# The Role of Visibility in Pursuit/Evasion Games

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## Abstract

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## 1. Introduction

## 2. Preliminaries

#### 2.1. Notation

- We use the following notations for sets:$\phantom{\rule{4pt}{0ex}}\mathbb{N}$ denotes $\left(\right)open="\{"\; close="\}">1,2,\dots $; ${\mathbb{N}}_{0}$ denotes $\left(\right)open="\{"\; close="\}">0,1,2,\dots $; $\left[K\right]$ denotes $\left(\right)open="\{"\; close="\}">1,\dots ,K$; $A-B=\left(\right)open="\{"\; close="\}">x:x\in A,x\notin B$; $\left|A\right|$ denotes the cardinality of A (i.e., the number of its elements).
- A graph $G=(V,E)$ consists of a node set V and an edge set E, where every $e\in E$ has the form $e=\left(\right)open="\{"\; close="\}">x,y\subseteq V$. In other words, we are concerned with finite, undirected, simple graphs; in addition we will always assume that G is connected and that G contains n nodes: $\left|V\right|=n$. Furthermore, we will assume, without loss of generality, that the node set is $V=\left(\right)open="\{"\; close="\}">1,2,\dots ,n$. We let ${V}^{K}=\underset{K\phantom{\rule{4.pt}{0ex}}\text{times}}{\underbrace{V\times V\times \dots \times V}}$. We also define ${V}_{D}^{2}\subseteq {V}^{2}$ by ${V}_{D}^{2}=\{(x,x):x\in V\}$ (it is the set of “diagonal” node pairs).
- A directed graph (digraph) $G=(V,E)$ consists of a node set V and an edge set E, where every $e\in E$ has the form $e=\left(\right)open="("\; close=")">x,y\in V\times V$. In other words, the edges of a digraph are ordered pairs.
- In graphs, the (open) neighborhood of some $x\in V$ is $N\left(x\right)=\left(\right)open="\{"\; close="\}">y:\left(\right)open="\{"\; close="\}">x,y\in E$; in digraphs it is $N\left(x\right)=\left(\right)open="\{"\; close="\}">y:\left(\right)open="("\; close=")">x,y\in E$. In both cases, the closed neighborhood of $x$ is $N\left[x\right]=N\left(x\right)\cup \left\{x\right\}$.
- Given a graph $G=\left(\right)open="("\; close=")">V,E$, its line graph $L\left(G\right)=\left(\right)open="("\; close=")">{V}^{\prime},{E}^{\prime}$ is defined as follows: the node set is ${V}^{\prime}=E$, i.e., it has one node for every edge of G; the edge set is defined by having the nodes $\left(\right)open="\{"\; close="\}">u,v,\left(\right)open="\{"\; close="\}">x,y$ connected by an edge $\left(\right)open="\{"\; close="\}">\left(\right)open="\{"\; close="\}">u,v$ if and only if $\left(\right)open="|"\; close="|">\left(\right)open="\{"\; close="\}">u,v\cap \left(\right)open="\{"\; close="\}">x,y$ (i.e., if the original edges of G are adjacent).
- We will write $f\left(n\right)=o\left(\right)open="("\; close=")">g\left(n\right)$ if and only if ${lim}_{n\to \infty}\frac{f\left(n\right)}{g\left(n\right)}=0$. Note that in this asymptotic notation n denotes the parameter with respect to which asymptotics are considered. So in later sections we will write $o\left(n\right)$, $o\left(M\right)$ etc.

#### 2.2. The CR Game Family

Adversarial Visible Robber | av-CR |

Adversarial Invisible Robber | ai-CR |

Drunk Visible Robber | dv-CR |

Drunk Invisible Robber | di-CR |

## 3. Cop Number and Capture Time

#### 3.1. The Node av-CR Game

**Theorem 1.**Let $\left(\right)open="("\; close=")">{\overline{V}}_{0},{\overline{V}}_{1},\overline{E},\overline{F}$ be a reachability game on the digraph $\overline{D}=\left(\right)open="("\; close=")">\overline{V},\overline{E}$. Then $\overline{V}$ can be partitioned into two sets ${\overline{W}}_{0}$ and ${\overline{W}}_{1}$ such that (for $i\in \left(\right)open="\{"\; close="\}">0,1$) player i has a memoryless strategy ${\sigma}_{i}$ which is winning whenever the game starts in $u\in {\overline{W}}_{i}$.

