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A Constant-Approximation Algorithm for Budgeted Sweep Coverage with Mobile Sensors

Wei Liang, Shaojie Tang, Zhao Zhang

TL;DR

This work tackles budgeted sweep coverage with mobile sensors by introducing the multi-orienteering problem (MOP) as a core subproblem. It develops a bicriteria approximation for $k$-MinWP$_m$ using prize-collecting variants (PCP$_m$, PCF$_m$) and a path-shortcutting technique, yielding a $0.035- ext{O}(ps)$-approximation for MOP. Building on this, the authors obtain a $0.0116- ext{O}(ps)$-approximation for the original BSC via a two-step process: vertex grouping by solving MOP instances and dynamic programming for optimal sensor allocation across paths. The methods offer a new, feasible constant-factor approach to deploying mobile sensors for periodic PoI sweep coverage, with practical impact for environmental monitoring, surveillance, and similar routing problems.

Abstract

In this paper, we present the first constant-approximation algorithm for {\em budgeted sweep coverage problem} (BSC). The BSC involves designing routes for a number of mobile sensors (a.k.a. robots) to periodically collect information as much as possible from points of interest (PoIs). To approach this problem, we propose to first examine the {\em multi-orienteering problem} (MOP). The MOP aims to find a set of $m$ vertex-disjoint paths that cover as many vertices as possible while adhering to a budget constraint $B$. We develop a constant-approximation algorithm for MOP and utilize it to achieve a constant-approximation for BSC. Our findings open new possibilities for optimizing mobile sensor deployments and related combinatorial optimization tasks.

A Constant-Approximation Algorithm for Budgeted Sweep Coverage with Mobile Sensors

TL;DR

This work tackles budgeted sweep coverage with mobile sensors by introducing the multi-orienteering problem (MOP) as a core subproblem. It develops a bicriteria approximation for -MinWP using prize-collecting variants (PCP, PCF) and a path-shortcutting technique, yielding a -approximation for MOP. Building on this, the authors obtain a -approximation for the original BSC via a two-step process: vertex grouping by solving MOP instances and dynamic programming for optimal sensor allocation across paths. The methods offer a new, feasible constant-factor approach to deploying mobile sensors for periodic PoI sweep coverage, with practical impact for environmental monitoring, surveillance, and similar routing problems.

Abstract

In this paper, we present the first constant-approximation algorithm for {\em budgeted sweep coverage problem} (BSC). The BSC involves designing routes for a number of mobile sensors (a.k.a. robots) to periodically collect information as much as possible from points of interest (PoIs). To approach this problem, we propose to first examine the {\em multi-orienteering problem} (MOP). The MOP aims to find a set of vertex-disjoint paths that cover as many vertices as possible while adhering to a budget constraint . We develop a constant-approximation algorithm for MOP and utilize it to achieve a constant-approximation for BSC. Our findings open new possibilities for optimizing mobile sensor deployments and related combinatorial optimization tasks.
Paper Structure (12 sections, 11 theorems, 38 equations, 2 figures, 4 algorithms)

This paper contains 12 sections, 11 theorems, 38 equations, 2 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Our Contribution
  4. Organization
  5. Problem formulation and preliminaries
  6. A Constant Approximation Algorithm for MOP
  7. A bicriteria algorithm for $k$-MinWP$_m$
  8. Constructing $m$-vertex disjoint paths $\mathcal{P}_m$
  9. Dealing with the case when $\mathcal{P}_m$ spans many vertices
  10. A $(0.035-O(\varepsilon))$-approximation for MOP
  11. Approximation algorithm for BSC
  12. Conclusion and Future Work

Key Result

Theorem 8

PCP$_m$ admits a $4$-LMP (denoted as $\mathcal{A}_{PC}$) and $\mathcal{A}_{PC}$ executes in time $O(|V|^4)$.

Figures (2)

  • Figure 1: The outline of finding a solution to BSC.
  • Figure 2: An illustration of $P^{(j)}$ and $w^{(j)}$ for $j\in\{0,1\}$.

Theorems & Definitions (28)

  • Definition 1: sweep coverage
  • Definition 2: budgeted sweep coverage (BSC)
  • Definition 3: multi-orienteering problem (MOP)
  • Definition 4: minimum weight vertex-disjoint multi-paths spanning at least $k$ vertices ($k$-MinWP$_m$)
  • Definition 5: prize-collecting vertex-disjoint multi-paths (PCP$_m$)
  • Definition 6: prize-collecting forest with $m$ components (PCF$_m$)
  • Definition 7: Lagrangian multiplier preserving algorithm with factor $r$ ($r$-LMP)
  • Theorem 8
  • proof
  • Lemma 9
  • ...and 18 more