Parameterized Edge-Hamiltonicity Valia Mitsou, JST ERATO Joint - - PowerPoint PPT Presentation

Parameterized Edge-Hamiltonicity

Valia Mitsou, JST ERATO

Michael Lampis, Kazuhisa Makino, Yushi Uno. RIMS, Kyoto University Osaka Prefecture University Joint with:

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Edge-Hamiltonian Path

Edge-Hamiltonian Path “Ordering of the edges such that consecutive edges share a common vertex.”

2 3 4 5

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Edge-Hamiltonian Path

Edge-Hamiltonian Path “Ordering of the edges such that consecutive edges share a common vertex.”

2 3 4 5 Edge-Hamiltonian Path: 14,

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Edge-Hamiltonian Path

Edge-Hamiltonian Path “Ordering of the edges such that consecutive edges share a common vertex.”

1 2 3 5 Edge-Hamiltonian Path: 14, 42,

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Edge-Hamiltonian Path

Edge-Hamiltonian Path “Ordering of the edges such that consecutive edges share a common vertex.”

1 2 3 4 5 Edge-Hamiltonian Path: 14, 42, 25,

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Edge-Hamiltonian Path

Edge-Hamiltonian Path “Ordering of the edges such that consecutive edges share a common vertex.”

1 2 3 4 5 Edge-Hamiltonian Path: 14, 42, 25, 51,

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Edge-Hamiltonian Path

Edge-Hamiltonian Path “Ordering of the edges such that consecutive edges share a common vertex.”

1 2 3 4 5 Edge-Hamiltonian Path: 14, 42, 25, 51, 53,

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Edge-Hamiltonian Path

Edge-Hamiltonian Path “Ordering of the edges such that consecutive edges share a common vertex.”

1 2 3 4 5 Edge-Hamiltonian Path: 14, 42, 25, 51, 53, 34,

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Edge-Hamiltonian Path

Edge-Hamiltonian Path “Ordering of the edges such that consecutive edges share a common vertex.”

1 2 3 4 5 Edge-Hamiltonian Path: 14, 42, 25, 51, 53, 34, 32

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1 2 3 4 5

24 15 25 14 35 23 34

Edge-Hamiltonian Path

Edge-Hamiltonian Path: 14, 42, 25, 51, 53, 34, 32

Edge-Hamiltonian Path is like Hamiltonian Path

• n Line Graphs.

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1 2 3 4 5

24 15 25 14 35 23 34

Edge-Hamiltonian Path

Edge-Hamiltonian Path: 14, 42, 25, 51, 53, 34, 32

Edge-Hamiltonian Path is like Hamiltonian Path

• n Line Graphs.

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1 2 3 4 5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Harary & Nash-Williams (1965): Edge-Hamiltonian Path is equivalent with Dominating Eulerian Subgraph.

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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1 2 3 4 5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Eulerian path: 53, 34

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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1 2 3 4

5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Eulerian path: 53, 34 Edge-Hamiltonian Path:

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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1 2 3 4

5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Eulerian path: 53, 34 Edge-Hamiltonian Path: 51

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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1 2 3 4

5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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1 2

3

4 5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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1 2

3

4 5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53, 32

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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1 2 3

4

5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53, 32, 34

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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1 2 3

4

5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53, 32, 34, 41

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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1 2 3

4

5

Edge-Hamiltonian Path as Dominating Eulerian Subgraph

Eulerian path: 53, 34 Edge-Hamiltonian Path: 51, 52, 53, 32, 34, 41, 42

Dominating Eulerian Subgraph “Find a connected subgraph G'

• f G, st:
• 1. G' is Eulerian;
• 2. All remaining edges of G are

covered by a vertex in G'.”

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EHP – related work

EHP is NP-Complete (Bertossi 1981);

– NP-Complete even on bipartite graphs (Lai and Wei

1993) or on graphs with maximum degree 3 (Ryjacek et al. 2011). Demaine et. al. 2014: EHP on bipartite graphs parameterized by the size of the smaller part is in XP. Motivation: UNO game.

