Generalizing Alcuin’s River Crossing Problem

Michael Lampis - Valia MitsouNational 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

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

Hardness and Approximation Results

• Ferry Cover is NP and APX-hard (like Vertex Cover [Håstad 1997]).

• 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

OPTFC (G) = OPTVC (G)

OPTVC (G) > 1

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

• For any other treeOPTFC (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

A family of graphs where OPTFC (G) = OPTVC (G)+1

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.

• 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)

ALGORITHM1. Load a vertex cover of size 2n/3.2. Unload n/3 vertices that form an

Independent Set and return.3. 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

Results for the Trip Constrained Ferry Cover

Problem

NP-hardTrivialX

0 1 2 ni: 2n-1n-1

NP-hard

≡ FC

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?