Variations on Cops and Robbers

We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R > 1 edges at a time, establishing a general upper bound of N / \alpha ^{(1-o(1))\sqrt{log_\alpha N}}, where \alpha = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng. We also show that in this case, the cop number of an N-vertex graph can be as large as N^{1 - 1/(R-2)} for finite R, but linear in N if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on N vertices is at most O(N(log log N)^2/log N). Our approach is based on expansion.Comment: 18 page

The kk-visibility Localization Game

We study a variant of the Localization game in which the cops have limited visibility, along with the corresponding optimization parameter, the $k$-visibility localization number $\zeta_k$, where $k$ is a non-negative integer. We give bounds on $k$-visibility localization numbers related to domination, maximum degree, and isoperimetric inequalities. For all $k$, we give a family of trees with unbounded $\zeta_k$ values. Extending results known for the localization number, we show that for $k\geq 2$, every tree contains a subdivision with $\zeta_k = 1$. For many $n$, we give the exact value of $\zeta_k$ for the $n \times n$ Cartesian grid graphs, with the remaining cases being one of two values as long as $n$ is sufficiently large. These examples also illustrate that $\zeta_i \neq \zeta_j$ for all distinct choices of $i$ and $j.

Search and Pursuit-Evasion in Mobile Robotics, A survey

This paper surveys recent results in pursuitevasion and autonomous search relevant to applications in mobile robotics. We provide a taxonomy of search problems that highlights the differences resulting from varying assumptions on the searchers, targets, and the environment. We then list a number of fundamental results in the areas of pursuit-evasion and probabilistic search, and we discuss field implementations on mobile robotic systems. In addition, we highlight current open problems in the area and explore avenues for future work

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

Geometric Pursuit Evasion

In this dissertation we investigate pursuit evasion problems set in geometric environments. These games model a variety of adversarial situations in which a team of agents, called pursuers, attempts to catch a rogue agent, called the evader. In particular, we consider the following problem: how many pursuers, each with the same maximum speed as the evader, are needed to guarantee a successful capture? Our primary focus is to provide combinatorial bounds on the number of pursuers that are necessary and sufficient to guarantee capture. The first problem we consider consists of an unpredictable evader that is free to move around a polygonal environment of arbitrary complexity. We assume that the pursuers have complete knowledge of the evader's location at all times, possibly obtained through a network of cameras placed in the environment. We show that regardless of the number of vertices and obstacles in the polygonal environment, three pursuers are always sufficient and sometimes necessary to capture the evader. We then consider several extensions of this problem to more complex environments. In particular, suppose the players move on the surface of a 3-dimensional polyhedral body; how many pursuers are required to capture the evader? We show that 4 pursuers always suffice (upper bound), and that 3 are sometimes necessary (lower bound), for any polyhedral surface with genus zero. Generalizing this bound to surfaces of genus g, we prove the sufficiency of (4g + 4) pursuers. Finally, we show that 4 pursuers also suffice under the "weighted region" constraints, where the movement costs through different regions of the (genus zero) surface have (different) multiplicative weights. Next we consider a more general problem with a less restrictive sensing model. The pursuers' sensors are visibility based, only providing the location of the evader if it is in direct line of sight. We begin my making only the minimalist assumption that pursuers and the evader have the same maximum speed. When the environment is a simply-connected (hole-free) polygon of n vertices, we show that Θ(n^1/2 ) pursuers are both necessary and sufficient in the worst-case. When the environment is a polygon with holes, we prove a lower bound of Ω(n^2/3 ) and an upper bound of O(n^5/6 ) pursuers, where n includes the vertices of the hole boundaries. However, we show that with realistic constraints on the polygonal environment these bounds can be drastically improved. Namely, if the players' movement speed is small compared to the features of the environment, we give an algorithm with a worst case upper bound of O(log n) pursuers for simply-connected n-gons and O(√h + log n) for polygons with h holes. The final problem we consider takes a small step toward addressing the fact that location sensing is noisy and imprecise in practice. Suppose a tracking agent wants to follow a moving target in the two-dimensional plane. We investigate what is the tracker's best strategy to follow the target and at what rate does the distance between the tracker and target grow under worst-case localization noise. We adopt a simple but realistic model of relative error in sensing noise: the localization error is proportional to the true distance between the tracker and the target. Under this model we are able to give tight upper and lower bounds for the worst-case tracking performance, both with or without obstacles in the Euclidean plane