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Solitaire UNO as Hamiltonicity

Deck of cards

• c colors;
• b numbers.

Matching rule: Cards agree either in number or in color.

➔ Given m cards, discard them

• ne by one, following the

matching rule.

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Solitaire UNO as Hamiltonicity

Deck of cards

• c colors;
• b numbers.

Matching rule: Cards agree either in number or in color.

➔ Given m cards, discard them

• ne by one, following the

matching rule.

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g b r y 2 3 4 5

Colors Numbers

Solitaire UNO as Edge-Hamiltonicity

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g b r y 2 3 4 5

Colors Numbers

Solitaire UNO as Edge-Hamiltonicity

Solution: 2, 5, 5, 5, 4, 2, 2, 3, 4, 4

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EHP – related work

EHP is NP-Complete (Bertossi 1981);

– NP-Complete even on bipartite graphs (Lai and Wei

1993) or on graphs with maximum degree 3 (Ryjacek et al. 2011). Demaine et. al. 2014: EHP on bipartite graphs parameterized by the size of the smaller part is in XP.

• Question: Is EHP on bipartite graphs FPT?

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Our Results

• EHP parameterized by size of a vertex cover admits a

cubic kernel;

– (Independently shown by Dei et. al. in FUN 2014)

• EHP on arbitrary hypergraphs parameterized by size
• f a hitting set is FPT;
• EHP parameterized by treewidth or cliquewidth is FPT.

– (Vertex HP is FPT for tw but W-hard for cw.) – Invoke theorem by Gursky & Wanke (2000): Graphs of

bounded cw and no large bi-cliques have bounded tw.

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Bipartite Graphs (one small part)

Colors (c) Numbers (b)

g b r y 1 2 3 4 5 6 7 8 9 IS IS

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Graphs of small Vertex Cover

Colors (c) Numbers (b)

g b r y 1 2 3 4 5 6 7 8 9 IS

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Kernelization algorithms

Given an instance of the problem, we construct (in polynomial time) an equivalent new instance with size that depends only on the parameter.

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EHP in bipartite graphs with one small part admits a cubic kernel

Given an instance of EHP with c colors and b numbers, we construct (in polynomial time) an equivalent new instance where:

– For each color, there will be at most O(c2) numbers

• f that color.

– Thus, there will be at most O(c3) numbers.

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EHP in bipartite graphs with one small part admits a cubic kernel

In an EHP, each color-group appears at most c times. SOL: 1 5 5 3 … 2 8 8 7

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EHP in bipartite graphs with one small part admits a cubic kernel

In an EHP, each color-group appears at most c times. SOL':1 5 8 2 … 3 5 8 7

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EHP in bipartite graphs with one small part admits a cubic kernel

In an EHP, each color-group appears at most c times. SOL': … … …

At most c At most c-1

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EHP in bipartite graphs with one small part admits a cubic kernel

Intuition: EHP has a small backbone, which we need to identify. All other edges can move freely in and out of the EHP. SOL': … … …

At most 2c 5

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EHP in bipartite graphs with one small part admits a cubic kernel

Intuition: EHP has a small backbone, which we need to identify. All other edges can move freely in and out of the EHP. SOL': … … …

At most 2c

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EHP in bipartite graphs with one small part admits a cubic kernel

Intuition: EHP has a small backbone, which we need to identify. All other edges can move freely in and out of the EHP. SOL': … … …

At most 2c 5

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers > 4c2 colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers > 4c2 S

S m a l l

• v

e r l a p

N(S) g

≤ 4c

colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers > 4c2 S

S m a l l

• v

e r l a p

N(S) 7 colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers > 4c2 S

S m a l l

• v

e r l a p

N(S) 7 colors

Graph that contains 7 has an EHP SOL Graph that doesn't contain 7 has an EHP SOL'.

