3}

1 3 2

{1,3} {1,2}

{2,3}

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Intervals have finite VC-dimension

Shattered set

1 3 2

Intervals 1 2 3 1

2 3 1

2 3

Objection {1,3} {1,3} {1,3}

1 3 2

{1,3}

{1,2}

{2,3}

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Intervals have finite VC-dimension

Shattered set

1 3 2

Intervals 1 2 3 1

2 3 1

2 3

Objection {1,3} {1,3} {1,3}

1 3 2

{1,3}

{1,2}

{2,3}

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Intervals have finite VC-dimension

Shattered set

1 3 2

Intervals 1 2 3 1

2 3 1

2 3

Objection {1,3} {1,3} {1,3}

1 3 2

{1,3}

{1,2}

{2,3}

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Part III

Identifying codes and VC-dimension

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Back to identifying codes

Graph class Lower bound (order) Approx

All logn log APX-h

Split logn log APX-h

Interval n^1/2 open

Unit Interval n 2

Bipartite logn log APX-h

Line graphs n^1/2 4

Chordal logn log APX-h

Planar n 7

Cograph n 1

VC dim)

∞

∞ 2 2

∞ 4

∞ 4 2

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Back to identifying codes

All logn log APX-h

Interval n^1/2 open

Unit Interval n 2

Line graphs n^1/2 4

Planar n 7

Cograph n 1

VC dim)

∞

∞ 2 2

∞ 4

∞ 4 2

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Back to identifying codes

All logn log APX-h

Interval n^1/2 open

Unit Interval n 2

Line graphs n^1/2 4

Planar n 7

Cograph n 1

VC dim)

∞

∞ 2 2

∞ 4

∞ 4 2

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Back to identifying codes

All logn log APX-h

Interval n^1/2 open

Unit Interval n 2

Line graphs n^1/2 4

Planar n 7

Cograph n 1

VC dim)

∞

∞ 2 2 4 4 2

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A dichotomy result

VC-dim of C ?

Infinite

⇓

There are infinitely manyG, γ^ID(G)≈log|V|

Finite d

⇓

For allG,

γ^ID(G)≥(|V| −1)^1/d Theorem

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Proof - Case with finite VC dimension

IfChasfinite VC-dimensiond,∀G ∈ C,γ^ID(G)≥(|V| −1)^1/d. Proposition

Proof: direct consequence of:

LetX be a subset of vertices of graphG of VC-dimensiond. The number ofdistinct tracesonX is at mostPd

i=1

|X| i

≤|X|^d+ 1. Sauer’s Lemma

X

Distinct traces≤ |X|^d+ 1

γ^ID(G)

≤γ^ID(G)^d+ 1 Identifying code

All vertices|V|

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Proof - Case with finite VC dimension

i=1

|X| i

≤|X|^d+ 1.

Sauer’s Lemma

X

γ^ID(G)

All vertices|V|

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Proof - Case with finite VC dimension

i=1

|X| i

≤|X|^d+ 1.

Sauer’s Lemma

X

γ^ID(G)

All vertices|V|

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Back to the table

Graph class Lower bound Approx

?

VC-dim

All logn log APX-h ∞

Split logn log APX-h ∞

Bipartite logn log APX-h ∞

Chordal logn log APX-h ∞

Interval n^1/2 open 2

Unit Interval n 2 2

Line graphs n^1/2 4 4

Planar n 7 4

Cograph n 1 2

Permutation n^1/3 open 3

Unit disk graphs n^1/3 open 3

• Lower bound not optimal (ex: Line graphs)

• What about approximation ?

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Back to the table

Graph class Lower bound Approx

?

VC-dim

Unit Interval n 2 2

Planar n 7 4

Cograph n 1 2

• Lower bound not optimal (ex: Line graphs)

• What about approximation ?

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Inapproximability in infinite VC dimension

IfChas∞VC-dimension,Min-Id-Codeis log-APX-hard onC.

