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

Michael Lampis - Valia Mitsou

National Technical University of Athens

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
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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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 remaining on the boat for all three trips

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F1

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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.
Step 3: If φ’ issatisfiable

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F1

Step 3: If φ’ issatisfiable

false

true

true

false

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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.

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?