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Multirobot Watchman Routes in a Simple Polygon

Joseph S. B. Mitchell, Linh Nguyen

TL;DR

The paper studies multirobot watchman routes in simple polygons, focusing on anchored $k$-Watchmen Routes ($k$-WRP) and Quota $k$-Watchmen Routes (Q$k$-WRP). It develops a pseudopolynomial-time dynamic programming algorithm for anchored $k$-WRP in simple orthogonal polygons under the $L_1$ metric using essential visibility cuts on the Hanan grid, and it extends this with an FPTAS for $L_1$ and a $(\sqrt{2}+\varepsilon)$-approximation for unrestricted motion. For the quota variant, it provides a constant-factor approximation by bounding a geodesic disk and constructing near-optimal quota routes, with a doubling-radius strategy to ensure area coverage and a total runtime of $O\left(\frac{n^5}{\varepsilon^6}\log\left(\frac{n}{\varepsilon}\right)\log n\right)$. The results yield practical, guaranteed-performance algorithms for multitarget surveillance in polygonal environments, applicable to axis-aligned and unrestricted motion scenarios with explicit approximation and complexity guarantees.

Abstract

The well-known \textsc{Watchman Route} problem seeks a shortest route in a polygonal domain from which every point of the domain can be seen. In this paper, we study the cooperative variant of the problem, namely the \textsc{$k$-Watchmen Routes} problem, in a simple polygon $P$. We look at both the version in which the $k$ watchmen must collectively see all of $P$, and the quota version in which they must see a predetermined fraction of $P$'s area. We give an exact pseudopolynomial time algorithm for the \textsc{$k$-Watchmen Routes} problem in a simple orthogonal polygon $P$ with the constraint that watchmen must move on axis-parallel segments, and there is a given common starting point on the boundary. Further, we give a fully polynomial-time approximation scheme and a constant-factor approximation for unconstrained movement. For the quota version, we give a constant-factor approximation in a simple polygon, utilizing the solution to the (single) \textsc{Quota Watchman Route} problem.

Multirobot Watchman Routes in a Simple Polygon

TL;DR

The paper studies multirobot watchman routes in simple polygons, focusing on anchored -Watchmen Routes (-WRP) and Quota -Watchmen Routes (Q-WRP). It develops a pseudopolynomial-time dynamic programming algorithm for anchored -WRP in simple orthogonal polygons under the metric using essential visibility cuts on the Hanan grid, and it extends this with an FPTAS for and a -approximation for unrestricted motion. For the quota variant, it provides a constant-factor approximation by bounding a geodesic disk and constructing near-optimal quota routes, with a doubling-radius strategy to ensure area coverage and a total runtime of . The results yield practical, guaranteed-performance algorithms for multitarget surveillance in polygonal environments, applicable to axis-aligned and unrestricted motion scenarios with explicit approximation and complexity guarantees.

Abstract

The well-known \textsc{Watchman Route} problem seeks a shortest route in a polygonal domain from which every point of the domain can be seen. In this paper, we study the cooperative variant of the problem, namely the \textsc{-Watchmen Routes} problem, in a simple polygon . We look at both the version in which the watchmen must collectively see all of , and the quota version in which they must see a predetermined fraction of 's area. We give an exact pseudopolynomial time algorithm for the \textsc{-Watchmen Routes} problem in a simple orthogonal polygon with the constraint that watchmen must move on axis-parallel segments, and there is a given common starting point on the boundary. Further, we give a fully polynomial-time approximation scheme and a constant-factor approximation for unconstrained movement. For the quota version, we give a constant-factor approximation in a simple polygon, utilizing the solution to the (single) \textsc{Quota Watchman Route} problem.
Paper Structure (12 sections, 8 theorems, 7 equations, 4 figures)

This paper contains 12 sections, 8 theorems, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $k$-Watchmen in a Simple Orthogonal Polygon
  4. Dynamic programming exact algorithm
  5. Proof of correctness
  6. Analysis of running time
  7. Fully polynomial-time approximation scheme
  8. Remark
  9. Quota $k$-Watchmen in a Simple Polygon
  10. Constant factor polynomial-time approximation
  11. Analysis of running time
  12. Improving the approximation factor

Key Result

Lemma 1

$\bigcup\limits_{i=1,\ldots,k}V(\gamma_i) = P$ if and only if $\{\gamma_i\}$ collectively visit all essential cuts of $P$.

Figures (4)

  • Figure 1: The essential cuts (dashed).
  • Figure 2: The Hanan grid formed by extensions of all edges in $P$.
  • Figure 3: An example subproblem $(c_3, p_1, l_1, p_2, l_2)$.
  • Figure 4: Left: $\gamma_B$ (red) is a tour no longer than $B$ within $C_g(r)$ (blue) that sees the most area. Right: enclosing $\gamma_B$ with a tour whose vertices are in $S_{\delta, r}$ seeing everything $\gamma_B$ sees (green).

Theorems & Definitions (8)

  • Lemma 1
  • Corollary 2
  • Lemma 3
  • Theorem 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 8