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The Ferry Cover Problem

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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  1. The Ferry Cover Problem Michael Lampis - Valia Mitsou National Technical University of Athens

  2. Wolf

  3. Goat

  4. Cabbage

  5. Guard

  6. Boat

  7. Previous Work • “Propositiones ad acuendos iuvenes”, Alcuin of York, 8th century A.D (in latin). • We propose a generalization of Alcuin’s puzzle

  8. Our generalization

  9. 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

  10. OPTFC (G) ≥ OPTVC (G)

  11. OPTFC (G) ≥ OPTVC (G)

  12. OPTFC (G) ≥ OPTVC (G)

  13. OPTFC (G) ≤ OPTVC (G) + 1

  14. OPTFC (G) ≤ OPTVC (G) + 1

  15. OPTFC (G) ≤ OPTVC (G) + 1

  16. OPTFC (G) ≤ OPTVC (G) + 1

  17. OPTFC (G) ≤ OPTVC (G) + 1

  18. OPTFC (G) ≤ OPTVC (G) + 1

  19. OPTFC (G) ≤ OPTVC (G) + 1

  20. OPTFC (G) ≤ OPTVC (G) + 1

  21. OPTFC (G) ≤ OPTVC (G) + 1

  22. OPTFC (G) ≤ OPTVC (G) + 1

  23. OPTFC (G) ≤ OPTVC (G) + 1

  24. OPTFC (G) ≤ OPTVC (G) + 1

  25. 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

  26. 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.

  27. Ferry Cover in other well known graph topologies

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