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Achromatic numbers NP-completeness
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Achromatic number of signed graphs Dimitri Lajou (LaBRI Bordeaux) e - - PowerPoint PPT Presentation
Achromatic number of signed graphs Dimitri Lajou (LaBRI Bordeaux) e - - PowerPoint PPT Presentation
Achromatic numbers NP-completeness Achromatic number of signed graphs Dimitri Lajou (LaBRI Bordeaux) e Hocquard and Supervised by Herv Eric Sopena 14 Novembre 2018 Achromatic numbers NP-completeness Overview Achromatic numbers 1
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Overview
1
Achromatic numbers Achromatic number of a graph (Harary and Hedetniemi (1970)) Achromatic number of a 2-edge-colored graph Achromatic number of a signed graph Other achromatic numbers
2
NP-completeness
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Overview
1
Achromatic numbers Achromatic number of a graph (Harary and Hedetniemi (1970)) Achromatic number of a 2-edge-colored graph Achromatic number of a signed graph Other achromatic numbers
2
NP-completeness
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5/28 Achromatic numbers NP-completeness
→
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5/28 Achromatic numbers NP-completeness
→ A surjective homomorphism is a sequence of identifications
- f non adjacent vertices.
→
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5/28 Achromatic numbers NP-completeness
→ A surjective homomorphism is a sequence of identifications
- f non adjacent vertices.
→ →
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5/28 Achromatic numbers NP-completeness
→ A surjective homomorphism is a sequence of identifications
- f non adjacent vertices.
→ → A clique is a graph in which we cannot identify vertices. →
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5/28 Achromatic numbers NP-completeness
→ A surjective homomorphism is a sequence of identifications
- f non adjacent vertices.
→ → A clique is a graph in which we cannot identify vertices. → χ(G) is the order of the smallest clique we can reach from G by a homomorphism.
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5/28 Achromatic numbers NP-completeness
→ A surjective homomorphism is a sequence of identifications
- f non adjacent vertices.
→ → A clique is a graph in which we cannot identify vertices. → χ(G) is the order of the smallest clique we can reach from G by a homomorphism. χ(G) = 3
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5/28 Achromatic numbers NP-completeness
→ A surjective homomorphism is a sequence of identifications
- f non adjacent vertices.
→ → A clique is a graph in which we cannot identify vertices. → χ(G) is the order of the smallest clique we can reach from G by a homomorphism. χ(G) = 3
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5/28 Achromatic numbers NP-completeness
→ A surjective homomorphism is a sequence of identifications
- f non adjacent vertices.
→ → A clique is a graph in which we cannot identify vertices. → χ(G) is the order of the smallest clique we can reach from G by a homomorphism. χ(G) = 3 →
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5/28 Achromatic numbers NP-completeness
→ A surjective homomorphism is a sequence of identifications
- f non adjacent vertices.
→ → A clique is a graph in which we cannot identify vertices. → χ(G) is the order of the smallest clique we can reach from G by a homomorphism. χ(G) = 3 → ψ(G) is the order of the largest clique we can reach from G by a surjective homomorphism.
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5/28 Achromatic numbers NP-completeness
→ A surjective homomorphism is a sequence of identifications
- f non adjacent vertices.
