CCC 2013
The length of vertex
pursuit games
Anthony Bonato
Ryerson University
Cops and Robbers
C
C
C
Vertex pursuit
Cops and Robbers
C
R
C
C
Vertex pursuit
Cops and Robbers
C
R
C
Vertex pursuit
C
Cops and Robbers
C
R
C
Vertex pursuit
C
Cops and Robbers
C
R
C
Vertex pursuit
C
Cops and Robbers
C
R
C
Vertex pursuit
C
Cops and Robbers
C
R
C
C
R loses
Vertex pursuit
Cops and Robbers
• played on reflexive undirected graphs G
• two players Cops C and robber R play at alternate
time-steps (cops first) with perfect information
• players move to vertices along edges; allowed to
moved to neighbors or pass
• cops try to capture (i.e. land on) the robber, while
robber tries to evade capture
• minimum number of cops needed to capture the
robber is the cop number c(G)
– well-defined as c(G) ≤ |V(G)|
Vertex pursuit
Bounds cop number
• c(G) ≤ γ(G) (the domination number of G)
– far from sharp: paths
• Meyniel’s conjecture: if G is connected, then
c(G) = O(|V(G)|1/2)
• best known upper bound: (Lu,Peng,13+), (Scott,
Sudakov,11), and (Frieze, Krivelevich, Loh, 11)
n

O (1o (1))
2
log2 n
Vertex pursuit

  n1o (1)