- Nodes of the form $u=\left(\right)open="("\; close=")">x,y,p$ correspond to positions (in the original CR game) with the cops located at $x\in {V}^{K}$, the robber at $y\in V$ and player $p\in \left(\right)open="\{"\; close="\}">C,R$ being next to move.
- There is single node $u=\left(\right)open="("\; close=")">\lambda ,\lambda ,C$ which corresponds to the starting position of the game: neither the cops nor the robber have been placed on G; it is C’s turn to move (recall that λ denotes the empty sequence).
- Finally, there exist n nodes of the form $u=\left(\right)open="("\; close=")">x,\lambda ,R$: the cops have just been placed in the graph (at positions $x\in {V}^{K}$) but the robber has not been placed yet; it is R’s turn to move.

**Definition 1.**The cop number of G is

**Theorem 2.**For every G, let $K\ge c\left(G\right)$ and consider the CR game played on G with K cops. There exists a a memoryless cop winning strategy ${\sigma}_{C}$ and a number $\overline{T}\left(\right)open="("\; close=")">K;G\infty $ such that, for every robber strategy ${s}_{R}$, C wins in no more than $\overline{T}\left(\right)open="("\; close=")">K;G$ rounds.

**Definition 2.**Given a graph G, some $K\in \mathbb{N}$ and strategies ${s}_{C}\in {\mathbf{S}}_{C}^{\left(K\right)}$, ${s}_{R}\in {\mathbf{S}}_{R}^{\left(K\right)}$ the av-CR capture time is defined by

**Definition 3.**For every graph G and $K\in \mathbb{N}$, the strategies ${s}_{C}^{\left(K\right)}\in {\mathbf{S}}_{C}^{\left(K\right)}$ and ${s}_{R}^{\left(K\right)}\in {\mathbf{S}}_{R}^{\left(K\right)}$ are a pair of optimal strategies if and only if

**Theorem 3.**Given any graph G and any $K\ge c\left(G\right)$, for the av-CR game there exists a pair $\left(\right)open="("\; close=")">{\sigma}_{C}^{\left(K\right)},{\sigma}_{R}^{\left(K\right)}\in {\tilde{\mathbf{S}}}_{C}^{\left(K\right)}\times {\tilde{\mathbf{S}}}_{R}^{\left(K\right)}$ of memoryless time optimal strategies such that

**Definition 4.**The adversarial visible capture time of G is

#### 3.2. The Node dv-CR Game

**Theorem 4.**Given any graph G and $K\in \mathbb{N}$, for the dv-CR game played on G with K cops there exists a memoryless strategy ${\sigma}_{C}^{\left(K\right)}\in {\tilde{\mathbf{S}}}_{C}^{\left(K\right)}$ such that

**Definition 5.**The drunk visible capture time of G is

#### 3.3. The Node ai-CR Game

**Theorem 5.**On any graph G let ${\overline{s}}_{C}^{\left(K\right)}$ denote the strategy in which K cops random-walk on G. Then

**Theorem 6.**Given any graph G and $K\ge c\left(G\right)$, the ai-CR game played on G with K cops has a value ${v}^{\left(K\right)}$ which satisfies

**Definition 6.**The adversarial invisible capture time of G is

#### 3.4. The Node di-CR Game

**Theorem 7.**Given any graph G and $K\in \mathbb{N}$, for the di-CR game played on G with K cops there exists a strategy ${s}_{C}^{\left(K\right)}\in {\overline{\mathbf{S}}}_{C}^{\left(K\right)}$ such that

**Definition 7.**The drunk invisible capture time of G is

#### 3.5. The Edge CR Games

## 4. The Cost of Visibility

#### 4.1. Cost of Visibility in the Node CR Games

**Definition 8.**For every G, the adversarial cost of visibility is ${H}_{a}\left(G\right)=\frac{c{t}_{i}\left(G\right)}{ct\left(G\right)}$ and the drunk cost of visibility is ${H}_{d}\left(G\right)=\frac{dc{t}_{i}\left(G\right)}{dct\left(G\right)}$.

