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
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Anthony Bonato
5
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
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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
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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
11
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
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Anthony Bonato
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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/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
15
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)
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Anthony Bonato
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Numerical bounds
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Anthony Bonato
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Infinite graphs
•
P
∞
•
“most” vertices saved
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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
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Anthony Bonato
24
Fighting Intelligent Fires
Anthony Bonato
25
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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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
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Anthony Bonato
31
¼

conjecture
32
Fighting Intelligent Fires
Anthony Bonato
Infinite hexagonal grid
•
can one cop
contain
the fire?
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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
36
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