Generalizing Alcuin’s River Crossing Problem

1. Generalizing Alcuin’s River Crossing Problem Michael Lampis - Valia Mitsou National Technical University of Athens

3. Wolf

4. Goat

5. Cabbage

9. Guard

10. Boat

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

27. Our generalization

28. 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

31. OPTFC (G) ≥ OPTVC (G)

32. OPTFC (G) ≥ OPTVC (G)

33. OPTFC (G) ≥ OPTVC (G)

34. OPTFC (G) ≤ OPTVC (G) + 1

35. OPTFC (G) ≤ OPTVC (G) + 1

36. OPTFC (G) ≤ OPTVC (G) + 1

37. OPTFC (G) ≤ OPTVC (G) + 1

38. OPTFC (G) ≤ OPTVC (G) + 1

39. OPTFC (G) ≤ OPTVC (G) + 1

40. OPTFC (G) ≤ OPTVC (G) + 1

41. OPTFC (G) ≤ OPTVC (G) + 1

42. OPTFC (G) ≤ OPTVC (G) + 1

43. OPTFC (G) ≤ OPTVC (G) + 1

44. OPTFC (G) ≤ OPTVC (G) + 1

45. OPTFC (G) ≤ OPTVC (G) + 1

46. The Ferry Cover Problem Lemma: OPTVC (G) ≤ OPTFC (G) ≤ OPTVC (G) + 1

47. 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.

48. Ferry Cover on Trees Lemma: For trees with OPTVC (G) > 1 OPTFC (G) = OPTVC (G)

49. u w v

50. u v w