→ → A clique is a graph in which we cannot identify vertices. → χ(G) is the order of the smallest clique we can reach from G by a homomorphism. χ(G) = 3 → ψ(G) is the order of the largest clique we can reach from G by a surjective homomorphism. ψ(G) = 4
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6/28 Achromatic numbers NP-completeness
Consider the following algorithm: Require: A graph G. Ensure: Returns an integer R(G). while there exist two non adjacent vertices do Choose randomly u and v such that uv / ∈ E(G). Identify u and v. end while return |G|
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6/28 Achromatic numbers NP-completeness
Consider the following algorithm: Require: A graph G. Ensure: Returns an integer R(G). while there exist two non adjacent vertices do Choose randomly u and v such that uv / ∈ E(G). Identify u and v. end while return |G| 1 2 3 4 5 6 7 χ(G) ψ(G)
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6/28 Achromatic numbers NP-completeness
Consider the following algorithm: Require: A graph G. Ensure: Returns an integer R(G). while there exist two non adjacent vertices do Choose randomly u and v such that uv / ∈ E(G). Identify u and v. end while return |G| 1 2 3 4 5 6 7 χ(G) ψ(G) R(G)
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7/28 Achromatic numbers NP-completeness
Problem: Achromatic number Instance: A graph G and an integer k Question: Is ψ(G) ≥ k? Theorem (Yannakakis and Gavril, 1980) The problem Achromatic number is NP-complete even for complements of bipartite graphs. Theorem (Bodlaender, 1989) The problem Achromatic number is NP-complete even for graphs that are both connected interval graphs and co-graphs.
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Overview
1
Achromatic numbers Achromatic number of a graph (Harary and Hedetniemi (1970)) Achromatic number of a 2-edge-colored graph Achromatic number of a signed graph Other achromatic numbers
2
NP-completeness
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Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges.
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Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges. →2ec
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9/28 Achromatic numbers NP-completeness
Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges. →2ec
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Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges.
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Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges. →2ec
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9/28 Achromatic numbers NP-completeness
Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges. →2ec
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9/28 Achromatic numbers NP-completeness
Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges.
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9/28 Achromatic numbers NP-completeness
Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges. →2ec
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Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges. →2ec A 2-edge-colored clique.
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Definition A 2-edge-colored graph (G, C) is a simple undirected graph where each edge can be either positive or negative. C is the set of negative edges. →2ec A 2-edge-colored clique. (G, C) →2ec (H, D) ⇐ ⇒ there exists a surjective homomorphism from (G, C) to (H, D).
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Definition For a 2-edge-colored graph (G, C), we define and note: χ2(G, C), the chromatic number of (G, C), is the order of the smallest 2-edge-colored clique (K, D) such that (G, C) →2ec (K, D), ψ2(G, C), the achromatic number of (G, C), is the order of the largest 2-edge-colored clique (K, D) such that (G, C) →2ec (K, D).
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Problem: 2-edge-colored graph achromatic number [2ec-an] Instance: A 2-edge-colored graph (G, C) and an integer k Question: Is ψ2(G, C) ≥ k? Theorem The problem 2ec-an is NP-complete even for graphs that are both connected interval graphs and co-graphs and for graphs that are complements of bipartite graphs.
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Overview
1
Achromatic numbers Achromatic number of a graph (Harary and Hedetniemi (1970)) Achromatic number of a 2-edge-colored graph Achromatic number of a signed graph Other achromatic numbers
2
NP-completeness
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13/28 Achromatic numbers NP-completeness
Definition A signed graph [G, Σ] is a graph where each edge can be either positive or negative. Moreover we can resign at each vertex v. Resigning at v consists in inverting the signs of all edges incident with v. Σ is the set of negative edges. resign →
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13/28 Achromatic numbers NP-completeness
Definition A signed graph [G, Σ] is a graph where each edge can be either positive or negative. Moreover we can resign at each vertex v. Resigning at v consists in inverting the signs of all edges incident with v. Σ is the set of negative edges. resign →
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13/28 Achromatic numbers NP-completeness
Definition A signed graph [G, Σ] is a graph where each edge can be either positive or negative. Moreover we can resign at each vertex v. Resigning at v consists in inverting the signs of all edges incident with v. Σ is the set of negative edges. resign →
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13/28 Achromatic numbers NP-completeness
Definition A signed graph [G, Σ] is a graph where each edge can be either positive or negative. Moreover we can resign at each vertex v. Resigning at v consists in inverting the signs of all edges incident with v. Σ is the set of negative edges. →s
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13/28 Achromatic numbers NP-completeness
Definition A signed graph [G, Σ] is a graph where each edge can be either positive or negative. Moreover we can resign at each vertex v. Resigning at v consists in inverting the signs of all edges incident with v. Σ is the set of negative edges. →s
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13/28 Achromatic numbers NP-completeness
Definition A signed graph [G, Σ] is a graph where each edge can be either positive or negative. Moreover we can resign at each vertex v. Resigning at v consists in inverting the signs of all edges incident with v. Σ is the set of negative edges. →s
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13/28 Achromatic numbers NP-completeness
Definition A signed graph [G, Σ] is a graph where each edge can be either positive or negative. Moreover we can resign at each vertex v. Resigning at v consists in inverting the signs of all edges incident with v. Σ is the set of negative edges. →s
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Definition A signed graph [G, Σ] is a graph where each edge can be either positive or negative. Moreover we can resign at each vertex v. Resigning at v consists in inverting the signs of all edges incident with v. Σ is the set of negative edges. →s A signed clique.
