Fighting Intelligent Fires

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29 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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Fighting Intelligent Fires
Anthony Bonato

1

Fighting Intelligent Fires

Anthony Bonato

Ryerson University

CMS Summer Meeting

June 5, 2010

Fighting Intelligent Fires
Anthony Bonato

2

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

3

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

4

Examples


sn(P
n
,v) = n
-
1
, if
v

is an end
-
vertex


= n
-
2
, else




sn(K
n
,v) = 1



(MacGillivray, P. Wang, 03)
:

sn(Q
n
,v) = n


Fighting Intelligent Fires
Anthony Bonato

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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

Fighting Intelligent Fires
Anthony Bonato

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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

Fighting Intelligent Fires
Anthony Bonato

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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



Fighting Intelligent Fires
Anthony Bonato

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Example: path

Fighting Intelligent Fires
Anthony Bonato

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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) ≥ 6/5
ε
,
where
0 <
ε

< 5/27



(Prałat, 10):


if
G

has size at most
(15/11


ε
)n
, then
ρ
(G) ≥
1/60
ε
, where
0 <
ε

< 1/2
(
15/11

best possible
)

Fighting Intelligent Fires
Anthony Bonato

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Intelligent fires


fire now
chooses

k

vertices to burn in each
round

Fighting Intelligent Fires
Anthony Bonato

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k=1

x

y

a

b

K
100

burns
51

vertices…

Intelligent fires


fire now
chooses

k

vertices to burn in each
round

Fighting Intelligent Fires
Anthony Bonato

12

k=1

x

y

a

b

K
100

burns two vertices

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

13

Bounds


Theorem
(BMP, 10)






ρ
(
G,k
) ≥ (1/(k+1))(1
-
1/n)





Theorem
(BMP,10)



ρ
(
G,k
) ≤ 1


2/n + 1/n
2

+ 1/n
2
(n
-
1)/(k+1)


Fighting Intelligent Fires
Anthony Bonato

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Eg:
Wheels

and
prisms

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Anthony Bonato

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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?

Fighting Intelligent Fires
Anthony Bonato

16

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

17

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

18

Sketch of proof


short cycle
: length
L = log
d
-
1
log
d
-
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

19

Stages


Stage I
: fire can spread only to
< k

nodes


up to round
t
0

=
Θ

(log k/d)
(constant)



Stage II
: no short cycles up to round


t
1
= 1/2L


burned subgraph is a tree order
(1+o(1))k t
1



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

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Numerical bounds

Fighting Intelligent Fires
Anthony Bonato

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Infinite graphs



P






“most” vertices saved

Fighting Intelligent Fires
Anthony Bonato

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Limiting surviving rate


ρ
(P

,k) = lim
n→∞

ρ
(
P
n
,k
) = 1



similarly,

ρ
(
K

,k
) = 1/(k+1)



how to define
ρ
(
G,k
)
for an infinite graph?


Fighting Intelligent Fires
Anthony Bonato

23

Chains


countably

infinite
G
, express as limit of
chain
C = (
G
n
: n ≥ 1),
where each
G
n

is
connected



ρ
C
(
G,k
)
= lim
n→∞

ρ
(
G
n
,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

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Fighting Intelligent Fires
Anthony Bonato

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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

Fighting Intelligent Fires
Anthony Bonato

26

Aside: cop
density


c(G)
=
cop number
of
G



if
C = (
G
n
: n ≥ 1)
is a chain of graphs with limit G,
then define the
cop density

D
C
(G
)
=
lim

c(
G
n
)/|V(
G
n
)|



(
B,Hahn,Wang
, 08):
There are chains
C

such
D
C
(R)

is any fixed real in
[0,1].


(Frankl,84):
if
G

connected, then
c(G) = o(|V(G)|)


D
C
(G) =0
for all chains
C



n→∞

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

ρ
C
(
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

27

Future research


ρ
(
G,k
)
in graph classes
, products



k
-
Firefighter as a combinatorial game?



grids? …

Fighting Intelligent Fires
Anthony Bonato

28

29

Fighting Intelligent Fires
Anthony Bonato

k = ∞
,
Cartesian
grid P
7


P
7

30

Fighting Intelligent Fires
Anthony Bonato

k = ∞
,
Cartesian grid
P
7


P
7

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

31

¼

-
conjecture

32

Fighting Intelligent Fires
Anthony Bonato

Infinite hexagonal grid


can one cop
contain

the fire?

Fighting Intelligent Fires
Anthony Bonato

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Fighting Intelligent Fires
Anthony Bonato

34


preprints, reprints, contact
:

Google:


Anthony Bonato


Graphs at Ryerson (
G@R
)

Fighting Intelligent Fires
Anthony Bonato

35

New Book


Cops and Robbers on Graphs


with Richard Nowakowski


expected release 2011…


Fighting Intelligent Fires
Anthony Bonato

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