[Paper Review] On the Problem of Wireless Scheduling with Linear Power Levels

Tigran Tonoyan|arXiv (Cornell University)|Jul 25, 2011

Mobile Ad Hoc Networks14 references3 citations

TL;DR

This paper presents a constant-factor deterministic approximation algorithm for wireless scheduling under linear power assignment in metric spaces, proving that the optimal schedule length is within a constant factor of the lower bound. It establishes NP-hardness and a 1.5-inapproximability threshold under standard complexity assumptions.

ABSTRACT

In this paper we consider the problem of communication scheduling in wireless networks with respect to the SINR(Signal to Interference plus Noise Ratio) constraint in metric spaces. The nodes are assigned linear powers, i.e. for each sender node the power is constant times the path loss between the sender and corresponding receiver. This is the minimal power for a successful transmission. We present a constant factor deterministic approximation algorithm, which works for at least Euclidean fading metrics. Simultaneously we obtain the approximate value of the optimal schedule length with error at most a constant factor. To give an insight into the complexity of the problem, we show that in some metric spaces the problem is NP-hard and cannot be approximated within a factor less than 1.5.

Motivation & Objective

• To develop a deterministic constant-factor approximation algorithm for wireless scheduling under linear power assignment in metric spaces.
• To establish the approximate value of the optimal schedule length within a constant factor of the true optimum.
• To analyze the computational complexity of the problem, showing it is NP-hard and cannot be approximated within a factor less than 1.5 unless P = NP.
• To extend prior work on SINR-based scheduling by focusing on linear power assignment, which minimizes transmit power while ensuring successful transmission.
• To demonstrate that the optimal schedule length is Θ(I), where I is a lower bound derived from interference constraints.

Proposed method

• Uses a SINR (Signal-to-Interference-plus-Noise Ratio) constraint model to define successful transmission in wireless networks.
• Applies linear power assignment, where each transmitter uses a power level equal to a constant multiple of the path loss to its receiver, ensuring minimal required power.
• Employs a metric space model where distances satisfy triangle inequality and other axioms, enabling generalization across network topologies.
• Constructs a reduction from the PARTITION problem to prove NP-hardness and inapproximability within a factor less than 1.5.
• Analyzes affectance (interference impact) of link sets on target links to determine feasibility of concurrent transmission.
• Uses asymptotic bounds and inequalities (e.g., Lemma 1 with s = α > 1) to evaluate total interference and ensure SINR constraints are met.

Experimental results

Research questions

• RQ1 Can a constant-factor deterministic approximation algorithm be designed for wireless scheduling under linear power assignment in metric spaces?
• RQ2 What is the relationship between the optimal schedule length and the lower bound I derived from interference constraints?
• RQ3 Is the problem NP-hard, and what is the best possible approximation ratio achievable?
• RQ4 Can the problem be reduced to known NP-hard problems like PARTITION to establish inapproximability bounds?
• RQ5 How does linear power assignment compare to other power schemes (e.g., uniform or mean power) in terms of schedule length and approximation guarantees?

Key findings

• A constant-factor deterministic approximation algorithm is proposed for wireless scheduling with linear power assignment in metric spaces.
• The optimal schedule length is proven to be Θ(I), where I is a lower bound on the number of time slots required.
• The problem is NP-hard and cannot be approximated within a factor less than 1.5 unless P = NP, establishing a tight inapproximability threshold.
• The reduction from the PARTITION problem to EQSCHEDULING with K=2 demonstrates that solving the scheduling problem within a factor < 1.5 would imply P = NP.
• The affectance on a central link l₀ by a set S of links is shown to be exactly 2/β, which is critical in linking the scheduling feasibility to the PARTITION problem.
• The results extend to randomized algorithms for cross-layer optimization, improving prior work by a factor of log n.