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Walks and games on graphs

Herrman, Rebekah Ph.D. The University of Memphis, May 2020. Walks and Games on Graphs. Major Professor: B\\u27ela Bollob\\u27as, Ph.D.Chapter 1 is joint work with Dr. Travis Humble and appears in the journal Physical Review A. In this work, we consider continuous-time quantum walks on dynamic graphs. Continuous-time quantum walks have been well studied on graphs that do not change as a function of time. We offer a mathematical formulation for how to express continuous-time quantum walks on graphs that can change in time, find a universal set of walks that can perform any operation, and use them to simulate basic quantum circuits. This work was supported in part by the Department of Energy Student Undergraduate Laboratory Internship and the National Science Foundation Mathematical Sciences Graduate Internship programs.The $(t,r)$ broadcast domination number of a graph $G$, $\gamma_{t,r}(G)$, is a generalization of the domination number of a graph. In Chapter 2, we consider the $(t,r)$ broadcast domination number on graphs, specifically powers of cycles, powers of paths, and infinite grids. This work is joint with Peter van Hintum and has been submitted to the journal Discrete Applied Mathematics.Bridge-burning cops and robbers is a variant of the cops and robbers game on graphs in which the robber removes an edge from the graph once it is traversed. In Chapter 3, we study the maximum time it takes the cops to capture the robber in this variant. This is joint with Peter van Hintum and Dr. Stephen Smith.In Chapter 4, we study a variant of the chip-firing game called the \emph{diffusion game}. In the diffusion game, we begin with some integer labelling of the vertices of a graph, interpreted as a number of chips on each vertex, and for each subsequent step every vertex simultaneously fires a chip to each neighbour with fewer chips. In general, this could result in negative vertex labels. Long and Narayanan asked whether there exists an $f(n)$ for each $n$, such that whenever we have a graph on $n$ vertices and an initial allocation with at least $f(n)$ chips on each vertex, then the number of chips on each vertex will remain non-negative. We answer their question in the affirmative, showing further that $f(n)=n-2$ is the best possible bound. We also consider the existence of a similar bound $g(d)$ for each $d$, where $d$ is the maximum degree of the graph. This work is joint with Andrew Carlotti and has been submitted to the journal Discrete Mathematics.In Chapter 5, we consider the eternal game chromatic number of random graphs. The eternal graph colouring problem, recently introduced by Klostermeyer and Mendoza \cite{klostermeyer}, is a version of the graph colouring game, where two players take turns properly colouring a graph. In this chapter, we show that with high probability $\chi_{g}^{\infty}(G_{n,p}) = (\frac{p}{2} + o(1))n$ for odd $n$, and also for even $n$ when $p=\frac{1}{k}$ for some $k \in \N$. This work is joint with Vojt\u{e}ch Dvo\u{r}\\u27ak and Peter van Hintum, and has been submitted to the European Journal of Combinatorics

