1. ETH Zurich – Distributed Computing – www.disco.ethz.ch Klaus-Tycho Förster, Rijad Nuridini, Jara Uitto, Roger Wattenhofer Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond

2. How to catch a robber on a graph? The game of Cops and Robbers

3. The rules of the game

4. The Cop is placed first C

5. The Robber may then choose a placement CR

6. Next, they alternate in moves CR

7. CR

8. CR

9. CR

10. The Cop won! C

11. C

12. How many moves does the cop need?

13. Catch multiple? RRRC

14. Upper bound to catch ℓ robbers

15. Lower bound to catch ℓ robbers … …

16. C … …

17. CR R R … …

18. CR R R … …

19. CR R R … …

20. R CR R … …

21. R CR R … …

22. CR R R … …

23. CR R R … …

24. R CR R … …

25. R C R … …

26. R C … …

27. R C … …

28. CR R … …

29. Summary so far

30. What about multiple cops and one robber?

31. Let’s start with two cops and one robber …

32. R C C … one cop not enough

33. Let’s start with two cops and one robber C R C …

34. Beyond two cops? … ??

36. Multiple cops and one robber …

37. Summary so far

38. What about multiple cops and multiple robbers?

39. Are we done? Multiple cops and multiple robbers … … … …

40. … … … …

41. … … … … C Prevents robbers from crossing 

42. C ……

43. C ……

44. C

45. C

46. C

47. Construction of the ring … … …… …

48. Robber placement Robbers choose side with less cops R R R

49. Robber strategy R R R

50. R R R

51. R R R C Can catch half of the robbers!

52. Robber strategy R R R C

53. R R R C

54. R R R C

55. R R R C

56. Summary

57. ETH Zurich – Distributed Computing – www.disco.ethz.ch Klaus-Tycho Förster, Rijad Nuridini, Jara Uitto, Roger Wattenhofer Lower Bounds for the Capture Time: Linear, Quadratic, and Beyond