A survey of some results on the Firefighter Problem

1. A survey of some results on the Firefighter Problem Wow! I need reinforcements! Kah Loon Ng DIMACS

2. A simple model

3. A simple model

4. A simple model

5. A simple model

6. A simple model

7. A simple model

8. A simple model

9. A simple model

10. Some questions that can be asked (but not necessarily answered!) • Can the fire be contained? • How many time steps is required before fire is contained? • How many firefighters per time step are necessary? • What fraction of all vertices will be saved (burnt)? • Does where the fire breaks out matter? • “Smart fires”? • Fire starting at more than 1 vertex? • Consider different graphs. Construction of (connected) graphs to minimize damage. • Complexity/Algorithmic issues

11. Some references • The firefighter problem for graphs of maximum of degree three (Finbow, King, MacGillivray, Rizzi) • Graph-theoretic models of spread and competition (Hartke) • On the firefighter problem (MacGillivray, Wang) • Catching the fire on grids (Fogarty) • Fire control on graphs (Wang, Moeller) • Firefighting on trees: How bad is the greedy algorithm? (Hartnell, Li) • On minimizing the effects of fire or a virus on a network (Finbow, Hartnell, Li, Schmeisser) • On designing a network to defend against random attacks of radius two (Finbow, Hartnell) • The optimum defense against random subversions in a network (Hartnell) • On minimizing the effects of betrayals in a resistance movement (Gunther, Hartnell)

12. Four general classes of problems • Containing fires in infinite grids where is the dimension.

13. Four general classes of problems 2. Saving vertices in finite grids of dimension 2 or 3.

14. Four general classes of problems 3. Firefighting on trees. Algorithmic and complexity issues.

15. Four general classes of problems 4. Construction of graphs that minimizes damage.

16. Containing fires in infinite grids Ld Fire starts at only one vertex: d= 1: Trivial. d = 2: Impossible to contain the fire with 1 firefighter per time step

17. 8 time steps 18 burnt vertices Containing fires in infinite grids Ld d = 2: Two firefighters per time step needed to contain the fire.

18. .…. Containing fires in infinite grids Ld Fact: If G is a k-regular graph, k – 1 firefighters per time step is always sufficient to contain any fire outbreak (at a single vertex) in G. d  3: ……

19. Containing fires in infinite grids Ld Fact: If G is a k-regular graph, k – 1 firefighters per time step is always sufficient to contain any fire outbreak (at a single vertex) in G. d  3: Shown: 2d – 2 firefighters per time step are not enough to contain an outbreak in Ld Thus, 2d – 1 firefighters per time step is the minimum number required to contain an outbreak in Ld and containment can be attained in 2 time steps.

20. Containing fires in infinite grids Ld Theorem (Hartke): Let be a rooted graph, a positive integer, and positive integers each at least such that the following holds: • Every nonempty satisfies • For , every where • satisfies • For , every such that • satisfies

21. Containing fires in infinite grids Ld Theorem (Hartke): Suppose that at most firefighters per time step are deployed. Then regardless of the sequence of firefighter placements. Specifically, firefighters per time step are insufficient to contain an outbreak that starts at the root vertex.

22. d 3: For any d  3, and any positive integer , firefighters per time step is not sufficient to contain all finite outbreaks in Ld. In other words, for d  3 and any positive integer , there is an outbreak such that firefighters per time step cannot contain the outbreak. Containing fires in infinite grids Ld Fire can start at more than one vertex. d = 2: Two firefighters per time step are sufficient to contain any outbreak at a finite number of vertices.

23. Saving vertices in finite grids G Assumptions: • 1 firefighter is deployed per time step • Fire starts at one vertex Let MVS(G, v) = maximum number of vertices that can can be saved in G if fire starts at v.

24. Saving vertices in finite grids G

25. Saving vertices in finite grids G

26. Saving vertices in finite grids G

27. Saving vertices in finite grids G

28. Saving vertices in finite grids G

29. Saving vertices in finite grids G

30. Saving vertices in

31. Saving vertices in If

32. For example, Some asymptotic results Let if fire starts at

33. For example, So for any Some asymptotic results Let if fire starts at

34. Some asymptotic results Fire starts at

35. Some asymptotic results

36. Some asymptotic results

37. Conjecture: Some asymptotic results

38. In fact, the optimal number of vertices that can be saved given an initial outbreak at (0,0,0) in when deploying one firefighter per time step is between Some asymptotic results Let be any vertex of Then the maximum number of vertices which can be saved by deploying one firefighter per time step with an initial outbreak at grows at most as In particular,

39. vertex is neither burning nor defended at time • At time no undefended vertex is adjacent to a burning vertex, and • At least vertices are saved at the end of time Algorithmic and Complexity matters FIREFIGHTER Instance: A rooted graph and an integer Question: Is That is, is there a finite sequence of vertices of such that if the fire breaks out at then,

40. EXACT COVER BY 3-SETS (X3C) Instance: A set with and a collection C of 3-element subsets of Question: Does C contain an exact cover for That is, is there a sub-collection C’C such that each element of occurs in exactly one member of C’ ? Algorithmic and Complexity matters FIREFIGHTER is NP-complete for bipartite graphs.

41. There is an exact cover of by elements of C At least vertices of can be saved Algorithmic and Complexity matters Suppose an instance of X3C ( C ) is given. We construct a rooted bipartite graph and a positive integer such that

42. for each element in C Ci Cj For each pair of Ci , Cj such that (Ci Cj =  ) join their respective vertices ( ) by paths of length two ( ) : : Note that the graph is bipartite. Algorithmic and Complexity matters

43. Ci Cj … save the vertices that corresponds to the subsets ( ) in the exact cover. : : Algorithmic and Complexity matters If has an exact cover…

44. If at least vertices can be saved… Ci Cj …at most of the vertices ( ) can be saved by time … : : …if , at most can be saved… ( + ) ( ) Algorithmic and Complexity matters

45. Algorithmic and Complexity matters Firefighting on Trees:

46. Algorithmic and Complexity matters Greedy algorithm: For each At each time step, save place firefighter at vertex that has not been saved such that weight (v) is maximized.

47. 26 Firefighting on Trees: 22 12 8 9 7 11 2 6 1 5 1 6 1 4 2 3 3 1 3 1 1 1 1 1 1 2 1 Algorithmic and Complexity matters

48. = 7 = 9 Algorithmic and Complexity matters Greedy Optimal

49. Soptimal A B Soptimal + Soptimal Algorithmic and Complexity matters Theorem: For any tree with one fire starting at the root and one firefighter to be deployed per time step, the greedy algorithm always saves more than ½ of the vertices that any algorithm saves. Sgreedy = number of vertices saved by greedy algorithm Soptimal = number of vertices saved by optimal algorithm number of vertices saved by optimal moves whose corresponding greedy moves performs no worse

50. why was this vertex chosen on second move and not this? B B B B Soptimal Soptimal Soptimal Soptimal Sgreedy  A A A Soptimal Soptimal + Soptimal B Soptimal Soptimal = Sgreedy …vertices saved by moves have already been saved by  Sgreedy > Algorithmic and Complexity matters * Greedy Optimal …because *’s ancestor has already been selected…  Sgreedy > ½ Soptimal