Vertex pursuit
R.J. Nowakowski, P. Winkler Vertex-tovertex pursuit in a graph, Discrete
Mathematics 43 (1983) 235-239.
• 5 pages
• > 200 citations (most for either author)
Vertex pursuit
The NW relation
• (Nowakowski,Winkler,83) introduced a
sequence of relations characterizing copwin graphs
• u ≤0 v if u = v
• u ≤i v if for all x in N[u], there is a y in N[v]
such that x ≤j y for some j < i.
Vertex pursuit
Characterization
• the relations are ≤i monotone increasing;
thus, there is an integer k such that
≤k = ≤k+1
– write:
≤k = ≤
Theorem (Nowakowski, Winkler, 83)
A cop has a winning strategy iff ≤ is
V(G) x V(G).
Vertex pursuit
k cops
• may define an analogous relation but in
V(G) x V(Gk) (categorical product)
• (Clarke,MacGillivray,12) k cops have a
winning strategy iff the relation ≤ is
V(G) x V(Gk).
Vertex pursuit
Axioms for pursuit games
(B, MacGillivray,13+)
• a pursuit game G is a discrete-time process satisfying the following:
1. Two players, Left L and Right R.
2. Perfect-information.
3. There is a set of allowed positions PL for L; similarly for Right.
4. For each state of the game and each player, there is a non-empty set of
allowed moves. Each allowed move leaves the position of the other player
unchanged.
5. There is a set of allowed start positions I a subset of PL x PR.
6. The game begins with L choosing some position pL and R choosing qR
such that (pL, qR) is in I.
7. After each side has chosen its initial position, the sides move alternately
with L moving first. Each side, on its turn, must choose an allowed move
from its current position.
8. There is a subset of final positions, F. Left wins if at any time, the current
position belongs to F. Right wins the current position never belongs to F.
Vertex pursuit
Examples of pursuit games
1. Cops and Robbers
–
–
2.
3.
4.
5.
6.
7.
8.
play on graphs, digraphs, orders, hypergraphs, etc.
play at different speeds, or on different edge sets
Cops and Robbers with traps
Distance k Cops and Robbers
Tandem-win Cops and Robbers
Helicopter Cops and Robbers
Maker-Breaker Games
Seepage
Scared Robber
Vertex pursuit
Relational characterization
• given a pursuit game G, we may define relations on
PL x PR as follows:
• pL ≤0 qR if (pL, qR) in F.
• pL ≤i qR if Right is on qR and for every xR in PR such that if
Right has an allowed move from (pL, qR) to (pL, xR), there
exists yL in PL such that xR ≤j yL for some j < i and Left
has an allowed move from (pL, xR) to (yL, xR).
• define ≤ analogously as before
Vertex pursuit
Characterization
Theorem (BM,13+) Left has a winning strategy in
the pursuit game G if and only if there exists pL in
PL, which is the first component of an ordered
pair in I, such that for all qR in PR with (pL, qR) in
I there exists wL in the set of allowed moves for
Left from pL such that qR ≤ wL.
• gives rise to a min/max expression for the length
of the game
Vertex pursuit
Length of game
• for an allowed start position (pL, qR), define
Corollary (BM,13+) If Left has a winning strategy in the a
pursuit game G, then assuming optimal play, the length
of the game is
min max ( pL , qR )
pL IL ( pL , qR )I
where IL is the set positions for Left which are the first
component of an ordered pair in I.
Vertex pursuit
Aside: position independence
• in case of position independence (eg Cops and
Robbers, but not Helicopter Cops and Robbers),
there is a characterization even more analogous
with that of NW and CM
• gives complexity bound on determining whether
L has a winning strategy
– in the case of Cops and Robbers with k cops
gives O(n2k+2) time algorithm, which matches
the best known complexity (CM,12)
Vertex pursuit
Capture time of a graph
• the length of Cops and Robbers was considered first as
capture time
• (B,Hahn,Golovach,Kratochvíl,09) capture time of G:
length of game with c(G) cops assuming optimal play,
written capt(G)
– if G is cop-win, then capt(G) ≤ n-4 if n ≥ 7
– capt(G) ≤ n/2 for many families of cop-win graphs including
chordal graphs
– examples of planar graphs with capt(G) = n-4
Vertex pursuit
Cop number > 1
• not much is know about capt(G) if c(G) > 1
• hypercubes? ...
Vertex pursuit
Vertex pursuit
Cop number of hypercubes
• (Maamoun,Meyniel,87): the cop number of the
Cartesian product of n trees is floor(n+1 / 2)
• no reference to the length of the game; i.e
capture time of the hypercube
• problem: determine capt(Qn)
Vertex pursuit
Capture time of grids
• (Merhabian,10): the capture time of the
Cartesian product of two trees T1 and T2 is
floor((diam(T1) + diam(T2)) / 2)
• in particular, the capture time of the m x n
Cartesian grid is floor((m + n)/2)-1
Vertex pursuit
Capture time of hypercubes
(B, Gordinowicz, Kinnersley, Pralat,13+) The
capture time of Qn is
Θ(nlog n).
• proof finds lower and upper bounds
• focus on lower bound
Vertex pursuit
Lower bound
• we prove something a little stronger than is
needed
Theorem (BGKP,13+) For d > 0 a constant, a
robber can escape nd cops for at least
(1-o(1))1/2 n log n rounds.
– probabilistic method: play with a random
robber
Vertex pursuit
Useful lemma
Lemma (BGKP,13+)
• lemma shows that, against a random robber, a cop
should keep the distance between the players even and,
subject to that, minimize the distance (ie play greedily)
Vertex pursuit
Coupon collector problem
• n coupons, all equally likely, drawn with
replacement
– how many do you need to draw before you
have collected all n of them?
• answer: (1+o(1))n log n
Vertex pursuit
Deviation bound
• we use the following generalization of a result of
(Doerr, 2011) to bound the probability that the actual
capturing time is significantly less than its expectation
Theorem (BGKP,13+)
Vertex pursuit
Proof of lower bound (sketch)
• let T= 1/2(n-1)log n, ε = ln((4d+1) ln n) / ln n = o(1).
• show that a random robber can play (1- ε)T rounds without
being captured
• can play initial round due to expansion
• next consider a single cop C playing greedily as in lemma
• can show process of C capturing R is equivalent to the
coupon collector problem
• using useful lemma and deviation bound, the probability
single cop captures robber is exp(-(n/2)ε/4); via union bound
for all nd cops this is o(1)
• hence, there is SOME deterministic strategy for the robber
to survive (1- ε)T rounds
Vertex pursuit
Problems and directions
• Conjecture: capt(Qn) = (1+o(1))1/2 n log n
• capture time for other graph families?
• Program: study capt(G) if number of cops varies
– for example, define captk(G), and consider
c(G) ≤ k ≤ γ(G)
Vertex pursuit
Vertex pursuit
Vertex pursuit
CGT
(Berlekamp, Conway, Guy, 82) A combinatorial game satisfies:
1.
2.
3.
4.
There are two players, Left and Right.
There is perfect information.
There is a set of allowed positions in the game.
The rules of the game specify how the game begins and, for each
player and each position, which moves to other positions are
allowed.
5. The players alternate moves.
6. The game ends when a position is reached where no moves are
possible for the player whose turn it is to move. In normal play the
last player to move wins.
Vertex pursuit
Example: NIM
Vertex pursuit
Pursuit → CGT
Theorem (BM,13+)
1. Every pursuit game is a combinatorial
game.
2. Not every combinatorial game is a pursuit
game.
•
•
uses characterization of (Smith, 66) via game digraphs
Nim is a counter-example for item (2)
Vertex pursuit