Fighting Intelligent Fires

1. CMS Summer Meeting June 5, 2010 Fighting Intelligent Fires Anthony Bonato Ryerson University Fighting Intelligent Fires Anthony Bonato

2. Fighting Intelligent Fires Anthony Bonato

3. Firefighter • G simple, undirected, connected graph • fire spreads from a vertex over discrete time-steps or rounds • vertices are burned, saved, or available • fire can spread to all available adjacent vertices • firefighter can save one vertex in each round • (Hartnell, 95) introduced Firefighter • simplified model for the spread of a fire/disease/virus in a network Fighting Intelligent Fires Anthony Bonato

4. Saving vertices • one-player game • firefighter aims to maximize the number of saved vertices • sn(G,v) = maximum number of saved vertices in G if a fire starts at v Fighting Intelligent Fires Anthony Bonato

5. Examples • sn(Pn,v) = n-1, if v is an end-vertex = n-2, else • sn(Kn,v) = 1 • (MacGillivray, P. Wang, 03):sn(Qn,v) = n Fighting Intelligent Fires Anthony Bonato

6. Previous Work • (MacGillivray, P. Wang, 03), (Messinger, 04), (Devlin,Hartke, 07), (Fogarty,07), (Cai, W. Wang, 09): finite and infinite grids: Cartesian, strong, triangular, higher dimensions • (Hartnell, Li, 00), (Cai et al, 10): trees • (Finbow et al, 10), (King, MacGillivray, 10): algorithms and complexity • (Cai, W. Wang, 07), (Finbow, P. Wang, W. Wang, 10), (Prałat, 10): surviving rate • (Finbow, MacGillivray, 10): survey Fighting Intelligent Fires Anthony Bonato

7. Complexity • (Finbow et al, 10) “Is sn(G,v) ≥ k?” NP-complete, if G is a tree with maximum degree 3 Fighting Intelligent Fires Anthony Bonato

8. Surviving rate • (Cai, W. Wang, 09) surviving rate of G, ρ(G) = expected percentage of vertices saved if fire starts at a random vertex Fighting Intelligent Fires Anthony Bonato

9. Example: path Fighting Intelligent Fires Anthony Bonato

10. Results on ρ(G) • (Cai, W. Wang, 10): ρ(G) ≥ 1 – Θ(log n /n) if G is outerplanar • (Finbow, P. Wang, W. Wang, 10): if G has size at most (4/3 – ε)n, then ρ(G) ≥ 6/5ε, where 0 < ε < 5/27 • (Prałat, 10): if G has size at most (15/11 – ε)n, then ρ(G) ≥ 1/60ε, where 0 < ε < 1/2 (15/11best possible) Fighting Intelligent Fires Anthony Bonato

11. Intelligent fires • fire now choosesk vertices to burn in each round k=1 K100 a x y b burns 51 vertices… Fighting Intelligent Fires Anthony Bonato

12. Intelligent fires • fire now choosesk vertices to burn in each round k=1 K100 a x y b burns two vertices Fighting Intelligent Fires Anthony Bonato

13. k-Firefighter (B, Messinger, Prałat, 10) • played similar to Firefighter, except now fire chooses at most k nodes to burn • two-player game: fire has strategy for optimal burning • proposed by (Devlin, Hartke,07) • k-surviving rate ρ(G,k) defined analogously Fighting Intelligent Fires Anthony Bonato

14. Bounds • Theorem (BMP, 10) ρ(G,k) ≥ (1/(k+1))(1-1/n) • Theorem (BMP,10) ρ(G,k) ≤ 1 – 2/n + 1/n2 + 1/n2(n-1)/(k+1) Fighting Intelligent Fires Anthony Bonato

15. Eg: Wheels and prisms Fighting Intelligent Fires Anthony Bonato

16. Questions • what is the value of ρ(G,k) in “typical” sparse graphs? • expect k-surviving rate to be high • ρ(G,k) in infinite graphs? • what is ρ(G,k) for the infinite random graph? Fighting Intelligent Fires Anthony Bonato

17. Random regular graphs • random d-regular graph, G(n,d) • d-regular graphs with uniform probability distribution • pairing model (Bollobás,Wormald) • G(n,d) is flammable for all k: for large n, high probability that a sizeable part of graph burns Fighting Intelligent Fires Anthony Bonato

