The Ferry Cover Problem
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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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The ferry cover problem

The Ferry Cover Problem

Michael Lampis - Valia Mitsou

National Technical University of Athens


The ferry cover problem

Wolf


The ferry cover problem

Goat


The ferry cover problem

Cabbage


The ferry cover problem

Guard


The ferry cover problem

Boat


Previous work

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

Our generalization


Our generalization1

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


The ferry cover problem

OPTFC (G) ≥ OPTVC (G)


The ferry cover problem

OPTFC (G) ≥ OPTVC (G)


The ferry cover problem

OPTFC (G) ≥ OPTVC (G)


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

OPTFC (G) ≤ OPTVC (G) + 1


The ferry cover problem

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

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 in other well known graph topologies

Ferry Cover in other well known graph topologies


Ferry cover on trees

Ferry Cover on Trees

Lemma:

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

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


The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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The ferry cover problem

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Ferry cover on trees1

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 trees2

Ferry Cover on Trees

Theorem:

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


The trip constrained ferry cover problem

The Trip Constrained Ferry Cover Problem


Trip constrained ferry cover

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).


The ferry cover problem

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.


Fc 1 is np hard

FC1 is NP-hard


H colorings

1

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 colorings1

H-Colorings

  • A constrained 3-coloring of graph G:

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

1

H :

2

3


H colorings2

1

2

3

H-Colorings

  • A loose 3-coloring of graph G:

    Vertices of color 2 can be connected with any vertex.

H :


Fc 1 as an f 1 coloring problem

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

1

2

F1

3


Fc 1 as an f 1 coloring problem1

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.

1

2

F1

3


Fc 1 is np hard1

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.


Reduction step 1

Reduction: Step 1

In NAESAT

For example:

Then:


Reduction step 2

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.


Reduction step 21

Reduction: Step 2

Example:


Reduction step 22

Reduction: Step 2

Example:


Reduction step 23

Reduction: Step 2

Example:


Reduction step 24

Reduction: Step 2

Example:


Step 3 if is satisfiable

Step 3: If φ’ issatisfiable

false

true

true

false

...

...

F1


Step 3 if is satisfiable1

Step 3: If φ’ issatisfiable

false

true

true

false

...

...

F1


Step 3 if is satisfiable2

Step 3: If φ’ issatisfiable

false

true

true

false

...

...

F1


Step 3 if is satisfiable3

Step 3: If φ’ issatisfiable

false

true

true

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

false

...

...


Step 3 if cost 7m

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 fc 1

Approximation algorithms for FC1


2 approximation

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 approximation for fc 1 on trees boat size 2n 3

(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.


4 3 approximation

4/3 Approximation


4 3 approximation1

4/3 Approximation


4 3 approximation2

4/3 Approximation


4 3 approximation3

4/3 Approximation


4 3 approximation4

4/3 Approximation


4 3 approximation5

4/3 Approximation


4 3 approximation6

4/3 Approximation


4 3 approximation7

4/3 Approximation


Optimal solution

Optimal Solution


Optimal solution1

Optimal Solution


Optimal solution2

Optimal Solution


Optimal solution3

Optimal Solution


Optimal solution4

Optimal Solution


Optimal solution5

Optimal Solution


Optimal solution6

Optimal Solution


Optimal solution7

Optimal Solution


Results for the trip constrained ferry cover problem

i:

0

1

2

n-2

n-1

2n-1

Results for the Trip Constrained Ferry Cover Problem

NP-hard

≡FC

?

Trivial

NP-hard


Further work

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 ferry cover problem

The End!


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