A probabilistic version of - PowerPoint Presentation

Slide1

A probabilistic version of the game of Zombies and Survivors on graphs

Anthony BonatoRyerson University

SIAM

DM

2016

Cops and Robbers and Pursuit-Evasion in Discrete StructuresSlide2
About the

minisymposiumCops and Robbers is a trending topic in graph theorystructural, probabilistic, topological, algorithmic…pursuit-evasion/graph searchingthree sessions and 14 speakers: Mon, Wed, ThursSlide3
Cops and Robbers

CC

RSlide4
Cops and Robbers

CC

RSlide5
Cops and Robbers

two-player game, perfect information, unit speed, played on vertices, move to neighbours or passminimum number of cops needed for a winning strategy to capture R in G = cop number of G, or c(G)Slide6

Meyniel’s Conjecture (85): c(G) = O(n1/2) if G is connectedcomputing c(G) ≤ k(Berrarducci, Intrigila

, 93), (Hahn,MacGillivray, 06), (B,Chiniforooshan, 09): polynomial time if

k

is fixed

(Kinnersley,15)

:

EXPTIME

-complete, if

k

not fixed

not know if it is in

NP

(Aigner,Fromme,84)

:

c(G)

≤

3

if

G

is planar

Schroeder’s conjecture (01)

:

c(G)

≤

g + 3

, if

g

is the genus of

GSlide7
Variants

power of cops:traps, photo radar, walls, capture from distance, teleportation, tandem-win, lazy copspower of robber:speed: fast or infinite, invisible, decoys, barricades, damage, capture copsvertex pursuit/search/good guys vs bad guys games:Firefighting, watchman’s problem, eternal domination, seepage, robot vacuum, robot crawler, graph cleaning, Prisoner’s dilemma, Walker-Breaker, Angel and Devil, …Slide8
Slide9

Zombies and SurvivorsSlide10

Zombies and SurvivorsSlide11

Zombie horde

Zombies and Survivors

u

p to

- 2

zombies on an induced path will never capture the survivor

 Slide12

Zombies and Survivorsset of zombies, one survivorplayers move at alternate ticks of the clock, from vertex to vertex along edgeszombies choose their initial locations u.a.r.at each step the zombies move along a shortest path connected to the survivorif more than one such path, then they choose one u.a.r.zombies

win if one or more can eat the survivorland on the survivor’s vertexotherwise, survivor winsNB: zombies have no strategy!

Zombies and SurvivorsSlide13

(B,Mitsche,Perez-Gimenez,Pralat,16+)sk(G): probability survivor wins if k zombies play, assuming optimal playsk+1 (G) ≤ sk (G) for all

k, and sk(G) → 0 as k → ∞

zombie number

of

G

is

z(G) = min{k ≥ c(G):

s

k

(G) ≤ ½}

well-defined

z(G)

represents the minimum number of zombies such that the probability that they eat the survivor is

> ½

note that

c(G) ≤ z(G)

Z(G)

= z(G) / c(G):

cost of being undead

Zombies and SurvivorsSlide14
Deterministic Zombies

deterministic version of the game: (Fitzpatrick, Howell, Messinger,Pike,16+) at the SIAM DM 2014 conference in Minneapolisin the deterministic version, the zombies choose their location, and can choose their geodesics if more than oneexample: where c(G) = 2 < z(G) = 3

bridged graphs, products, hypercubes (z(G) - c(G) → ∞)

Zombies and SurvivorsSlide15

Zombies and Survivors

c(G) = 2

with probability

all

k

zombies begin

outside

the cycle

implies:

a

nd so

G

n-5

leavesSlide16
Zombie number of cycles

Theorem (BMPGP,16+)If n ≥ 27, then z(Cn) = 4, so Z(Cn) = 2.

