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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).

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Generalizing Alcuin’s River Crossing Problem

Michael Lampis - Valia Mitsou

National Technical University of Athens

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 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
The Ferry Cover Problem

Lemma:

OPTVC (G) ≤ 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 on trees
Ferry Cover on Trees

Lemma:

For trees with

OPTVC (G) > 1

OPTFC (G) = OPTVC (G)

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ferry cover on trees1
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 trees2
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 trees3
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
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
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).
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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 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.
results for the trip constrained ferry cover problem

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
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?