**Theorem 8.**For every $N\in \mathbb{N}$ we have ${H}_{a}\left(\right)open="("\; close=")">{S}_{N,1}=N$.

**Proof.**$\underset{}{\mathbf{(i)}\mathbf{Computing}\mathit{ct}\left(\right)open="("\; close=")">{\mathit{S}}_{\mathit{N},\mathbf{1}}}\xaf$. In av-CR, for every $N\in \mathbb{N}$ we have $\mathit{ct}\left(\right)open="("\; close=")">{S}_{N,1}=1$: the cop starts at ${X}_{0}=0$, the robber starts at some ${Y}_{0}=u\ne 0$ and, at $t=1$, he is captured by the cop moving into $u$; i.e., $\mathit{ct}\left(\right)open="("\; close=")">{S}_{N,1}\le 1$; on the other hand, since there are at least two vertices ($N\ge 1$), clearly $\mathit{ct}\left(\right)open="("\; close=")">{S}_{N,1}\ge 1$.

**Theorem 9.**For every $N\in \mathbb{N}-\left\{1\right\}$ we have

**Proof.**$\underset{}{\mathbf{(i)}\mathbf{Computing}\mathit{dct}\left(\right)open="("\; close=")">{\mathit{S}}_{\mathit{N},\mathit{M}}}\xaf$. We will first show that, for any $N\in \mathbb{N}$, we have $\mathit{dct}\left(\right)open="("\; close=")">{S}_{N,M}=\left(\right)open="("\; close=")">1+o\left(1\right)$ (recall that the parameter $N$ is a fixed constant whereas $M\to \infty $.) Suppose that the cop starts on the $i$-th ray, at distance $(1+o(1\left)\right)\mathit{cM}$ from the center (for some constant $c\in [0,1]$). The robber starts at a random vertex. It follows that for any $j$ such that $1\le j\le N$, the robber starts on the $j$-th ray with probability $(1+o(1\left)\right)/N$. It is a straightforward application of Chernoff bounds to show that with probability $1+o\left(1\right)$ the robber will not move by more than $o\left(M\right)$ in the next $O\left(\mathit{MN}\right)=O\left(M\right)$ steps, which suffice to finish the game. This is so because, if $X$ has a binomial distribution $\mathit{Bin}(n,p)$, then $\mathit{Pr}\left(\right|X-\mathit{np}|\ge \u03f5\mathit{np})\le 2exp(-{\u03f5}^{2}\mathit{np}/3)$ for any $\u03f5\le 3/2$. Now suppose the robber starts at distance $\omega \left({M}^{2/3}\right)$ from the center. During $N=O\left(M\right)$ steps the robber makes in expectation $N/2$ steps towards the center, and $N/2$ steps towards the end of the ray. The probability to make during $N$ steps more than $N/2+{M}^{2/3}$ steps towards the center, say, is thus at most ${e}^{-{\mathit{cM}}^{1/3}}$, and the same holds also by taking a union bound over all $O\left(M\right)$ steps. Hence, with probability at least $1-{e}^{-{\mathit{cM}}^{1/3}}$ he will throughout $O\left(M\right)$ steps remain at distance $O\left({M}^{2/3}\right)$ from his initial position. In short, the expected capture time is easy to calculate.

- With probability $(1-c+o(1\left)\right)/N$, the robber starts on the same ray as the cop but farther away from the center. Conditioning on this event, the expected capture time is $M(1-c+o(1\left)\right)/2$.
- With probability $(c+o(1\left)\right)/N$, the robber starts on the same ray as the cop but closer to the center. Conditioning on this event, the expected capture time is $M(c+o(1\left)\right)/2$.
- With probability $(N-1+o(1\left)\right)/N$, the robber starts on different ray than the cop. Conditioning on this event, the expected capture time is $(c+o(1\left)\right)M+M(1/2+o(1\left)\right)$.

