[Paper Review] Optimally Approximating the Lifetime of Wireless Sensor Networks.

Vivek Bagaria, Ashwin Pananjady+1 more|arXiv (Cornell University)|Jul 19, 2013

Energy Efficient Wireless Sensor Networks8 citations

TL;DR

This paper addresses the lifetime maximization of wireless sensor networks by proving that the Maximum Lifetime Coverage Problem (MLCP) cannot be approximated within a factor better than $\ln n$, and presents a new $\ln n$-approximation algorithm via the Maximum Disjoint Set Cover Problem (DSCP). It further improves this bound using network expansiveness and provides polynomial-time solutions for 1D and $1+\epsilon$-approximation for 2D sensor coverage with circular ranges.

ABSTRACT

We consider the problem of maximizing the lifetime of coverage (MLCP) of targets in a wireless sensor network with battery-limited sensors. We first show that the MLCP cannot be approximated within a factor less than $\ln n$ by any polynomial time algorithm, where $n$ is the number of targets. This provides closure to the long-standing open problem of showing optimality of previously known $\ln n$ approximation algorithms. We also derive a new $\ln n$ approximation to the MLCP by showing a $\ln n$ approximation to the maximum disjoint set cover problem (DSCP), which has many advantages over previous MLCP algorithms, including an easy extension to the $k$-coverage problem. We then present an improvement (in certain cases) to the $\ln n$ algorithm in terms of a newly defined quantity expansiveness of the network. For the special one-dimensional case, where each sensor can monitor a contiguous region of possibly different lengths, we show that the MLCP solution is equal to the DSCP solution, and can be found in polynomial time. Finally, for the special two-dimensional case, where each sensor can monitor a circular area with a given radius around itself, we combine existing results to derive a $1+\epsilon$ approximation algorithm for solving MLCP for any $\epsilon >0$.

Motivation & Objective

• To resolve the long-standing open problem of whether the Maximum Lifetime Coverage Problem (MLCP) can be approximated better than $\ln n$.
• To develop a new $\ln n$-approximation algorithm for MLCP by reducing it to the Maximum Disjoint Set Cover Problem (DSCP).
• To improve the approximation factor using a new network property called expansiveness, particularly in specific network configurations.
• To provide a polynomial-time exact algorithm for the 1D case where sensors cover contiguous intervals.
• To design a $1+\epsilon$-approximation algorithm for the 2D case with circular sensing ranges.

Proposed method

• Prove that MLCP cannot be approximated within a factor less than $\ln n$ unless P=NP, establishing tightness of existing $\ln n$ algorithms.
• Reduce MLCP to the DSCP problem and derive a $\ln n$-approximation for DSCP, which enables a new, extensible algorithm for MLCP.
• Introduce the concept of network expansiveness to refine the approximation ratio beyond $\ln n$ in favorable network topologies.
• Show that in the 1D case, MLCP is equivalent to DSCP and can be solved optimally in polynomial time using interval scheduling techniques.
• Combine geometric and combinatorial results to design a $1+\epsilon$-approximation algorithm for 2D circular coverage using a hierarchical decomposition and rounding strategy.

Experimental results

Research questions

• RQ1 Can the Maximum Lifetime Coverage Problem (MLCP) be approximated within a factor better than $\ln n$?
• RQ2 Is there a new algorithmic framework for MLCP that improves upon existing approaches in terms of extensibility and structure?
• RQ3 Can the approximation ratio be improved beyond $\ln n$ using a new network property such as expansiveness?
• RQ4 Is there a polynomial-time exact algorithm for the 1D version of MLCP?
• RQ5 Can a $1+\epsilon$-approximation be achieved for the 2D case with circular sensing ranges?

Key findings

• The Maximum Lifetime Coverage Problem (MLCP) cannot be approximated within a factor less than $\ln n$ by any polynomial-time algorithm, proving the optimality of existing $\ln n$ algorithms.
• A new $\ln n$-approximation algorithm for MLCP is developed via a reduction to the Maximum Disjoint Set Cover Problem (DSCP), which is more modular and extensible.
• The approximation factor can be improved beyond $\ln n$ in certain network configurations by leveraging the newly defined network expansiveness property.
• For the 1D case, where sensors cover contiguous intervals, MLCP is equivalent to DSCP and solvable in polynomial time.
• For the 2D case with circular sensing ranges, a $1+\epsilon$-approximation algorithm is constructed for any $\epsilon > 0$ using geometric decomposition and rounding techniques.