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

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


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 in other well known graph topologies


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.


The Trip Constrained Ferry Cover Problem


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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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

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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

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.


Reduction: Step 2

Example:


Reduction: Step 2

Example:


Reduction: Step 2

Example:


Reduction: Step 2

Example:


Step 3: If φ’ issatisfiable

false

true

true

false

...

...

F1


Step 3: If φ’ issatisfiable

false

true

true

false

...

...

F1


Step 3: If φ’ issatisfiable

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.


4/3 Approximation


4/3 Approximation


4/3 Approximation


4/3 Approximation


4/3 Approximation


4/3 Approximation


4/3 Approximation


4/3 Approximation


Optimal Solution


Optimal Solution


Optimal Solution


Optimal Solution


Optimal Solution


Optimal Solution


Optimal Solution


Optimal Solution


i:

0

1

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n-2

n-1

2n-1

Results for the Trip Constrained Ferry Cover Problem

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!


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