Minimizing Expected Intrusion Detection Time
in Adversarial Patrolling

By the definition of $\operatorname{Val}$, there exist a vertex $v$ and an infinite sequence $\Gamma=\gamma_{1},\gamma_{2},\ldots$ of Defender’s strategies such that $\operatorname{Val}(\gamma_{i})=\sup_{\pi}\mathbb{E}^{\gamma_{i},v}[\mathcal{D}^{\pi}]$ for all $i\geq 1$, and the infinite sequence $\operatorname{Val}(\gamma_{1}),\operatorname{Val}(\gamma_{2}),\ldots$ converges to $\operatorname{Val}$.

Let $\textit{Histories}=h_{1},h_{2},\ldots$ be a sequence where every history occurs exactly once (without any special ordering). Let $\Gamma_{0}=\Gamma$. For every $i\geq 1$, we inductively define an infinite sequence of strategies $\Gamma_{i}$ and a probability distribution $\gamma(h_{i})$ over $V$, assuming that $\Gamma_{i-1}$ has already been defined. Let $t$ be the last vertex of $h_{i}$, and let $\{u_{1},\ldots,u_{k}\}$ be the set of all immediate successors of $t$ in $G$ (i.e., $t\to u_{j}$ for all $j\leq k$). Since every bounded infinite sequence of real numbers contains an infinite convergent subsequence, there exists an infinite subsequence $\Gamma_{i}=\varrho_{1},\varrho_{2},\ldots$ of $\Gamma_{i-1}$ such that the sequence

is convergent for every $j\leq k$. We put

It is easy to check that $\sum_{j=1}^{k}\gamma(h_{i})(u_{j})=1$, i.e., $\gamma(h_{i})$ is indeed a distribution on $V$. Hence, the function $\gamma$ is a Defender’s strategy, and we show that $\operatorname{Val}(\gamma)=\operatorname{Val}$.

For the sake of contradiction, suppose $\sup_{\pi}\mathbb{E}^{\gamma,v}[\mathcal{D}^{\pi}]-\operatorname{Val}=\delta>0$. Let $\varepsilon=\delta/4$. Then, there exists an Attackers strategy $\pi^{*}$ such that

Let $\mathit{Att}(\Omega)\subseteq\Omega$ be the set of all observations $o$ such that $\pi^{*}(o)=\mathit{attack}_{\tau}$ for some $\tau\in T$. For every $o=v_{1},\ldots,v_{n},v_{n}{\to}v_{n+1}\in\mathit{Att}(\Omega)$, let $\mathit{walk}(o)$ be the set of all walks starting with $v_{1},\ldots,v_{n+1}$. Observe that if $o,o^{\prime}\in\mathit{Att}(\Omega)$ and $o\neq o^{\prime}$, then $\mathit{walk}(o)\cap\mathit{walk}(o^{\prime})=\emptyset$. Furthermore, for every $w\in\mathcal{W}\smallsetminus\bigcup_{o\in\mathit{Att}(\Omega)}\mathit{walk}(o)$, we have that $\mathcal{D}^{\pi^{*}}(w)=0$. Hence, we obtain

where $\mathbb{E}^{\gamma,v}[\mathcal{D}^{\pi^{*}}\mid\mathit{walk}(o)]$ is the conditional expected value of $\mathcal{D}^{\pi^{*}}$ under the condition that a walk starts with $o$. (Note that the conditional expectation is undefined when $\mathbb{P}^{\gamma,v}(\mathit{walk}(o))=0$. In that case, “$0\cdot\textrm{undefined}$” is interpreted as $0$.) Hence, there exists a finite set of observations $O\subseteq\mathit{Att}(\Omega)$ such that

For each $o=v_{1},\ldots,v_{n},v_{n}{\to}v_{n+1}\in O$, let $\tau_{o}$ be the target attacked by $\pi$ after observing $o$, and let $\mathcal{H}(o)$ be the set of all histories $h$ initiated in $v_{n+1}$ such that $\tau_{o}$ is the last vertex of $h$ and $\tau_{o}$ occurs exactly once in $h$. We use $o\odot h$ to denote the history $v_{1},\ldots,v_{n},h$. We have that $\mathbb{E}^{\gamma,v}[\mathcal{D}^{\pi^{*}}\mid\mathit{walk}(o)]$ is equal to

where $\mathbb{P}^{\gamma,v}\big{(}\mathit{walk}(o\odot h)\mid\mathit{walk}(o)\big{)}$ is the conditional probability of performing a walk starting with $o\odot h$ under the condition that a walk starting with $o$ is performed. Clearly, there exists a finite $H(o)\subseteq\mathcal{H}(o)$ such that the sum (6) decreases at most by $\varepsilon$ when $h$ ranges over $H(o)$ instead of $\mathcal{H}(o)$. For short, we use $E^{\gamma,v}(o)$ to denote the sum

