Cops and Robbers from a Distance Miniconference on the Mathematics of Computation EURO’09 July 2009 Cops and Robbers from a Distance Anthony Bonato Ryerson University Toronto, Canada

Game of Cops and Robbers R loses 12/13/2019 Anthony Bonato

cop (or search) number of G, c(G) (Aigner, Fromme,84) some graphs need more than one cop cop (or search) number of G, c(G) (Aigner, Fromme,84) girth(G) ≥ 5, then c(G) ≥ δ(G) if G is planar, then c(G) ≤ 3 structure of cop-win (c(G) = 1) finite graphs is well-known and beautiful (Nowakowski, Winkler, 83), (Quilliot, 78) C 12/13/2019 Anthony Bonato

Cops and Robber - Anthony Bonato The infinite case dismantling characterization, however, does not apply in the infinite case eg: ray is dismantlable, but has infinite cop number 12/13/2019 Cops and Robber - Anthony Bonato

Cops and Robbers is a simplified model for network security: cops (authorities/network administrator/anti-virus software) guard network from the robber (intruder/hacker/virus) the larger the cop number, the less secure the network large literature on the game and its variants: surveys: Alspach, 2006; Hahn, 2007; Fomin, 2008 12/13/2019 Anthony Bonato

Application: multiple-agent moving-target search example: in video games, player controls robber, while cops are computer generated agents octile connected maps 12/13/2019 Anthony Bonato

agents must be smart and perform calculations quickly (Greiner et al, 08): problem in AI agents must be smart and perform calculations quickly 12/13/2019 Anthony Bonato

Complexity (Berrarducci, Intrigila, 93), (Goldstein, Reingold,95), (Hahn, MacGillivray, 03) 1. “c(G) ≤ s?” s fixed: running time O(n2s+3), n = |V(G)| (Fomin, Golovach, Kratochvíl, Nisse, Suchan, 08): computing the cop number of a given graph is: NP-hard; W[2]-hard subexponential time: 2o(n)nO(1) 12/13/2019 Anthony Bonato

How big can c(G) be? Meyniel Conjecture: For a connected graph G of order n, c(G) = O(n1/2). (Chiniforooshan, 08): c(G) = O(n / log n) (Lu, Peng, 09?): c(G) =O(n / 2(1-o(1))(log n)1/2) Soft Meyniel Conjecture: c(G) = O(n1-c), for some c > 0 12/13/2019 Anthony Bonato

Miniconference on the Mathematics of Computation G(n,p) (Erdős, Rényi, 63) n a positive integer, p = p(n) a real number in (0,1) G(n,p): probability space on graphs with nodes {1,…,n}, two nodes joined independently and with probability p 1 2 3 4 5 12/13/2019 Anthony Bonato

Cop number of random graphs (Bonato, Hahn, Wang, 07): for p constant, a.a.s. (with probability tending to 1 as n → ∞) c(G(n,p)) = Θ(log n) (Bollobás, Kun, Leader, 08): for p = p(n), c(G(n,p)) ≤ n1/2 + o(1) - bound attained by sparse random graphs (Bonato, Prałat, Wang, 09): dense random graphs, and random power law graphs (Prałat, Łuczak, 09): zig-zag theorem 12/13/2019 Anthony Bonato

Graphs realizing the lower bound consider a finite projective plane P with n = q2+q+1 points, where q is a prime power define a 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 line observation: c(G(P)) = Θ(n1/2) folklore; see also (Prałat, 09) 12/13/2019 Anthony Bonato

Variations on the game players may have variable speed cops have partial information players can move on edges (edge searching), or must clear edges (brush number) players must move according to certain rules for eg, cops must be distance at most 2 apart (tandem win graphs) 12/13/2019 Anthony Bonato

New variation: Distance k Cops and Robber cops can “shoot” robber at some specified distance k play as in classical game, but capture includes case when robber is distance k from the cops k = 0 is the classical game C k = 1 R 12/13/2019 Anthony Bonato

A new parameter: ck(G) ck(G) = minimum number of cops needed to capture robber at distance at most k G connected implies ck(G) ≤ diam(G) – 1 for all k ≥ 1, ck(G) ≤ ck-1(G) 12/13/2019 Anthony Bonato

Example: k = 1 C c1(G) > 1 R 12/13/2019 Anthony Bonato

Example c1(G) = 2 R C C 12/13/2019 Anthony Bonato

Polytime algorithm Theorem (Bonato,Chiniforooshan,09) Given G as input with k ≥ 0 and s > 0 integers, there is a O(n2s+3) algorithm to determine if ck(G) ≤ s. generalizes algorithm in case k = 0 12/13/2019 Anthony Bonato

Strong products sth strong power of G: vertices: s-tuples from V(G) edges: two s-tuples are joined if they are equal or adjacent in each coordinate idea: set of s cops moving in G move as one cop moving in the sth strong power of G 12/13/2019 Anthony Bonato

Example: s = 2, G = P3 12 13 11 C C 1 C 22 C 2 C 21 23 3 C 31 32 33 12/13/2019 Anthony Bonato

Characterization Theorem. Suppose that k, s ≥ 0. Then ck(G) > s iff there is a function such that 12/13/2019 Anthony Bonato

Algorithm finds a function Ψ from satisfying (1), (2) from the theorem at each step, for any function Ψ’ satisfying (1), (2) of Theorem, Ψ’(T) is a subset of Ψ(T) for all T ck(G) > s iff final value of Ψ satisfies (1), (2) 12/13/2019 Anthony Bonato

ck(n) ck(n) = maximum value of ck(G) over connected G of order n Meyniel conjecture: c0(n) = O(n1/2). 12/13/2019 Anthony Bonato

Upper bound Theorem (BC,09) For n > 0 and k ≥ 0, recovers upper bound from k=0: c0(n) = O(n / log n) 12/13/2019 Anthony Bonato

Lower bound, continued Theorem (BC,Prałat,09) For k ≥ 0, sketch of proof: form Gk by subdividing k times each edge of G a lemma: ck(G2k+1) ≥ c(G) (Bollobás, Kun, Leader, 08) if p = 3log(n) / n, then a.a.s. c(G(n,p))=n1/2+o(1); now use lemma 12/13/2019 Anthony Bonato

Random graphs for random graphs G(n,p) with p = p(n), the behaviour of distance k cop number is complicated Theorem (BCP,09) 12/13/2019 Anthony Bonato

fk(x) = log E(ck(G(n,nx-1))) / log n Zigzag functions for x in (0,1), define fk(x) = log E(ck(G(n,nx-1))) / log n 12/13/2019 Anthony Bonato

Open problems distance k Meyniel conjecture: ck(n) = Θ((n/k)1/2)? structure of ck(G) = 1 graphs, k ≥ 1: dismantling/local-type characterization? Goldstein, Reingold conjecture: computing the cop number of a given graph is EXPTIME-complete is computing ck(G) EXPTIME-complete? is computing ck(G) in FPT? Likely not, as computing c0(G) is W[2]-hard. 12/13/2019 Anthony Bonato

12/13/2019 Anthony Bonato

Google: “Anthony Bonato” preprints, reprints, contact: Google: “Anthony Bonato” 12/13/2019 Anthony Bonato