The length of vertex pursuit games

Anthony BonatoRyerson University

CCC 2013

Cops and Robbers

Vertex pursuit

C

CC

Cops and Robbers

Vertex pursuit

C

CC

R

Cops and Robbers

Vertex pursuit

C

CC

R

Cops and Robbers

Vertex pursuit

C

CC

R

Cops and Robbers

Vertex pursuit

C

CC

R

Cops and Robbers

Vertex pursuit

C

CC

R

Cops and Robbers

Vertex pursuit

C

CC

R

R loses

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

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)

)1(1log))1(1( 22

ono

nnO

Vertex pursuit

R.J. Nowakowski, P. Winkler Vertex-to-vertex 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 cop-win 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 games1. Cops and Robbers

– play on graphs, digraphs, orders, hypergraphs, etc.– play at different speeds, or on different edge sets

2. Cops and Robbers with traps3. Distance k Cops and Robbers4. Tandem-win Cops and Robbers5. Helicopter Cops and Robbers6. Maker-Breaker Games7. Seepage8. 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 beforeVertex 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

where IL is the set positions for Left which are the first component of an ordered pair in I.

Vertex pursuit

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II

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 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

Lemma (BGKP,13+)

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

Vertex pursuit

Theorem (BGKP,13+)

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. There are two players, Left and Right.2. There is perfect information.3. There is a set of allowed positions in the game.4. 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