18. Main result Theorem (BMP,10) A.a.s. (i.e. with probability tending to 1 as n →∞) ρ(G,k) ≤ (1 + O(d-1/2))/ k+1 → 1/(k+1) as d →∞ • eg: for k=1, fire can burn about ½ of graph! Fighting Intelligent Fires Anthony Bonato

19. Sketch of proof • short cycle: length L = logd-1logd-1 n • a.a.s. most nodes not in a short cycle • wlog focus on such nodes: U • fire starts at u in U, and spreads in three stages: Fighting Intelligent Fires Anthony Bonato

20. Stages • Stage I: fire can spread only to < k nodes • up to round t0 = Θ (log k/d) (constant) • Stage II: no short cycles up to round t1 = 1/2L • burned subgraph is a tree order (1+o(1))k t1 • Stage III: there are short cycles, but many nodes burning • firefighter cannot contain fire effectively • spectral bounds (Friedman, 10), expander mixing lemma (Alon, Chung,88) Fighting Intelligent Fires Anthony Bonato

21. Numerical bounds Fighting Intelligent Fires Anthony Bonato

22. Infinite graphs • P∞ • “most” vertices saved … Fighting Intelligent Fires Anthony Bonato

23. Limiting surviving rate • ρ(P∞,k) = limn→∞ρ(Pn,k) = 1 • similarly, ρ(K∞,k) = 1/(k+1) • how to define ρ(G,k) for an infinite graph? Fighting Intelligent Fires Anthony Bonato

24. Chains • countably infinite G, express as limit of chain C = (Gn: n ≥ 1), where each Gn is connected • ρC(G,k) = limn→∞ρ(Gn,k) • real number in [0,1], when it exists • does not depend on C for paths, cliques, will depend on C, in general Fighting Intelligent Fires Anthony Bonato

25. The infinite random graph Theorem (Erdős,Rényi, 63): With probability 1, any two graphs sampled from G(N,p) are isomorphic. • isotype R unique with the e.c. property: B For all finite A there exists z Fighting Intelligent Fires Anthony Bonato

26. Aside: cop density • c(G) = cop number of G • if C = (Gn: n ≥ 1) is a chain of graphs with limit G, then define the cop density DC(G) = lim c(Gn)/|V(Gn)| • (B,Hahn,Wang, 08): There are chains C such DC(R) is any fixed real in [0,1]. • (Frankl,84): if G connected, then c(G) = o(|V(G)|) • DC(G) =0 for all chains C n→∞ Fighting Intelligent Fires Anthony Bonato

27. Limiting surviving rate of R • Theorem (BMP,10): For every real number r in [1/k+1,1], there is a chain C such that ρC(R,k) = r. • Proof ideas: • use e.c. property • many extensions lower k-surviving rate • add long path to increase it Fighting Intelligent Fires Anthony Bonato

28. Future research • ρ(G,k) in graph classes, products • k-Firefighter as a combinatorial game? • grids? … Fighting Intelligent Fires Anthony Bonato

29. k = ∞, Cartesian grid P7P7 Fighting Intelligent Fires Anthony Bonato

30. k = ∞, Cartesian grid P7P7 Fighting Intelligent Fires Anthony Bonato

31. MW Strategy • (MacGillivray, Wang, 03): MW Strategy: If fire breaks out at (r,c), 1≤r≤c≤n/2, save vertices in following order: (r + 1, c), (r + 1, c + 1), (r + 2, c - 1), (r + 2, c + 2), (r + 3, c -2),(r + 3, c - 3), ..., (r + c, 1), (r + c, 2c), (r + c, 2c + 1), ..., (r + c, n) • MW strategy saves n(n-r)-(c-1)(n-c) vertices • MW is optimal strategy assuming fire breaks out in columns (rows) 1,2, n-1, n Fighting Intelligent Fires Anthony Bonato

32. ¼ -conjecture Fighting Intelligent Fires Anthony Bonato

33. Infinite hexagonal grid • can one cop contain the fire? Fighting Intelligent Fires Anthony Bonato

34. preprints, reprints, contact: Google: “Anthony Bonato” Fighting Intelligent Fires Anthony Bonato

35. Graphs at Ryerson (G@R) Fighting Intelligent Fires Anthony Bonato

36. New Book • Cops and Robbers on Graphs • with Richard Nowakowski • expected release 2011… Fighting Intelligent Fires Anthony Bonato