Combinatorics and Metric Geometry

This thesis consists of an introduction and seven chapters, each devoted to a different combinatorial problem. In Chapters 1 and 2, we consider the main subject of this thesis; the sharp stability of the Brunn-Minkowski inequality (BM). This celebrated theorem from the 19th century asserts that for bodies A,B $\subset$ $\mathbb{R}$$^{k}$, we have |A + B|$^{1/k}$ $\geq$ |A|$^{1/k}$ + |B|$^{1/k}$, where |$\cdot$| is the Lebesgue measure and A + B := {a + b : a $\in$ A, b $\in$ B} is the Minkowski sum. Moreover, we have equality if and only if A,B are homothetic convex sets. The stability question, studied in many papers, asks how the distance to equality in BM relates to the distance from A,B to homothetic convex sets. In particular, given Brunn-Minkowsi deficit $\delta$ := |A+B|$^{1/k}$ / |A|$^{1/k}$ + |B|$^{1/k}$ -1, and normalized volume ratio $\textit{t}$ := |A|$^{1/k}$ / |A|$^{1/k}$ + |B|$^{1/k}$, what is the best bound one can find on $\omega$ := |K$_{A}$ \ A| / |A| + |K$_{B}$ \ B| / |B|, where K$_{A}$ $\supset$ A, and K$_{B}$ $\supset$ B are homothetic convex sets of minimal size? In Chapter 2, we prove a conjecture by Figalli and Jerison establishing the sharp stability for homothetic sets. In particular, we show that for homothetic sets, we have $\omega$ = O$_{k}$($\delta$t$^{-1}$), for $\delta$ sufficiently small. In Chapter 3, we establish the sharp stability for planar sets, i.e. we show that for planar sets and $\delta$ sufficiently small, we have $\omega$ = O($\delta$$^{1/2}$t$^{-1/2}$). A crucial result in Chapter 3 shows that for any $\epsilon$ > 0, if $\delta$ is sufficiently small, then we have |co(A + B) \ (A + B)| $\leq$ (1 + $\epsilon$)(|co(A) \ A| + |co(B) \ B|). In Chapter 4, we consider a reconstruction problem for functions on graphs. Given a function $\textit{f}$:V(G) $\rightarrow$ [k] on the vertices of a graph G and a random walk (U$_{i}$)$^{{\infty}}_{i = 1}$ on that graph, can we reconstruct $\textit{f}$ (up to automorphisms) based on just ($\textit{f}$(U$_{i}$)$^{{\infty}}_{i = 1}$? Gross and Grupel showed this was not generally possible on the hypercube, by constructing non-isomorphic $\textit{locally p-biased}$ sets $\textit{X}$, so that for each vertex $\textit{v}$ the fraction of neighbours which is in $\textit{X}$ is exactly $\textit{p}$. Answering a question of Gross and Grupel, we construct uncountably many non-isomorphic partitions of $\mathbb{Z}$$^{k}$ into 2k parts such that every element of $\mathbb{Z}$$^{k}$ has exactly one neighbour in each part. As a result, we find $\textit{locally p-biased}$ sets for all $\textit{p = c/2n}$ with $\textit{c}$ $\in$ {0, ... , 2n}. In Chapter 5, we prove the complete graph case of the bunkbed conjecture. Given a graph G, let the bunkbed graph BB(G) be the graph G$\Box$K$_{2}$, i.e. the graph obtained from considering two copies of G and connecting equivalent vertices with an edge. The bunkbed conjecture posed by Kasteleyn in 1985 asserts the very intuitive statement that when considering percolation with uniform parameter p, we have $\mathbb{P}$(u$_{1}$ $\leftrightarrow$ v$_{1}$) $\geq$ $\mathbb{P}$(u$_{1}$ $\leftrightarrow$ v$_{2}$), i.e. a vertex has a higher probability of being connected to a vertex in the same copy of G than being connected to the equivalent vertex in the other copy of G. In Chapter 6, we consider the (t,r) broadcast domination number, a generalisation of the domination number in graphs. In this form of domination, we consider a set T $\subset$ V(G) of towers which broadcast at strength t, where broadcast strength decays linearly with distance in the graph. A set of towers is (t,r) broadcast dominating if every vertex in the graph receives at least r signal from all towers combined. More formally, the (t,r) broadcast domination number of a graph G is the minimal cardinality of a set T $\subset$ V(G) such that for every vertex v $\in$ V(G), we have $^{{\Sigma}}_{u {{\in}} T}$ max{t - d(u,v),0} $\geq$ r. Proving a conjecture by Drews, Harris, and Randolph, we establish that the minimal asymptotical density of (t,3) broadcasting subset of $\mathbb{Z}$$^{2}$ is the same as the minimal asymptotical density of a (t-1,1) broadcasting subset of $\mathbb{Z}$$^{2}$. In Chapter 7, we consider the eternal game chromatic number, a version of the game chromatic number in which the game continues after all vertices have been coloured. We show that with high probability $\chi$$^\infty_g$ (G$_{n,p}$) = (p/2 + o(1))n for odd n, and also for even n when p = 1/k for some k $\in$ $\mathbb{N}$. The upper bound applies for even n and any other value of p as well, but we conjecture in this case this upper bound is not sharp. Finally, we answer a question posed by Klostermeyer and Mendoza. In Chapter 8, we consider the bridge-burning cops and robbers game, a version of the game where after a robber moves over an edge, the edge is removed from the graph. Proving a generalization of a conjecture by Kinnersley and Peterson, we establish the asymptotically maximal capture time in this game for graphs with bridge-burning cops number at least three. In particular, we show that this maximal capture time grows as k$^{-O(k)}$n$^{k+2}$, where k $\geq$ 3 is the bridge burning cop number and n is the number of vertices of the graph