Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximation Algorithms for Minimum Sum of Moving-Distance and Opening-Costs Target Coverage Problem

Lei Zhao, Zhao Zhang

TL;DR

The paper studies the Minimum Sum of Moving-Distance and Opening-Costs Target Coverage problem (MinMD+OCTC), where opening base stations incurs costs and emitted mobile sensors with radius r must cover all targets while minimizing total cost. It develops a dynamic-programming framework that yields an exact polynomial-time solution when all targets lie on a line (MinMD+OCTC_line) and extends to a general-line setting, albeit with higher complexity. For planar targets, it provides an 8.928-approximation for a special case (far-away sensors) by reducing to an uncapped facility location problem on a hexagonal grid and leveraging a 1.488-approximation, combined with a geometric covering argument. The results leverage structural properties such as consecutive sensor intervals per base station and base-station ordering, and they reveal both NP-hardness and practical approximation guarantees, outlining future work on reducing radius-distance constraints and handling diverse sensor radii.

Abstract

In this paper, we study the Minimum Sum of Moving-Distance and Opening-Costs Target Coverage problem (MinMD$+$OCTC). Given a set of targets and a set of base stations on the plane, an opening cost function for every base station, the opened base stations can emit mobile sensors with a radius of $r$ from base station to cover the targets. The goal of MinMD$+$OCTC is to cover all the targets and minimize the sum of the opening cost and the moving distance of mobile sensors. We give the optimal solution in polynomial time for the MinMD$+$OCTC problem with targets on a straight line, and present a 8.928 approximation algorithm for a special case of the MinMD$+$OCTC problem with the targets on the plane.

Approximation Algorithms for Minimum Sum of Moving-Distance and Opening-Costs Target Coverage Problem

TL;DR

The paper studies the Minimum Sum of Moving-Distance and Opening-Costs Target Coverage problem (MinMD+OCTC), where opening base stations incurs costs and emitted mobile sensors with radius r must cover all targets while minimizing total cost. It develops a dynamic-programming framework that yields an exact polynomial-time solution when all targets lie on a line (MinMD+OCTC_line) and extends to a general-line setting, albeit with higher complexity. For planar targets, it provides an 8.928-approximation for a special case (far-away sensors) by reducing to an uncapped facility location problem on a hexagonal grid and leveraging a 1.488-approximation, combined with a geometric covering argument. The results leverage structural properties such as consecutive sensor intervals per base station and base-station ordering, and they reveal both NP-hardness and practical approximation guarantees, outlining future work on reducing radius-distance constraints and handling diverse sensor radii.

Abstract

In this paper, we study the Minimum Sum of Moving-Distance and Opening-Costs Target Coverage problem (MinMDOCTC). Given a set of targets and a set of base stations on the plane, an opening cost function for every base station, the opened base stations can emit mobile sensors with a radius of from base station to cover the targets. The goal of MinMDOCTC is to cover all the targets and minimize the sum of the opening cost and the moving distance of mobile sensors. We give the optimal solution in polynomial time for the MinMDOCTC problem with targets on a straight line, and present a 8.928 approximation algorithm for a special case of the MinMDOCTC problem with the targets on the plane.
Paper Structure (8 sections, 6 theorems, 13 equations, 6 figures)

This paper contains 8 sections, 6 theorems, 13 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. Contribution
  4. The Algorithm
  5. Polynomial-Time Algorithm for MinMD$+$OCTC$_{line}$
  6. Polynomial-Time Algorithm for General MinMD$+$OCTC$_{line}$
  7. Approximation algorithm for a special case of MinMD$+$OCTC
  8. Conclusion

Key Result

Lemma 2.2

Given an instance of the MinMD$+$OCTC$_{line}$ problem $\mathcal{I}=(\mathcal{T},\mathcal{B})$, where the coordinates of base station $b_k$ are $(x(b_k),y(b_k))$, and the opening cost is $c_k$. Construct a new instance $\mathcal{I}'=(\mathcal{T},\mathcal{B}')$, where the coordinates of base station

Figures (6)

  • Figure 1: An illustration for the proof of consecutive arrangement of sensors.
  • Figure 2: An illustration for the proof of sensors orderliness.
  • Figure 3: An illustration of the number of sensor locations being polynomial.
  • Figure 4:
  • Figure 5:
  • ...and 1 more figures

Theorems & Definitions (15)

  • Definition 2.1: MinMD$+$OCTC
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof
  • Theorem 2.6
  • proof
  • proof
  • Lemma 2.9
  • ...and 5 more