Hence,

By combining (5) and (8), we obtain

Since $\varepsilon=\delta/4$, this implies

Let $H=\bigcup_{o\in O}H(o)$. Since $H$ is finite, there exists an index $\ell$ such that all elements of $H$ appear among the first $\ell$ elements of Histories. Consider the sequence $\Gamma_{\ell}=\varrho_{1},\varrho_{2},\ldots$ and observe that, for all $i\geq 1$,

Since the sequence of distributions $\varrho_{1}(h),\varrho_{2}(h),\ldots$ converges to $\gamma(h)$ for every $h\in H$, we also obtain that

By combining (10), (11), and (22), we obtain $\operatorname{Val}(\varrho_{j})\geq\operatorname{Val}+\varepsilon/2$ for all sufficiently large $j$. This means that the sequence $\operatorname{Val}(\varrho_{1}),\operatorname{Val}(\varrho_{2}),\ldots$ does not converge to $\operatorname{Val}$, and we have a contradiction. ∎

By Theorem 3.1, there exists a Defender’s strategy $\gamma$ such that $\operatorname{Val}(\gamma)=\operatorname{Val}$. We show that for every $\varepsilon>0$, there exist sufficiently large $d,\ell\in\mathbb{N}$ such that $\operatorname{Val}(\sigma_{d,\ell})\leq\operatorname{Val}(\gamma)+\varepsilon$, where $\sigma_{\delta,\ell}$ is a regular strategy obtained by $d$-discretization and $\ell$-folding of $\gamma$.

A $d$-discretization of $\gamma$ is a strategy $\gamma_{d}$ where for all $h\in\mathcal{H}$ and $v\in V$, the following conditions are satisfied:

• •

  $\gamma_{d}(h)(v)=k/d$ for some $k\in\{0,\ldots,d\}$;

• •

  $\gamma_{d}(h)(v)=0$ iff $\gamma(h)(v)=0$;

• •

  $|\gamma_{d}(h)(v)-\gamma(h)(v)|\leq|V|/d$.

Observe that a $d$-discretization of $\gamma$ exists for every $d\geq|V|$.

The regular strategy $\sigma_{d,\ell}$ is obtained by $\ell$-folding the strategy $\gamma_{d}$ constructed for a sufficiently large $d$. We can view $\gamma_{d}$ as an infinite tree $T$ where the nodes are histories and $h\stackrel{{\scriptstyle\raisebox{-0.90417pt}{\scriptsize$x$}}}{{\rightarrow}}hu$ iff $\gamma(h)(u)=x$. Furthermore, we label each node $h$ of $T$ with the last vertex of $h$. Since the edge probabilities range over finitely many values, the tree $T$ contains only finitely many subtrees of height $\ell$ up to isomorphism preserving both node and edge labels. For a given history $h=v_{0},\ldots,v_{n}$, let $f_{h},s_{h}\in\mathbb{N}$ be the lexicographically smallest pair of indexes such that $f_{h}<s_{h}$, both $f_{h}$ and $s_{h}$ are integer multiples of $\ell$, and the subtrees of height $\ell$ rooted by $v_{f_{h}}$ and $v_{s_{h}}$ are isomorphic. If no such $f_{h},s_{h}$ exist, we say that $h$ does not contain a folding pair.

Note that there exists a constant $c_{\ell}$ independent of $h$ such that every history $h$ of length at least $c_{\ell}$ contains a folding pair $f_{h},s_{h}$ with both components bounded by $c_{\ell}$. We define a strategy $\gamma_{d,\ell}$ as follows:

• •

  $\gamma_{d,\ell}(h)=\gamma_{d}(h)$ for every history $h$ without a folding pair.

• •

  $\gamma_{d,\ell}(h)=\gamma_{d,\ell}(h^{\prime})$ for every history $h$ with a folding pair $f_{h},s_{h}$, where the $h^{\prime}$ is obtained from $h=v_{0},\ldots,v_{n}$ by deleting the subpath $v_{f_{h}},\ldots,v_{s_{h}-1}$.

Note that $\gamma_{d,\ell}$ can be equivalently represented as a finite-memory strategy $\sigma_{d,\ell}$ where the memory elements correspond to the (finitely many) histories without a folding pair.

It remains to show that for every $\varepsilon>0$ there are sufficiently large $d,\ell$ such that $\operatorname{Val}(\gamma_{d,\ell})\leq\operatorname{Val}+\varepsilon$.

We start by observing important structural properties of the optimal strategy $\gamma$. Let $v^{*}\in V$ such that $\sup_{\pi}\mathbb{E}^{\gamma,v^{*}}[\mathcal{D}^{\pi}]=\operatorname{Val}$. A history $h$ is $\gamma$-eligible if $\mathbb{P}^{\gamma,v^{*}}(\mathit{walk}(h))>0$. For every $\gamma$-eligible history $h$, let $\gamma[h]$ be a strategy such that $\gamma[h](h^{\prime})=\gamma(h\odot h^{\prime})$ for every $h^{\prime}\in\mathcal{H}$ initiated in the last vertex $u$ of $h$ (for the other $h^{\prime}$, the strategy $\gamma[h]$ is defined arbitrarily). We show that