**(ii.1)**Suppose C starts at the end of one ray (chosen arbitrarily), goes to the center, and then successively checks the remaining rays without repetition, with probability at least $1-O\left({M}^{-1/3}\right)$, the robber will be caught. If the robber is caught (this implies that the robber did not switch rays), the capture time is calculated as follows:

- With probability $(1+o(1\left)\right)/N$, the robber starts on the same ray as the cop. Conditioning on this event, the expected capture time is $(1+o(1\left)\right)M/2$.
- With probability $(1+o(1\left)\right)/N$, the robber starts on the $j$-th ray visited by the cop. Conditioning on this event, the expected capture time is $(1+o(1\left)\right)(M+2M(j-2)+M/2)$. ($M$ steps are required to move from the end of the first ray to the center, $2M$ steps are `wasted’ to check $j-2$ rays, and $M/2$ steps are needed to catch the robber on the $j$-th ray, on expectation.)

**(ii.2)**Now suppose $C$ starts at the center of the ray, rather than the end, and checks all rays from there. By the same arguments as before, the capture time is

**(ii.3)**Similarly, suppose the cop starts at distance $\mathit{cM}$ from the center, for some $c\in [0,1]$. If he first goes to the center of the ray, and then checks all rays (suppose the one he came from is the last to be checked), then the capture time is

#### 4.2. Cost of Visibility in the Edge CR Games

**Definition 9.**For every G, the edge adversarial cost of visibility is ${\overline{H}}_{a}\left(G\right)=\frac{{\overline{ct}}_{i}\left(G\right)}{\overline{ct}\left(G\right)}$ and the edge drunk cost of visibility is defined as ${\overline{H}}_{d}\left(G\right)=\frac{{\overline{dct}}_{i}\left(G\right)}{\overline{dct}\left(G\right)}$.

**Theorem 10.**For every $N\in \mathbb{N}-\left\{1\right\}$ we have ${\overline{H}}_{a}\left(\right)open="("\; close=")">{S}_{N,1}=N-1$.

**Proof.**We have ${\overline{H}}_{a}\left({S}_{N,1}\right)=\frac{{\overline{ct}}_{i}\left({S}_{N,1}\right)}{\overline{ct}\left({S}_{N,1}\right)}=\frac{c{t}_{i}\left({K}_{N}\right)}{ct\left({K}_{N}\right)}$ and, since $N\ge 2$, clearly $ct\left({K}_{N}\right)=1$. Let us now compute $c{t}_{i}\left({K}_{N}\right)$.

**Theorem 11.**For every $N\in \mathbb{N}-\left\{1\right\}$ we have ${\overline{H}}_{d}\left(\right)open="("\; close>{S}_{N,1})=\frac{N(N-1)}{2N-3}$.

**Proof.**This is quite similar to the adversarial case. We have ${\overline{H}}_{d}\left({S}_{N,1}\right)=\frac{{\overline{dct}}_{i}\left({S}_{N,1}\right)}{\overline{dct}\left({S}_{N,1}\right)}=\frac{dc{t}_{i}\left({K}_{N}\right)}{dct\left({K}_{N}\right)}$. Clearly we have $dct\left({K}_{N}\right)=1-1/N$ (with probability $1/N$ the robber selects the same vertex to start with as the cop and is caught before the game actually starts; otherwise is caught in the first round).

## 5. Algorithms for COV Computation

#### 5.1. Algorithms for Visible Robbers

#### 5.1.1. Algorithm for Adversarial Robber

- $C\left(\right)open="("\; close=")">x,y$, the optimal game duration when the cop/robber configuration is $(x,y)$ and it is C’s turn to play;
- $R\left(\right)open="("\; close=")">x,y$, the optimal game duration when the cop/robber configuration is $(x,y)$ and it is R’s turn to play.