Zombies and SurvivorsSlide17
Sketch of proof

survivor wins iff the zombies are on a subpath of order at most - 2overcount these configurations:

distinguishing two zombies one at each end of the subpathk(k-1) ways to select the two zombies on subpath with r ≤

-

2

vertices

n

choices for position of the path

two zombies start in right place with probability

(1/n)

2

remaining zombies on

subpath

, with probability

(r/n)

k-2

Zombies and SurvivorsSlide18
Sketch of proof

hence,k = 4 gives sk(Cn) ≤

½lower bound analogous

Zombies and SurvivorsSlide19

Zombies and Survivorsfinite projective plane P:two lines meet in a unique pointtwo points determine a unique lineexist 4 points, no line contains more than two of themq2+q+1

points; each line (point) contains (is incident with) q+1 points (lines)incidence graph (IG) of P:

bipartite graph

G(P)

with

red

nodes the points of

P

and

blue

nodes the lines of

P

a point is joined to a line if it is on that lineSlide20

ExampleZombies and SurvivorsFano plane

Heawood graphSlide21

Cop number of planesIG of projective planes: girth 6, (q+1)-regular, so have cop number ≥ q+1order 2(q2+q+1)

(Prałat,10): cop number = q+1Zombies and SurvivorsSlide22
Zombies on planes

Theorem (BMPGP,16+)For large q,

).

Hence,

Z(

) = 2.

Zombies and SurvivorsSlide23

(Maamoun,Meyniel,87)

Theorem (

BMPGP,16+)

For large

n

,

).

Hence,

Z(

) = 4/3.

Zombies and SurvivorsSlide24
Sketch of proof of lower bound

show a.a.s. survivor can escape

zombies

vertices

: binary bit strings

Chernoff’s

bound

:

zombies begin with an even number of ones in their binary string

this holds throughout the game, as zombies always move

the survivor, independently, always has an even distance to less than

n/3

zombies, and an odd distance to less than

n/3

zombies

Zombies and SurvivorsSlide25
Sketch of proof of lower bound

survivor strategy: pick an arbitrary vertex distance at least 2 from all the zombiesnote that there are at most k(n+1) < 2n vertices at distance at most 1 from some zombieif after the zombie’s move, no zombie is distance 1 to the survivor, then the survivor stands stillotherwise, survivor must move to a safe vertexeach zombie distance

1 (< n/3) forbids a coordinate, and each zombie at distance 2 (< n/3) forbids two coordinates

so less than

n

coordinates forbidden, so survivor can move

Zombies and SurvivorsSlide26
Cartesian grids

(Tosic, 87) c(G H) ≤ c(G) + c(H)Theorem (BMPGP,16+)

For n ≥ 2, z(Pn Pn

) = 2

,

so

Z(P

n

P

n

) =1

.

Zombies and SurvivorsSlide27
Toroidal grids

Tn = Cn Cn(Neufeld, 90): c(

Tn) = 3Theorem (BMPGP,16+)Let

ω

=

ω

(n)

be a function tending to infinity with

n

.

Then

a.a.s

.

/ (

ω

log n).

Zombies and SurvivorsSlide28
Toroidal grids, continued

despite the lower bound, no known upper bound is known for the zombie number of toroidal graphs!Zombies and SurvivorsSlide29

Necromancers, Zombies, and Survivorsset of j necromancers (i.e. cops, who can access all strategies) and k zombies play vs survivor(j,k)-necro-win: bad guys win with probability > 1/2

Theorem (BMPGPR,16+)For large m and n

, toroidal grids

C

m

C

n

are

(2,1)

-

necro

-win.

If

gcd

(

m,n

) ≤ 3

, then

C

m

C

n

is

(1,2)

-

necro

-win

.

Zombies and SurvivorsSlide30
Problems

zombie number of toroidal gridupper bound?other grids? graph products?necromancer + k zombies: how large must k be to ensure win on square toroidal grids?random structures? binomial random graph, G(n,r), random regular…

Zombies and SurvivorsSlide31
Deterministic case

(FHMP,16+) characterize zombie-win graphsconjecture:

lower bound tight; best upper bound:

n-1

how large can

z(G)/c(C)

be?

Zombies and SurvivorsSlide32

CanaDAM 2017Ryerson University, TorontoSlide33
Math PhD program at Ryerson

starts Fall 2016emphasis on modelling, applicationsdiscrete math/graph theory is a fieldtaking applications in October for Fall 2017http://math.ryerson.ca/graduate/overview-phd.htmlZombies and Survivors