Generalizing Alcuin’s River Crossing Problem

1 / 116

# Generalizing Alcuin’s River Crossing Problem - PowerPoint PPT Presentation

Generalizing Alcuin’s River Crossing 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).

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'Generalizing Alcuin’s River Crossing Problem' - cooper-allison

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Generalizing Alcuin’s River Crossing 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

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

OPTFC (G) = OPTVC (G)

u

w

v

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

v

w

u

u

w

v

u

w

v

u

w

v

Ferry Cover on Trees

But what happens when OPTVC (G) = 1 ?

• We have seen that for a star with two leaves

OPTFC (G) = OPTVC (G) = 1

• For a star with three or more leaves…
Ferry Cover on Trees
• For a star with three or more leaves

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

• For any other tree

OPTFC (G) = OPTVC (G)

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.

Ferry Cover in other well known graph topologies
• OPTFC (Pn) = OPTVC (Pn)
• OPTFC (Cn) = OPTVC (Cn)
• OPTFC (Kn) = OPTVC (Kn)
• OPTFC (Sn) = OPTVC (Sn) + 1
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.
• Trivial 2-approximation for general graphs.
• A (4/3+ε)-approximation for trees is possible.
(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-1

n

2n-1

Results for the Trip Constrained Ferry Cover Problem

NP-hard

≡FC

X

Trivial

NP-hard

Further Work
• Is it NP-hard to determine whether OPTFC(G) = OPTVC(G) or

OPTFC(G) = OPTVC(G) +1

• Is FC equivalent to FCn?
• Is FCi for 1 < i < n polynomially solved?
• Can we have an efficient approximation of FC1 in the general case?