Algorithm 1: Cops Against Adversarial Robber (CAAR) |

Input: $G=(V,E)$ |

01 For All $\left(\right)open="("\; close=")">x,y\in {V}_{D}^{2}$ |

02 ${C}^{\left(0\right)}\left(\right)open="("\; close=")">x,y=0$ |

03 ${R}^{\left(0\right)}\left(\right)open="("\; close=")">x,y=0$ |

04 EndFor |

05 For All $\left(\right)open="("\; close=")">x,y\in {V}^{2}-{V}_{D}^{2}$ |

06 ${C}^{\left(0\right)}\left(\right)open="("\; close=")">x,y=\infty $ |

07 ${R}^{\left(0\right)}\left(\right)open="("\; close=")">x,y=\infty $ |

08 EndFor |

09 $i=1$ |

10 While $1>0$ |

11 For All $\left(\right)open="("\; close=")">x,y\in {V}^{2}-{V}_{D}^{2}$ |

12 ${C}^{\left(i\right)}\left(\right)open="("\; close=")">x,y=1+{min}_{{x}^{\prime}\in N\left[x\right]}{R}^{\left(\right)}$ |

13 ${R}^{\left(i\right)}\left(\right)open="("\; close=")">x,y=1+{max}_{{y}^{\prime}\in N\left[y\right]}{C}^{\left(i\right)}\left(\right)open="("\; close=")">x,{y}^{\prime}$ |

14 EndFor |

15 If ${C}^{\left(i\right)}={C}^{\left(\right)}$ And $\phantom{\rule{4pt}{0ex}}{R}^{\left(i\right)}={R}^{\left(\right)}$ |

16 Break |

17 EndIf |

18 $i\leftarrow i+1$ |

19 EndWhile |

20 $C={C}^{\left(i\right)}$ |

21 $R={R}^{\left(i\right)}$ |

Output: C, R |

#### 5.1.2. Algorithm for Drunk Robber

Algorithm 2: Cops Against Drunk Robber (CADR) |

Input: $G=(V,E)$, ε |

01 For All $\left(\right)open="("\; close=")">x,y\in {V}_{D}^{2}$ |

02 $\phantom{\rule{4pt}{0ex}}{C}^{\left(0\right)}\left(\right)open="("\; close=")">x,y=0$ |

03 EndFor |

04 For All $\left(\right)open="("\; close=")">x,y\in V-{V}_{D}^{2}$ |

05 $\phantom{\rule{4pt}{0ex}}{C}^{\left(0\right)}\left(\right)open="("\; close=")">x,y=\infty $ |

06 EndFor |

07 $i=1$ |

08 While $1>0$ |

09 For All $\left(\right)open="("\; close=")">x,y\in V-{V}_{D}^{2}$ |

10 ${C}^{\left(i\right)}\left(\right)open="("\; close=")">x,y=1+{min}_{{x}^{\prime}\in N\left[x\right]}{\sum}_{{y}^{\prime}\in V}P\left(\right)open="("\; close=")">\left(\right)open="("\; close=")">{x}^{\prime},y$ |

11 EndFor |

12 If ${max}_{\left(\right)open="("\; close=")">x,y}\left(\right)open="|"\; close="|">{C}^{\left(i\right)}\left(\right)open="("\; close=")">x,y$ |

13 Break |

14 EndIf |

15 $i\leftarrow i+1$ |

16 EndWhile |

17 $C={C}^{\left(i\right)}$ |

Output: C |

#### 5.2. Algorithms for Invisible Robbers

#### 5.2.1. Algorithms for Adversarial Robber

#### 5.2.2. Algorithm for Drunk Robber

Algorithm 3: Pruned Cop Search (PCS) |

Input: $G=(V,E)$, ${x}_{0}$, ${J}_{max}$, ε |

01 $t=0$ |

02 $S.\mathbf{x}={x}_{0}$, $S.\mathbf{p}=Pr\left({y}_{0}\right|{x}_{0})$, $S.C=0$ |