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers ≥ 4c2 S

S m a l l

• v

e r l a p

N(S) 7

SOL':

At most c

colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers ≥ 4c2 S

S m a l l

• v

e r l a p

N(S) 7

SOL': 7

At most c

colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers > 4c2 S

S m a l l

• v

e r l a p

N(S) 7

SOL: … 7 …

colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers S

S m a l l

• v

e r l a p

N(S) 7

SOL: … 4 7 9 …

colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers S

S m a l l

• v

e r l a p

N(S)

SOL': … 4 7 9 …

colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers S

S m a l l

• v

e r l a p

N(S) 7

SOL: … 4 7 7 …

colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers S

S m a l l

• v

e r l a p

N(S)

> 4c

7

SOL: … 4 7 7 …

y colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers S

S m a l l

• v

e r l a p

N(S)

> 4c

7

SOL: … 4 7 7 …

y

At most 2c At most 2c

colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers S

S m a l l

• v

e r l a p

N(S) 7 2

SOL: … 4 7 7 … 2 2

y colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers S

S m a l l

• v

e r l a p

N(S) 7 2

SOL: … 4 7 7 … 2 2

y colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers S

S m a l l

• v

e r l a p

N(S)

SOL': … 4 2 2 … 7

colors 7 y

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EHP in bipartite graphs with one small part admits a cubic kernel

b Blue numbers S

S m a l l

• v

e r l a p

N(S)

Ø

colors

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EHP in bipartite graphs with one small part admits a cubic kernel

b

S m a l l

• v

e r l a p

… Blue numbers ≤ 4c2 t p g colors

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EHP on r-uniform hypergraphs

Colors (c)

• Attr. A (b1)

g b r y 1 2 3 4 5 6 7 8 9

• Attr. B (b2)

1 2 3 4 5 6 7 8 9

• Attr. C (b3)

1 2 3 4 5 6 7 8 9

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EHP on r-uniform hypergraphs

Colors (c)

• Attr. A (b1)

g b r y 1 2 3 4 5 6 7 8 9

• Attr. B (b2)

1 2 3 4 5 6 7 8 9

• Attr. C (b3)

1 2 3 4 5 6 7 8 9

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EHP on arbitrary hypergraphs with small hitting set

Colors (c)

• Attr. A (b1)

g b r y 1 2 3 4 5 6 7 8 9

• Attr. B (b2)

1 2 3 4 5 6 7 8 9

• Attr. C (b3)

1 2 3 4 5 6 7 8 9

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EHP on arbitrary hypergraphs with small hitting set

Colors (c)

• Attr. A (b1)

g b r y 1 2 3 4 5 6 7 8 9

• Attr. B (b2)

1 2 3 4 5 6 7 8 9

• Attr. C (b3)

1 2 3 4 5 6 7 8 9

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EHP on h-graphs parameterized by Hitting Set is FPT

Idea:

– Few backbone hyperedges. – Color-coding helps identify them.

Bonus: Vertex HP parameterized by the complement chromatic number is FPT.

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EHP parameterized by tw is FPT

Dominating Eulerian Subgraph: ∃ V'⊂V, E'⊂E s.t

– V' is a vertex cover of G(V,E); – G'(V',E') is connected; – G' is Eulerian.

All the above properties can be expressed in CMSO2.

• Dominating Eulerian Subgraph (and Edge-Hamiltonian

Path) is FPT when parameterized by tw.

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EHP parameterized by cw is FPT

Theorem (Gursky & Wanke 2000): If G has cw k and does not contain Kt,t as a subgraph, then G has tw at most 3kt.

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EHP parameterized by cw is FPT

• G has Eulerian Dominating Subgraph iff G' does;
• If cw(G) = k, then cw(G') = k+2.

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Conclusions

Edge-Hamiltonian Path is FPT for the following parameters:

• Vertex Cover (it even admits a cubic kernel)
• Hitting Set
• Treewidth
• Cliquewidth

M M e e r r c c i i b b e e a a u u c c

• u

u p p ! !