[Paper Review] Barrier Coverage with Non-uniform Lengths to Minimize Aggregate Movements

Serge Gaspers, Joachim Gudmundsson+2 more|arXiv (Cornell University)|Sep 29, 2017

Indoor and Outdoor Localization Technologies2 references3 citations

TL;DR

This paper studies the barrier coverage problem on a line where sensors with non-uniform radii must be repositioned to cover a target interval [0,L] while minimizing total movement. It presents a ((2+ε)ρ + 2/ε)-approximation algorithm in O(n³/ε²) time and proves that no ρ¹⁻⁰ approximation exists unless P=NP, with fixed-parameter tractability when parameterized by the total movement budget B.

ABSTRACT

Given a line segment $I=[0,L]$, the so-called barrier, and a set of $n$ sensors with varying ranges positioned on the line containing $I$, the barrier coverage problem is to move the sensors so that they cover $I$, while minimising the total movement. In the case when all the sensors have the same radius the problem can be solved in $O(n \log n)$ time (Andrews and Wang, Algorithmica 2017). If the sensors have different radii the problem is known to be NP-hard to approximate within a constant factor (Czyzowicz et al., ADHOC-NOW 2009). We strengthen this result and prove that no polynomial time $ρ^{1-\varepsilon}$-approximation algorithm exists unless $P=NP$, where $ρ$ is the ratio between the largest radius and the smallest radius. Even when we restrict the number of sensors that are allowed to move by a parameter $k$, the problem turns out to be W[1]-hard. On the positive side we show that a $((2+\varepsilon)ρ+2/\varepsilon)$-approximation can be computed in $O(n^3/\varepsilon^2)$ time and we prove fixed-parameter tractability when parameterized by the total movement assuming all numbers in the input are integers.

Motivation & Objective

• To address the NP-hard problem of minimizing aggregate sensor movement in barrier coverage with non-uniform sensor radii.
• To analyze the approximation hardness of the problem in terms of the ratio ρ between largest and smallest sensor radii.
• To develop a polynomial-time approximation algorithm whose performance depends on ρ and ε.
• To investigate parameterized complexity, particularly when the number of moved sensors or the total movement budget is small.
• To establish fixed-parameter tractability when the total movement budget B is the parameter, under integer input constraints.

Proposed method

• Propose an order-preserving solution strategy, where sensor order along the line is maintained in the final configuration.
• Design a (2+ε)ρ + 2/ε-approximation algorithm by discretizing candidate positions and bounding the number of relevant sensors per gap using budget constraints.
• Use a branching algorithm that considers only sensors within distance B of each gap, and limit consideration to the B+1 longest intervals per position to bound search space.
• Prove that optimal solutions can be restricted to integer positions, enabling exact computation via branching.
• Apply parameterized complexity techniques to show fixed-parameter tractability when parameterized by the total movement budget B.
• Leverage structural properties of sensor intervals and gaps to bound the number of candidate sensor movements and positions.

Experimental results

Research questions

• RQ1 Can a polynomial-time approximation algorithm be designed for non-uniform barrier coverage whose approximation ratio depends on the ratio ρ between largest and smallest sensor radii?
• RQ2 Is there a fundamental limit to the approximation ratio achievable in polynomial time, and does it depend on ρ?
• RQ3 Is the problem fixed-parameter tractable when parameterized by the total movement budget B, assuming integer inputs?
• RQ4 Can the approximation ratio be improved beyond (2+ε)ρ + 2/ε, ideally to ρ + O(1)?
• RQ5 Is the problem still fixed-parameter tractable when parameterized by the number of sensors that move, k?

Key findings

• No polynomial-time ρ¹⁻⁰-approximation algorithm exists unless P=NP, establishing a strong inapproximability threshold.
• An O(n³/ε²)-time ((2+ε)ρ + 2/ε)-approximation algorithm is achievable for any ε>0.
• The problem is W[1]-hard when parameterized by the number of moved sensors k.
• The problem is fixed-parameter tractable when parameterized by the total movement budget B, solvable in 2^{2B²(B+1)} · n^{O(1)} time.
• Optimal solutions can be restricted to integer positions, enabling exact branching algorithms.
• The approximation ratio can be improved to ρ + O(1) if order-preserving solutions are restricted to a tighter analysis, though this remains open.