03 $\mathbf{S}=\left\{S\right\}$ |

04 ${C}_{best}^{old}=0$ |

05 While $1>0$ |

06 $\tilde{\mathbf{S}}=\varnothing $ |

07 For All $S\in \mathbf{S}$ |

08 $\mathbf{x}=S.\mathbf{x}$, $\mathbf{p}=S.\mathbf{p}$, $C=S.C$ |

09 For All $v\in N\left(\right)open="["\; close="]">{x}_{t}$ |

10 ${\mathbf{x}}^{\prime}=\mathbf{x}\&v$ |

11 ${\mathbf{p}}^{\prime}=\mathbf{p}\xb7P\left(v\right)$ |

12 ${C}^{\prime}=\mathbf{Cost}({\mathbf{x}}^{\prime},{\mathbf{p}}^{\prime},C)$ |

13 ${S}^{\prime}.\mathbf{x}={\mathbf{x}}^{\prime}$, ${S}^{\prime}.\mathbf{p}={\mathbf{p}}^{\prime}$, ${S}^{\prime}.C={C}^{\prime}$ |

14 $\tilde{\mathbf{S}}=\tilde{\mathbf{S}}\cup \left\{{S}^{\prime}\right\}$ |

15 EndFor |

16 EndFor |

17 $\mathbf{S}=\mathbf{Prune}(\tilde{\mathbf{S}},{J}_{max})$ |

18 $[{\mathbf{x}}_{best},{C}_{best}]=\mathbf{Best}\left(\mathbf{S}\right)$ |

19 If $|{C}_{best}-{C}_{best}^{old}|<\epsilon $ |

20 Break |

21 Else |

22 ${C}_{best}^{old}={C}_{best}$ |

23 $t\leftarrow t+1$ |

24 EndIf |

25 EndWhile |

Output: ${\mathbf{x}}_{best}$, ${C}_{best}=C\left(\right)open="("\; close=")">{\mathbf{x}}_{best}$. |

**Prune**which computes “best” in terms of smallest $C\left(\mathbf{x}\right)$). Finally, the subroutine $\mathbf{Best}$ in line 18 computes the overall smallest expected capture time ${C}_{best}=C\left(\right)open="("\; close=")">{\mathbf{x}}_{best}$. The procedure is repeated until the termination criterion $|{C}_{best}-{C}_{best}^{old}|<\epsilon $ is satisfied. As explained above, the criterion is expected to be always eventually satisfied because of Equation (20).

## 6. Experimental Estimation of the Cost of Visibility

#### 6.1. Experiments with Node Games

**Figure 4.**$dct\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

**Figure 5.**$dc{t}_{i}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

**Figure 6.**${H}_{d}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

#### 6.2. Experiments with Edge Games

**Figure 8.**$\overline{dct}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

**Figure 9.**${\overline{dct}}_{i}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

**Figure 10.**${\overline{H}}_{d}\left(G\right)$ curves for floorplans with n = 30 or n = 28 cells. Each curve corresponds to a fixed $(M,N)$ pair. The horizontal axis corresponds to the edge insertion probability ${p}_{0}$.

