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The Ferry Cover Problem. Michael Lampis - Valia Mitsou National Technical University of Athens. Wolf. Goat. Cabbage. Guard. Boat. Previous Work. “Propositiones ad acuendos iuvenes”, Alcuin of York, 8th century A.D (in latin). We propose a generalization of Alcuin’s puzzle.

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The Ferry Cover Problem

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#### Presentation Transcript

The Ferry Cover Problem

Michael Lampis - Valia Mitsou

National Technical University of Athens

Wolf

Goat

Cabbage

Guard

Boat

### Previous Work

• “Propositiones ad acuendos iuvenes”, Alcuin of York, 8th century A.D (in latin).

• We propose a generalization of Alcuin’s puzzle

### Our generalization

• We seek to transport n items, given their incompatibility graph.

• Objective: Minimize the size of the boat

• We call this the Ferry Cover Problem

OPTFC (G) ≥ OPTVC (G)

OPTFC (G) ≥ OPTVC (G)

OPTFC (G) ≥ OPTVC (G)

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

OPTFC (G) ≤ OPTVC (G) + 1

### The Ferry Cover Problem

Lemma:

OPTVC (G) ≤ OPTFC (G) ≤ OPTVC (G) + 1

Graphs are divided into two categories:

• Type-0, ifOPTFC (G) = OPTVC (G)

• Type-1, if OPTFC (G) = OPTVC (G) + 1

### Hardness and Approximation Results

• Ferry Cover is NP and APX-hard (like Vertex Cover [Håstad1997]).

• A ρ-approximation algorithm for Vertex Cover yields a (ρ+1/ OPTFC)-approximation algorithm for Ferry Cover.

### Ferry Cover on Trees

Lemma:

For trees with OPTVC (G) > 1 (i.e. not stars)

OPTFC (G) = OPTVC (G) (Type-0)

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### Ferry Cover on Trees

• For a star with three or more leaves

OPTFC (G) = OPTVC (G)+1 = 2 (Type-1)

• For any other tree

OPTFC (G) = OPTVC (G) (Type-0)

Fact:

The Vertex Cover Problem can be solved in Polynomial time on trees.

### Ferry Cover on Trees

Theorem:

The Ferry Cover Problem can be solved in polynomial time on trees.

### Trip Constrained Ferry Cover

• Variation of Ferry Cover: we are also given a trip constraint. We seek to minimize the size of the boat s.t. there is a solution within this constraint.

• Definition: FCi→ determine the minimum boat size s.t. there is a solution with at most 2i+1 trips (i round-trips).

### FC1

• An interesting special case: only one round-trip allowed.

• FC1 is NP-hard.

• 2-approximation for general graphs.

• A (4/3+ε)-approximation for trees.

## FC1 is NP-hard

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2

3

### H-Colorings

• A traditional 3-coloring of graph G:

Vertices of color 2 are connected with vertices of colors 1 or 3

H :

### H-Colorings

• A constrained 3-coloring of graph G:

Vertices of color 2 are only connected with vertices of color 3

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H :

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3

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3

### H-Colorings

• A loose 3-coloring of graph G:

Vertices of color 2 can be connected with any vertex.

H :

### FC1 as an F1-Coloring problem

Vertices are partitioned into 3 groups:

• Those loaded and unloaded on the first trip

• Those remaining on the boat for all three trips

• Those loaded and unloaded on the third trip

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F1

3

### FC1 as an F1-Coloring problem

• Boat size is |V2|+ max{|V1|, |V3|}

• FC1 is equivalent to finding an F1-coloring that minimizes the above function.

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F1

3

### FC1 is NP-hard

Via a reduction from NAE3SAT

Sketch:

• Given a NAE3SAT formula φ with m clauses, create a new formula φ’

• From φ’ create a graph G

• G has an F1-coloring of cost 7m iff φ is satisfiable.

In NAESAT

For example:

Then:

### Reduction: Step 2

• For every clause construct a triangle.

• For every variable construct a complete bipartite graph.

• Connect each triangle vertex to one corresponding bipartite vertex.

Example:

Example:

Example:

Example:

false

true

true

false

...

...

F1

false

true

true

false

...

...

F1

false

true

true

false

...

...

F1

### Step 3: If φ’ issatisfiable

false

true

true

Cost = 7m = 2m + 2m + 3m

false

...

...

### Step 3: If cost=7m

• Observe that 7m is the minimum possible cost.

• It is possible to show that a coloring of this cost is a coloring of the previous form.

• Therefore, φ is satisfiable.

• Bonus: This reduction is also gap-preserving. Therefore, FC1 is APX-hard.

## Approximation algorithms for FC1

### 2-approximation

• The boat arrives to the destination bank twice.

• Therefore, its size must be at least n/2

• A boat of size n is a 2-approximation!

### (4/3+ε)– approximation for FC1 on trees (boat size 2n/3)

Fact:For a tree G OPTVC(G) ≤ n/2 (because tree is a bipartite graph)

ALGORITHM

• Load a vertex cover of size 2n/3.

• Unload n/3 vertices that form an Independent Set and return.

• Load the remaining vertices and transfer all of them to the destination.

i:

0

1

2

n-2

n-1

2n-1

NP-hard

≡FC

?

Trivial

NP-hard

### Further Work

• Is it NP-hard to determine whether a graph G is Type-0 or Type-1?

• Is FC equivalent to FCn?

• Is FCi for 1 < i < n-1 polynomially solved?

• Can we have an efficient approximation of FC1 in the general case?

The End!