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Definition A signed graph [G, Σ] is a graph where each edge can be either positive or negative. Moreover we can resign at each vertex v. Resigning at v consists in inverting the signs of all edges incident with v. Σ is the set of negative edges. →s A signed clique. [G, Σ] →s [H, Π] ⇐ ⇒ there exists a surjective homomorphism (identifications and resignings) from [G, Σ] to [H, Π].
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This graph is a 2-edge-colored clique but not a signed clique. This graph is a 2-edge-colored clique and a signed clique.
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Definition For a signed graph [G, Σ], we define and note: χs[G, Σ], the chromatic number of [G, Σ], is the order of the smallest signed clique [K, Π] such that [G, Σ] →s [K, Π], ψs[G, Σ], the achromatic number of [G, Σ], is the order of the largest signed clique [K, Π] such that [G, Σ] →s [K, Π].
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Problem: Signed graph achromatic number [Signed-an] Instance: A signed graph [G, Σ] and an integer k Question: Is ψs[G, Σ] ≥ k? Theorem The problem Signed-an is NP-complete even for graphs that are both connected interval graphs and co-graphs and for graphs that are complements of bipartite graphs. G
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16/28 Achromatic numbers NP-completeness
Problem: Signed graph achromatic number [Signed-an] Instance: A signed graph [G, Σ] and an integer k Question: Is ψs[G, Σ] ≥ k? Theorem The problem Signed-an is NP-complete even for graphs that are both connected interval graphs and co-graphs and for graphs that are complements of bipartite graphs. G z
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16/28 Achromatic numbers NP-completeness
Problem: Signed graph achromatic number [Signed-an] Instance: A signed graph [G, Σ] and an integer k Question: Is ψs[G, Σ] ≥ k? Theorem The problem Signed-an is NP-complete even for graphs that are both connected interval graphs and co-graphs and for graphs that are complements of bipartite graphs. G z
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16/28 Achromatic numbers NP-completeness
Problem: Signed graph achromatic number [Signed-an] Instance: A signed graph [G, Σ] and an integer k Question: Is ψs[G, Σ] ≥ k? Theorem The problem Signed-an is NP-complete even for graphs that are both connected interval graphs and co-graphs and for graphs that are complements of bipartite graphs. G z
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Problem: Signed graph achromatic number [Signed-an] Instance: A signed graph [G, Σ] and an integer k Question: Is ψs[G, Σ] ≥ k? Theorem The problem Signed-an is NP-complete even for graphs that are both connected interval graphs and co-graphs and for graphs that are complements of bipartite graphs. G z G → K ⇐ ⇒ [G + z, ∅] →s [K + z, ∅]
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Overview
1
Achromatic numbers Achromatic number of a graph (Harary and Hedetniemi (1970)) Achromatic number of a 2-edge-colored graph Achromatic number of a signed graph Other achromatic numbers
2
NP-completeness
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Definition For a graph G and a signed graph [G, Σ]: ψmin[G, Σ] = min {ψ2(G, C) | (G, C) ∈ [G, Σ]}, ψmax[G, Σ] = max {ψ2(G, C) | (G, C) ∈ [G, Σ]}, ψ2ec
max(G) is the order of the greatest 2ec clique (H, D) such
that (G, C) →2ec (H, D), ψsigned
max (G) is the order of the greatest signed clique [H, Π]
such that [G, Σ] →s [H, Π].