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

## References

- Chung, T.H.; Hollinger, G.A.; Isler, V. Search and pursuit-evasion in mobile robotics. Auton. Robots
**2011**, 31, 299–316. [Google Scholar] [CrossRef] - Isler, V.; Karnad, N. The role of information in the cop-robber game. Theor. Comput. Sci.
**2008**, 399, 179–190. [Google Scholar] [CrossRef] - Alspach, B. Searching and sweeping graphs: A brief survey. Le Matematiche
**2006**, 59, 5–37. [Google Scholar] - Bonato, A.; Nowakowski, R. The Game of Cops and Robbers on Graphs; AMS: Providence, RI, USA, 2011. [Google Scholar]
- Fomin, F.V.; Thilikos, D.M. An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci.
**2008**, 399, 236–245. [Google Scholar] [CrossRef] - Nowakowski, R.; Winkler, P. Vertex-to-vertex pursuit in a graph. Discret. Math.
**1983**, 43, 235–239. [Google Scholar] [CrossRef] - Dereniowski, D.; Dyer, D.; Tifenbach, R.M.; Yang, B. Zero-visibility cops and robber game on a graph. In Frontiers in Algorithmics and Algorithmic Aspects in Information and Management; Springer: Berlin, Germany, 2013; pp. 175–186. [Google Scholar]
- Isler, V.; Kannan, S.; Khanna, S. Randomized pursuit-evasion with local visibility. SIAM J. Discret. Math.
**2007**, 20, 26–41. [Google Scholar] [CrossRef] - Kehagias, A.; Mitsche, D.; Prałat, P. Cops and invisible robbers: The cost of drunkenness. Theor. Comput. Sci.
**2013**, 481, 100–120. [Google Scholar] [CrossRef] - Adler, M.; Racke, H.; Sivadasan, N.; Sohler, C.; Vocking, B. Randomized pursuit-evasion in graphs. Lect. Notes Comput. Sci.
**2002**, 2380, 901–912. [Google Scholar] - Vieira, M.; Govindan, R.; Sukhatme, G.S. Scalable and practical pursuit-evasion. In Proceedings of the 2009 IEEE Second International Conference on Robot Communication and Coordination (ROBOCOMM’09), Odense, Denmark, 31 March–2 April 2009; pp. 1–6.
- Gerkey, B.; Thrun, S.; Gordon, G. Parallel stochastic hill-climbing with small teams. In Multi-Robot Systems. From Swarms to Intelligent Automata; Springer: Dordrecht, Netherlands, 2005; Volume III, pp. 65–77. [Google Scholar]
- Hollinger, G.; Singh, S.; Djugash, J.; Kehagias, A. Efficient multi-robot search for a moving target. Int. J. Robot. Res.
**2009**, 28, 201–219. [Google Scholar] [CrossRef] - Hollinger, G.; Singh, S.; Kehagias, A. Improving the efficiency of clearing with multi-agent teams. Int. J. Robot. Res.
**2010**, 29, 1088–1105. [Google Scholar] [CrossRef] - Lau, H.; Huang, S.; Dissanayake, G. Probabilistic search for a moving target in an indoor environment. In Proceedings of the 2006 IEEE/RSJ International Conference on Intelligent Robots and Systems, Beijing, China, 9–15 October 2006; pp. 3393–3398.
- Sarmiento, A.; Murrieta, R.; Hutchinson, S.A. An efficient strategy for rapidly finding an object in a polygonal world. In Proceedings of the 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems(IROS 2003), Las Vegas, NV, USA, 27–31 October 2003; Volume 2, pp. 1153–1158.
- Hsu, D.; Lee, W.S.; Rong, N. A point-based POMDP planner for target tracking. In Proceedings of the 2008 IEEE International Conference on Robotics and Automation (ICRA 2008), Pasadena, CA, USA, 19–23 May 2008; pp. 2644–2650.
- Kurniawati, H.; Hsu, D.; Lee, W.S. Sarsop: Efficient point-based POMDP planning by approximating optimally reachable belief spaces. In Proceedings of Robotics: Science and Systems, Zurich, Switzerland, 25–28 June 2008.
- Pineau, J.; Gordon, G. POMDP planning for robust robot control. Robot. Res.
**2007**, 28, 69–82. [Google Scholar] - Smith, T.; Simmons, R. Heuristic search value iteration for POMDPs. In Proceedings of the 20th Conference on Uncertainty in Artificial Intelligence, Banff, Canada, 7–11 July 2004; pp. 520–527.
- Spaan, M.T.J.; Vlassis, N. Perseus: Randomized point-based value iteration for POMDPs. J. Artif. Intel. Res.
