Identifying codes and VC-dimension Aline Parreau University of Li` ege, Belgium Joint work with: N. Bousquet, A. Lagoutte, Z. Li and S. Tomass´ e Combgraph Seminar - October 3, 2013 1/27
Contents VC-dimension Identifying codes ? 2/27
Part I Identifying codes 3/27
Fire detection in a museum? 4/27
Fire detection in a museum? • Detector can detect fire in their room or in their neighborhood. 4/27
Fire detection in a museum? • Detector can detect fire in their room or in their neighborhood. 4/27
Fire detection in a museum? • Detector can detect fire in their room or in their neighborhood. 4/27
Fire detection in a museum? • Detector can detect fire in their room or in their neighborhood. 4/27
Fire detection in a museum? • Detector can detect fire in their room or in their neighborhood. • Each room must contain a detector or have a detector in a neighboring room. 4/27
Fire detection in a museum? • Detector can detect fire in their room or in their neighborhood. • Each room must contain a detector or have a detector in a neighboring room. 4/27
Modelization with a graph • Vertices V : rooms • Edges E : between two neighboring rooms 5/27
Modelization with a graph • Vertices V : rooms • Edges E : between two neighboring rooms 5/27
Modelization with a graph • Vertices V : rooms • Edges E : between two neighboring rooms • Set of detectors = dominating set S : ∀ u ∈ V , N [ u ] ∩ S � = ∅ 5/27
Modelization with a graph • Vertices V : rooms • Edges E : between two neighboring rooms • Set of detectors = dominating set S : ∀ u ∈ V , N [ u ] ∩ S � = ∅ 5/27
Back to the museum 6/27
Back to the museum Where is the fire ? 6/27
Back to the museum Where is the fire ? 6/27
Back to the museum Where is the fire ? 6/27
Back to the museum ? ? Where is the fire ? To locate the fire, we need more detectors. 6/27
Identifying where is the fire a c d b 6/27
Identifying where is the fire a c d a,b b,c,d c,d b b a,b,c b,c In each room, the set of detectors in the neighborhood is unique. 6/27
Modelization with a graph Identifying code C = subset of vertices which is • dominating : ∀ u ∈ V , N [ u ] ∩ C � = ∅ , • separating : ∀ u , v ∈ V , N [ u ] ∩ C � = N [ v ] ∩ C . V \ C a b c d d 1 • • - - a c 1 1 6 6 4 4 • 2 - - - b 3 - • • - 5 5 • • 4 - - 2 3 5 • • • - 6 - • • • Given a graph G , what is the size γ ID ( G ) of minimum identifying code ? 7/27
Modelization with a graph Identifying code C = subset of vertices which is • dominating : ∀ u ∈ V , N [ u ] ∩ C � = ∅ , • separating : ∀ u , v ∈ V , N [ u ] ∩ C � = N [ v ] ∩ C . V \ C a b c d d 1 • • - - a c 1 1 6 6 4 4 • 2 - - - b 3 - • • - 5 5 • • 4 - - 2 3 5 • • • - 6 - • • • Given a graph G , what is the size γ ID ( G ) of minimum identifying code ? 7/27
Modelization with a graph Identifying code C = subset of vertices which is • dominating : ∀ u ∈ V , N [ u ] ∩ C � = ∅ , • separating : ∀ u , v ∈ V , N [ u ] ∩ C � = N [ v ] ∩ C . V \ C a b c d d 1 • • - - a c 1 1 6 6 4 4 • 2 - - - b 3 - • • - 5 5 • • 4 - - 2 3 5 • • • - 6 - • • • Given a graph G , what is the size γ ID ( G ) of minimum identifying code ? 7/27
Modelization with a graph Identifying code C = subset of vertices which is • dominating : ∀ u ∈ V , N [ u ] ∩ C � = ∅ , • separating : ∀ u , v ∈ V , N [ u ] ∩ C � = N [ v ] ∩ C . V \ C a b c d d 1 • • - - a c 1 1 6 6 4 4 • 2 - - - b 3 - • • - 5 5 • • 4 - - 2 3 5 • • • - 6 - • • • Given a graph G , what is the size γ ID ( G ) of minimum identifying code ? 7/27
Some facts about identifying codes • Introduced in 1998 by Karpvosky, Chakrabarty and Levitin • Exists iff there is no twins u Twins: N [ u ] = N [ v ] v 8/27
Some facts about identifying codes • Introduced in 1998 by Karpvosky, Chakrabarty and Levitin • Exists iff there is no twins • NP-complete (Charon, Hudry, Lobstein, 2001) 8/27