Clearly,

for every $\gamma$-eligible $h$ because otherwise we have a contradiction with the definition of $\operatorname{Val}$. Now suppose $\sup_{\pi}\mathbb{E}^{\gamma[h],u}[\mathcal{D}^{\pi}]>\operatorname{Val}$ for some $\gamma$-eligible $h=v_{1},\ldots,v_{n}$. We show that then there is an Attacker’s strategy $\pi$ such that $\mathbb{E}^{\gamma,v^{*}}[\mathcal{D}^{\pi}]>\operatorname{Val}$, which contradicts the optimality of $\gamma$. Let $H$ be the set of all $\gamma$-eligible histories of the form $v_{1},\ldots,v_{k},t$ where $k<n$ and $t\neq v_{k+1}$. Furthermore, let $\kappa=\sup_{\pi}\mathbb{E}^{\gamma[h],u}[\mathcal{D}^{\pi}]-\operatorname{Val}$, and let $\delta>0$ be a constant satisfying

For every $h^{\prime}\in H$, let $\pi_{h^{\prime}}$ be an Attacker’s strategy such that $\mathbb{E}^{\gamma[h^{\prime}],u}[\mathcal{D}^{\pi_{h^{\prime}}}]\geq\operatorname{Val}-\delta$, and we also fix an Attacker’s strategy $\pi_{h}$ such that $\mathbb{E}^{\gamma[h],v_{n}}[\mathcal{D}^{\pi_{h^{\prime}}}]\geq\operatorname{Val}+\frac{2}{3}\kappa$. The strategy $\pi$ is defined as follows. For every $\hbar\in H\cup\{h\}$ and every observation of the form $h^{\prime\prime},u\to t$ such that $h^{\prime\prime}t$ is $\gamma[\hbar]$-eligible and $\pi_{\hbar}(h^{\prime\prime},u{\to}t)=\mathit{attack}_{\tau}$ for some $\tau\in T$, we put $\pi(\hbar\odot h^{\prime\prime},u{\to}t)=\mathit{attack}_{\tau}$. Thus, we obtain $\mathbb{E}^{\gamma,v^{*}}[\mathcal{D}^{\pi}]\geq\operatorname{Val}+\kappa/3$.

For every target $\tau$, let $\pi[\tau]$ be the Attacker’s strategy where $\pi[\sigma](u,u\to v)=\mathit{attack}_{\tau}$ for all $u,v\in V$, i.e., $\pi[\tau]$ attacks $\tau$ right after the Defender starts its walk. An immediate consequence of (13) is that for every $\gamma$-eligible history $h$ ending in a vertex $u$ and every target $\tau$, we have that

Let $R_{\tau}$ be a function assigning to every walk $w=v_{0},v_{1},\ldots$ the least $n$ such that $v_{n}=\tau$. If there is no such $n$, we put $R_{\tau}(w)=\infty$. Since $R_{\tau}\leq\mathcal{D}^{\pi[\tau]}$, we have that

by (15) and Markov inequality.

Note that (16) holds for all $\gamma$-eligible $h$ and $\tau$. Hence, we have that

because $R_{\tau}\geq i\cdot 2\operatorname{Val}$ requires success of $i$ consecutive independent experiments where each experiment succeeds with probability bounded by $1/2$.

Now we show that for every $\varepsilon>0$, there exist sufficiently large $d,\ell$ such that $\operatorname{Val}(\gamma_{d,\ell})\leq\operatorname{Val}(\gamma)+\varepsilon$, which proves our theorem.

For the rest of this proof, we fix $\varepsilon>0$. Furthermore, we fix $k\in\mathbb{N}$ satisfying

Here $\mathit{tm}_{\max}=\max\{\mathit{tm}(e)\mid e\in E\}$ and $\alpha_{\max}=\max\{\alpha(\tau)\mid\tau\in T\}$. Note that such a $k$ exists because the above sum converges to $0$ as $k\to\infty$. For every $i\in\mathbb{N}$, let $\mathcal{I}_{i}$ be the set of all integers $j$ satisfying

We have that

Observe

Inequality (20) is trivial, and (21) follows from (17). Using (18), we obtain

Consider a strategy $\gamma_{d}$ where $d\geq|V|$. Then every $\gamma$-eligible history is $\gamma_{d}$-eligible, and vice-versa. Furthermore, for every $\delta>0$, there exists a sufficiently large $d$ such that, for all $h$, $\tau$, and $i$,

Consequently, we can fix a sufficiently large $d$ such that the values of

increase at most by $\varepsilon/4$ when $\gamma$ is replaced with $\gamma_{d}$. Then,

Now consider a strategy $\gamma_{d,\ell}$ where $\ell=k\cdot 2\operatorname{Val}$, and let $h$ be a $\gamma_{d,\ell}$-eligible history. Then $\mathbb{E}^{\gamma_{d,\ell}[h],u}[\mathcal{D}^{\pi[\tau]}]$ is equal to

By definition of $\ell$-folding, for every $i\geq 1$ there is a $\gamma_{d}$-eligible history $h^{\prime}$ without a folding pair such that