**2005**, 24, 195–220. [Google Scholar] - Hauskrecht, M. Value-function approximations for partially observable Markov decision processes. J. Artif. Intel. Res.
**2000**, 13, 33–94. [Google Scholar] - Littman, M.L.; Cassandra, A.R.; Kaelbling, L.P. Efficient Dynamic-Programming Updates in Partially Observable Markov Decision Processes; Technical Report CS-95-19; Brown University: Providence, RI, USA, 1996. [Google Scholar]
- Monahan, G.E. A survey of partially observable Markov decision processes: Theory, models, and algorithms. Manag. Sci.
**1982**, 28, 1–16. [Google Scholar] [CrossRef] - Canepa, D.; Potop-Butucaru, M.G. Stabilizing Flocking Via Leader Election in Robot Networks. In Proceedings of the 9th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS 2007), Paris, France, 14–16 November 2007; pp. 52–66.
- Gervasi, V.; Prencipe, G. Robotic Cops: The Intruder Problem. In Proceedings of the 2003 IEEE Conference on Systems, Man and Cybernetics (SMC 2003), Washington, DC, USA, 5–8 October 2003; pp. 2284–2289.
- Prencipe, G. The effect of synchronicity on the behavior of autonomous mobile robots. Theory Comput. Syst.
**2005**, 38, 539–558. [Google Scholar] [CrossRef] - Dudek, A.; Gordinowicz, P.; Pralat, P. Cops and robbers playing on edges. J. Comb.
**2013**, 5, 131–153. [Google Scholar] [CrossRef] - Kuhn, H.W. Extensive games. Proc. Natl. Acad. Sci. USA
**1950**, 36, 570–576. [Google Scholar] [CrossRef] [PubMed] - Bonato, A.Y.; Macgillivray, G. A General Framework for Discrete-Time Pursuit Games, preprint.
- Hahn, G.; MacGillivray, G. A note on k-cop, l-robber games on graphs. Discret. Math.
**2006**, 306, 2492–2497. [Google Scholar] [CrossRef] - Berwanger, D. Graph Games with Perfect Information, preprint.
- Mazala, R. Infinite games. Automata, Logics and Infinite Games
**2002**, 2500, 23–38. [Google Scholar] - Aigner, M.; Fromme, M. A game of cops and robbers. Discret. App. Math.
**1984**, 8, 1–12. [Google Scholar] [CrossRef] - Osborne, M.J. A Course in Game Theory; MIT Press: Cambridge, MA, USA, 1994. [Google Scholar]
- Puterman, M.L. Markov Decision Processes: Discrete Stochastic Dynamic Programming; John Wiley & Sons, Inc.: New York, NY, USA, 1994. [Google Scholar]
- Kehagias, A.; Prałat, P. Some remarks on cops and drunk robbers. Theor. Comput. Sci.
**2012**, 463, 133–147. [Google Scholar] [CrossRef] - De la Barrière, R.P. Optimal Control Theory: A Course in Automatic Control Theory; Dover Pubns: New York, NY, USA, 1980. [Google Scholar]
- Eaton, J.H.; Zadeh, L.A. Optimal pursuit strategies in discrete-state probabilistic systems. Trans. ASME Ser. D J. Basic Eng.
**1962**, 84, 23–29. [Google Scholar] [CrossRef] - Howard, R.A. Dynamic Probabilistic Systems, Volume Ii: Semi-Markov and Decision Processes; Dover Publications: New York, NY, USA, 1971. [Google Scholar]
- Raghavan, T.E.S.; Filar, J.A. Algorithms for stochastic games—A survey. Math. Methods Oper. Res.
**1991**, 35, 437–472. [Google Scholar] [CrossRef]

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## Share and Cite

**MDPI and ACS Style**

Kehagias, A.; Mitsche, D.; Prałat, P.
The Role of Visibility in Pursuit/Evasion Games. *Robotics* **2014**, *3*, 371-399.
https://doi.org/10.3390/robotics3040371

**AMA Style**

Kehagias A, Mitsche D, Prałat P.
The Role of Visibility in Pursuit/Evasion Games. *Robotics*. 2014; 3(4):371-399.
https://doi.org/10.3390/robotics3040371

**Chicago/Turabian Style**

Kehagias, Athanasios, Dieter Mitsche, and Paweł Prałat.
2014. "The Role of Visibility in Pursuit/Evasion Games" *Robotics* 3, no. 4: 371-399.
https://doi.org/10.3390/robotics3040371