[Paper Review] Wireless Scheduling with Power Control

Magnús M. Halldórsson|arXiv (Cornell University)|Oct 17, 2010

Mobile Ad Hoc Networks33 references4 citations

TL;DR

This paper presents an approximation algorithm for wireless scheduling with power control under the physical (SINR) model, achieving an $O(\log n \cdot \log\log\Delta)$-approximation ratio using an oblivious power assignment. The key contribution is a polylogarithmic approximation in input size under binary length representation, with a matching lower bound showing $\Omega(\log\log\Delta)$ is unavoidable for any well-behaved oblivious power function.

ABSTRACT

We consider the scheduling of arbitrary wireless links in the physical model of interference to minimize the time for satisfying all requests. We study here the combined problem of scheduling and power control, where we seek both an assignment of power settings and a partition of the links so that each set satisfies the signal-to-interference-plus-noise (SINR) constraints. We give an algorithm that attains an approximation ratio of $O(\log n \cdot \log\log Δ)$, where $n$ is the number of links and $Δ$ is the ratio between the longest and the shortest link length. Under the natural assumption that lengths are represented in binary, this gives the first approximation ratio that is polylogarithmic in the size of the input. The algorithm has the desirable property of using an oblivious power assignment, where the power assigned to a sender depends only on the length of the link. We give evidence that this dependence on $Δ$ is unavoidable, showing that any reasonably-behaving oblivious power assignment results in a $Ω(\log\log Δ)$-approximation. These results hold also for the (weighted) capacity problem of finding a maximum (weighted) subset of links that can be scheduled in a single time slot. In addition, we obtain improved approximation for a bidirectional variant of the scheduling problem, give partial answers to questions about the utility of graphs for modeling physical interference, and generalize the setting from the standard 2-dimensional Euclidean plane to doubling metrics. Finally, we explore the utility of graph models in capturing wireless interference.

Motivation & Objective

• To address the fundamental challenge of minimizing scheduling time in wireless networks under the physical interference model with joint power control and link assignment.
• To analyze the performance limits of oblivious power assignments—where power depends only on link length—in the context of SINR-based scheduling and capacity maximization.
• To establish tight theoretical bounds on the approximation ratio achievable by such power assignments, particularly in terms of the dynamic range $\Delta$ of link lengths.
• To evaluate the utility of graph-based models in capturing physical-layer interference, especially in comparison to the more accurate SINR model.
• To extend results beyond the 2D Euclidean plane to doubling metrics and consider bidirectional communication variants.

Proposed method

• Proposes a new oblivious power assignment function $\Psi(\ell_v) = \ell_v^\alpha / \log \ell_v$ that ensures good scheduling performance across all link lengths.
• Uses a constructive lower bound argument based on a carefully arranged linear configuration of links with exponentially increasing lengths to prove inherent limitations of well-behaved power functions.
• Applies SINR feasibility analysis to show that the constructed instance can be scheduled in a single time slot under the proposed power assignment, while any alternative well-behaved function forces $\Omega(\log\log\Delta)$ slots.
• Employs a signal-to-interference-plus-noise ratio (SINR) constraint model where each link must achieve a minimum SINR threshold to be considered successfully scheduled.
• Analyzes interference accumulation by bounding the interference power from each link on others using distance-dependent path loss with exponent $\alpha$.
• Introduces a novel technique to bound total interference per receiver using geometric series and logarithmic scaling, enabling a constant total interference factor under the new power law.

Experimental results

Research questions

• RQ1 What is the best possible approximation ratio achievable by any oblivious power assignment for wireless scheduling under the SINR model?
• RQ2 Can the dependence on $\Delta$, the ratio between longest and shortest link, be reduced to polylogarithmic in the input size?
• RQ3 Is the $\log\log\Delta$ factor unavoidable for any well-behaved oblivious power function?
• RQ4 How does the performance of uniform or linear power assignments compare in online and distributed settings?
• RQ5 To what extent can graph-based models accurately represent physical-layer interference in wireless networks?

Key findings

• The proposed algorithm achieves an $O(\log n \cdot \log\log\Delta)$-approximation for the PC-Scheduling problem, which is polylogarithmic in the input size under binary length representation.
• A lower bound of $\Omega(\log\log\Delta)$ is proven for any well-behaved oblivious power function, showing that the $\log\log\Delta$ factor is unavoidable.
• The mean power assignment $\Psi(\ell_v) = \ell_v^\alpha / \log \ell_v$ enables a single-slot schedule for a hard instance, demonstrating its optimality up to constant factors.
• For the bidirectional variant, the algorithm achieves an $O(\log n)$-approximation, improving upon previous $O(\log^c n)$ results with $c > 5$.
• The construction proves that any well-behaved power function (e.g., polynomial) cannot achieve better than $\Omega(\log\log\Delta)$-approximation in the worst case.
• The results extend to doubling metrics and show that the SINR model's complexity cannot be fully captured by graph-based interference models without significant overhead.