Theorem

Consequence of:

IfC has infinite VC-dimension,C contains:

• allbipartitegraphs, or

• allsplit graphs, or

• allcobipartite graphs. Proposition

and

Min-Id-Codeis log-APX-hard on bipartite, split and cobipartite graphs.

Theorem Foucaud, 2013

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Inapproximability in infinite VC dimension

IfChas∞VC-dimension,Min-Id-Codeis log-APX-hard onC.

Theorem

Consequence of:

IfC has infinite VC-dimension,C contains:

• allbipartitegraphs, or

• allsplit graphs, or

• allcobipartitegraphs.

Proposition

and

Min-Id-Codeis log-APX-hard on bipartite, split and cobipartite graphs.

TheoremFoucaud, 2013

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In the finite case ?

Graph class Lower bound Approx VC-dim

Unit Interval n 2 2

Planar n 7 4

Cograph n 1 2

Is there a constant approximation in finite VC-dimension?

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A class of finite VC-dimension with no good approximation

Min-ID-Codecannot be approximed within ao(log|V|) factor in polynomial time for the class ofbipartiteC4-freegraphs.

Theorem

• Class of VC-dimension 2

• Reduction fromSet covering with intersection 1.

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What about open approximation ?

Unit Interval n 2 2

Planar n 7 4

Cograph n 1 2

Thank you for your attention !

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What about open approximation ?

Unit Interval n 2 2

Planar n 7 4

Cograph n 1 2

Thank you for your attention !

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What about open approximation ?

Interval n^1/2 6 2

Unit Interval n 2 2

Planar n 7 4

Cograph n 1 2

Thank you for your attention !

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What about open approximation ?

Interval n^1/2 6 2

Unit Interval n 2 2

Planar n 7 4

Cograph n 1 2

Thank you for your attention !

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Identifying codes and VC-dimension

Identifying codes and VC-dimension

Aur´ elie Lagoutte

Contents

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Part I

Identifying codes

Modelization with a graph

Modelization with a graph

Modelization with a graph

Modelization with a graph

Some facts about identifying codes

Some facts about identifying codes

Some facts about identifying codes

Some facts about identifying codes

Some facts about identifying codes

In restricted classes of graphs?

In restricted classes of graphs?

In restricted classes of graphs?

In restricted classes of graphs?

Part II

VC-dimension

Shattered set

Vapnik Chervonenkis (VC) dimension of an hypergraph

Vapnik Chervonenkis (VC) dimension of an hypergraph

Vapnik Chervonenkis (VC) dimension of an hypergraph

Vapnik Chervonenkis (VC) dimension of an hypergraph

Vapnik Chervonenkis (VC) dimension of an hypergraph

Vapnik Chervonenkis (VC) dimension of an hypergraph

Vapnik Chervonenkis (VC) dimension of an hypergraph

VC dimension of a graph / of a class of graph

⇒

VC dimension of a graph / of a class of graph

⇒

Split graphs have infinite VC-dimension

Split graphs have infinite VC-dimension

Intervals have finite VC-dimension

Intervals have finite VC-dimension

Intervals have finite VC-dimension

Intervals have finite VC-dimension

Intervals have finite VC-dimension

Part III

Identifying codes and VC-dimension

Back to identifying codes

Back to identifying codes

Back to identifying codes

Back to identifying codes

A dichotomy result

VC-dim of C ?

Infinite

⇓

Finite d

⇓

Proof - Case with finite VC dimension

Proof - Case with finite VC dimension

Proof - Case with finite VC dimension

Back to the table

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Back to the table

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Inapproximability in infinite VC dimension

Inapproximability in infinite VC dimension

In the finite case ?

A class of finite VC-dimension with no good approximation

What about open approximation ?

Thank you for your attention !

What about open approximation ?

Thank you for your attention !

What about open approximation ?

Thank you for your attention !

What about open approximation ?

Thank you for your attention !