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Definition For a graph G and a signed graph [G, Σ]: ψmin[G, Σ] = min {ψ2(G, C) | (G, C) ∈ [G, Σ]}, ψmax[G, Σ] = max {ψ2(G, C) | (G, C) ∈ [G, Σ]}, ψ2ec
max(G) is the order of the greatest 2ec clique (H, D) such
that (G, C) →2ec (H, D), ψsigned
max (G) is the order of the greatest signed clique [H, Π]
such that [G, Σ] →s [H, Π]. 1 2 3 4 5 6 7 8 χs[G, Σ] ψs[G, Σ] R[G, Σ]
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Definition For a graph G and a signed graph [G, Σ]: ψmin[G, Σ] = min {ψ2(G, C) | (G, C) ∈ [G, Σ]}, ψmax[G, Σ] = max {ψ2(G, C) | (G, C) ∈ [G, Σ]}, ψ2ec
max(G) is the order of the greatest 2ec clique (H, D) such
that (G, C) →2ec (H, D), ψsigned
max (G) is the order of the greatest signed clique [H, Π]
such that [G, Σ] →s [H, Π]. 1 2 3 4 5 6 7 8 χs[G, Σ] ψs[G, Σ] R[G, Σ] ψmax[G, Σ]
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χs[G, Σ] χ2(G, C) ψ2(G, C) ψs[G, Σ] ψmin[G, Σ] ψmax[G, Σ] A B means A ≤ B.
Figure: Relationship between some numbers for every signed graph [G, Σ] and every 2-edge-colored graph (G, C) ∈ [G, Σ].
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Observation In a 2-edge-colored clique (K, D), every two vertices u and v verify at least one of the following: either uv ∈ E(K) or u and v are joined by a path +− or −+. Theorem (Naserasr, Rollov´ a and Sopena, 2014) In a signed clique [K, Π], every two vertices u and v verify at least
- ne of the following: either uv ∈ E(K) or u and v belong to an
UC4. UC4
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Corollary A 2-edge-colored clique (K, D) has diameter 2.
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Overview
1
Achromatic numbers Achromatic number of a graph (Harary and Hedetniemi (1970)) Achromatic number of a 2-edge-colored graph Achromatic number of a signed graph Other achromatic numbers
2
NP-completeness
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Problem: Signed graph max-achromatic number [Signed-max-an] Instance: A signed graph [G, Σ] and an integer k Question: Is ψmax[G, Σ] ≥ k? Problem: Graph 2-edge-colored max-achromatic number [Max-2ec-an] Instance: A graph G and an integer k Question: Is ψ2ec
max(G) ≥ k?
Problem: Graph signed max-achromatic number[Max-signed-an] Instance: A graph G and an integer k Question: Is ψsigned
max (G) ≥ k?
Theorem The following problems are NP-complete: Signed-max-an, even for connected diamond-free perfect graphs, Max-2ec-an, even for connected diamond-free perfect graphs, Max-signed-an, even for connected perfect graphs.
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Problem: 3-partition Instance: A set A = {a1, . . . , a3m} ∈ N3m and an integer B such that B
4 < ai < B 2 for every i, 1 ≤ i ≤ m.
Question: Is there a partition {P1, . . . Pm} of A such that |Pi| = 3 and
aj∈Pi
aj = B for every i, 1 ≤ i ≤ m? Theorem (Garey and Johnson, 1990) The problem 3-partition is NP-complete.
