Fighting Intelligent Fires

Fighting Intelligent Fires Anthony Bonato

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Fighting Intelligent Fires

Anthony BonatoRyerson University

CMS Summer MeetingJune 5, 2010


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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


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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


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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


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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


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Complexity

• (Finbow et al, 10)“Is sn(G,v) ≥ k?” NP-complete, if G is a tree

with maximum degree 3


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Surviving rate

• (Cai, W. Wang, 09) surviving rate of G, ρ(G) = expected percentage of vertices

saved if fire starts at a random vertex

)(

2 ),(1)(GVv

vGsnn

G


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Example: path

2

2

2

221

)1(2)2)(2(1

),(1)(

nn

nnnn

vPsnn

PVv

nn


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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 (15/11 best possible)


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Intelligent fires

• fire now chooses k vertices to burn in each round

k=1

x ya

b

K100

burns 51 vertices…


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Intelligent fires

• fire now chooses k vertices to burn in each round

k=1

x ya

b

K100

burns two vertices


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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


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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)


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Eg: Wheels and prisms


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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?


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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


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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!


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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:


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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)


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Numerical bounds


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Infinite graphs

• P∞

• “most” vertices saved

…


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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?


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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


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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:

The infinite random graph

For all finite A B

there exists z


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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→∞


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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


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Future research

• ρ(G,k) in graph classes, products

• k-Firefighter as a combinatorial game?

• grids? …


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k = ∞, Cartesian grid P7 P7


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k = ∞, Cartesian grid P7 P7


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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


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¼ -conjecture

nPPsn nnnn 4

1)),(, (

n largefor ,0every For

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Infinite hexagonal grid

• can one cop contain the fire?


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• preprints, reprints, contact:Google: “Anthony Bonato”


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Graphs at Ryerson (G@R)


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New Book

• Cops and Robbers on Graphs– with Richard Nowakowski– expected release 2011…

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Fighting Intelligent Fires

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