Some facts about identifying codes • Introduced in 1998 by Karpvosky, Chakrabarty and Levitin • Exists iff there is no twins • NP-complete (Charon, Hudry, Lobstein, 2001) • Hard to approximate: best approximation factor log( | V | ) 8/27
Some facts about identifying codes • Introduced in 1998 by Karpvosky, Chakrabarty and Levitin • Exists iff there is no twins • NP-complete (Charon, Hudry, Lobstein, 2001) • Hard to approximate: best approximation factor log( | V | ) • Lower bound: → A vertex is identified by a nonempty subset of C ⇒ | V | ≤ 2 γ ID ( G ) − 1 γ ID ( G ) ≥ log( | V | + 1) Tight example: a c b 8/27
Some facts about identifying codes • Introduced in 1998 by Karpvosky, Chakrabarty and Levitin • Exists iff there is no twins • NP-complete (Charon, Hudry, Lobstein, 2001) • Hard to approximate: best approximation factor log( | V | ) • Lower bound: → A vertex is identified by a nonempty subset of C ⇒ | V | ≤ 2 γ ID ( G ) − 1 γ ID ( G ) ≥ log( | V | + 1) Tight example: a c b ab abc bc ac 8/27
In restricted classes of graphs? Example 1: Class of interval graphs 3 I 3 I 2 I 5 2 5 I 1 I 4 1 4 9/27
In restricted classes of graphs? Example 1: Class of interval graphs 3 I 3 I 2 I 5 2 5 I 1 I 4 1 4 Proposition Foucaud, Naserasr, P., Valicov, 2012+ If G is an interval graph, γ ID ( G ) ≥ � 2 | V | . 9/27
In restricted classes of graphs? Example 2: Class of split graphs Clique Stable set 10/27
In restricted classes of graphs? Example 2: Class of split graphs Clique Stable set Proposition For infinitely many split graphs G , γ ID ( G ) = log( | V | + 1). Proposition Foucaud, 2013 Min-Id-Code is log-APX-hard for split graphs. 10/27
Some known results in restricted classes of graphs Restriction: classes of graphs closed by induced subgraphs Graph class lower bound (order) Approximation All log n log APX-h Split log n log APX-h n 1 / 2 Interval open Unit Interval n 2 Bipartite log n log APX-h n 1 / 2 Line graphs 4 Chordal log n log APX-h Planar n 7 Cograph n 1 11/27
Part II VC-dimension 12/27
Shattered set • H = ( V , E ) an hypergraph • A set X ⊆ V is shattered if for all Y ⊆ X , there exists e ∈ E , s.t e ∩ X = Y . A 2-shattered set A 3-shattered set 13/27
Vapnik Chervonenkis (VC) dimension of an hypergraph • A set X is shattered if ∀ Y ⊆ X , ∃ e ∈ E , s.t e ∩ X = Y . • VC-dimension of H : largest size of a shattered set. 14/27
Vapnik Chervonenkis (VC) dimension of an hypergraph • A set X is shattered if ∀ Y ⊆ X , ∃ e ∈ E , s.t e ∩ X = Y . • VC-dimension of H : largest size of a shattered set. No 3-shattered set ⇒ VC-dim ≤ 2 14/27
Vapnik Chervonenkis (VC) dimension of an hypergraph • A set X is shattered if ∀ Y ⊆ X , ∃ e ∈ E , s.t e ∩ 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 14/27
Vapnik Chervonenkis (VC) dimension of an hypergraph • A set X is shattered if ∀ Y ⊆ X , ∃ e ∈ E , s.t e ∩ 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 14/27
Vapnik Chervonenkis (VC) dimension of an hypergraph • A set X is shattered if ∀ Y ⊆ X , ∃ e ∈ E , s.t e ∩ 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 14/27
Vapnik Chervonenkis (VC) dimension of an hypergraph • A set X is shattered if ∀ Y ⊆ X , ∃ e ∈ E , s.t e ∩ 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 14/27
Vapnik Chervonenkis (VC) dimension of an hypergraph • A set X is shattered if ∀ Y ⊆ X , ∃ e ∈ E , s.t e ∩ 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 14/27
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 15/27
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 over C ◮ Class of interval graphs has VC-dimension 2. ◮ Class of split graphs has infinite VC-dimension. 15/27
Split graphs have infinite VC-dimension For any k , there is a split graph with VC-dimension k . 1 2 Stable set Clique of size k of size 2 k 3 4 16/27
Split graphs have infinite VC-dimension For any k , there is a split graph with VC-dimension k . 1 2 Stable set Clique of size k of size 2 k 3 4 { 1 , 2 , 4 } 16/27
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