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T G t1 t2 tm−1 tm C1 C2 Cm−1 Cm Cp−1 Cp L1 L2 LB ... ... ... . . . s1 s2 s3m 1 2 a1 − 1 a1 1 2 a2 − 1 a2 1 2 a3m − 1 a3m S
a positive complete subgraph on the vertices a negative complete subgraph on the vertices a complete bipartite positive graph between the left nodes and the right nodes
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T G t1 t2 tm−1 tm C1 C2 Cm−1 Cm Cp−1 Cp L1 L2 LB ... ... . . . 1 2 B − 1 B 1 2 B − 1 B S m stars
a positive complete subgraph on the vertices a negative complete subgraph on the vertices a complete bipartite positive graph between the left nodes and the right nodes
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T G t1 t2 tm−1 tm C1 C2 Cm−1 Cm Cp−1 Cp L1 L2 LB
a positive complete subgraph on the vertices a negative complete subgraph on the vertices a complete bipartite positive graph between the left nodes and the right nodes
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T G t1 t2 tm−1 tm C1 C2 Cm−1 Cm Cp−1 Cp L1 L2 LB
If 3-partition has a solution then ψmax[G, Σ] ≥ |T | + |G|.
a positive complete subgraph on the vertices a negative complete subgraph on the vertices a complete bipartite positive graph between the left nodes and the right nodes
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T G t1 t2 tm−1 tm C1 C2 Cm−1 Cm Cp−1 Cp L1 L2 LB ... ... ... . . . s1 s2 s3m 1 2 a1 − 1 a1 1 2 a2 − 1 a2 1 2 a3m − 1 a3m S
a positive complete subgraph on the vertices a negative complete subgraph on the vertices a complete bipartite positive graph between the left nodes and the right nodes
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Claim If (G, C) →2ec (K, D) where (K, D) is a 2-edge-colored clique of size greater than |T | + |G| then G → K ′ where K ′ has order greater than |T | + |G| and K ′ has diameter 2.
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T G t1 t2 tm−1 tm C1 C2 Cm−1 Cm Cp−1 Cp L1 L2 LB LB+1 LB+2 LB+r LB+r+1 LB+r+2 LB+r+q ... ... ... . . . s1 s2 s3m 1 2 a1 − 1 a1 1 2 a2 − 1 a2 1 2 a3m − 1 a3m S
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28/28 Achromatic numbers NP-completeness
Ordinary graphs 2-edge-colored graphs Signed graphs ψ NP-complete ψ2 NP-complete ψs NP-complete ψmax NP-complete NP-complete ψmin Π2 (complete ?) Π2 (complete ?) ψ2ec
max
NP-complete ψsigned
max
NP-complete
Table: Decision problems related to achromatic numbers.
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Ordinary graphs 2-edge-colored graphs Signed graphs ψ NP-complete ψ2 NP-complete ψs NP-complete ψmax NP-complete NP-complete ψmin Π2 (complete ?) Π2 (complete ?) ψ2ec
max
NP-complete ψsigned
max
NP-complete
Table: Decision problems related to achromatic numbers.
Thank you for your attention!
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References I
Hans L. Bodlaender. Achromatic number is NP-complete for cographs and interval graphs. Information Processing Letters, 31(3):135–138, 1989. Michael R. Garey and David S. Johnson. Computers and Intractability; A Guide to the Theory of NP-Completeness.
- W. H. Freeman & Co., New York, NY, USA, 1990.
Frank Harary and Stephen Hedetniemi. The achromatic number of a graph. Journal of Combinatorial Theory, 8(2):154–161, 1970.
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References II
Reza Naserasr, Edita Rollov´ a, and ´ Eric Sopena. Homomorphisms of signed graphs. Journal of Graph Theory, 79(3):178–212, 2015.
- M. Yannakakis and F. Gavril.