Limitations of current wireless link scheduling algorithms

Abstract

We consider the following basic scheduling problem in wireless networks: partition a given set of unit demand communication links into the minimum number of feasible subsets. A subset is feasible if all communications can be done simultaneously, subject to mutual interference. We use the so-called physical model to formulate feasibility.

We consider the two families of approximation algorithms that are known to guarantee $O(\log n)$ approximation for the scheduling problem, where n is the number of links. We present network constructions showing that the approximation ratios of those algorithms are no better than logarithmic, both in n and in Δ, where Δ is a geometric parameter – the ratio of the maximum and minimum link lengths.

Introduction

The task of the MAC layer in TDMA-based (time-division multiple access) wireless networks is to determine which nodes can communicate in which time-frequency slot. A scheduler aims to optimize criteria involving throughput and fairness. This requires obtaining effective spatial reuse while satisfying the interference constraints. We treat the fundamental scheduling problem of partitioning a given set of communication links into the fewest possible feasible sets or slots.

We adopt the physical or SINR model of communication, where signal decays (loses power) as it travels and a transmission is successful if its strength at the receiver exceeds the accumulated signal strength of interfering transmission by a sufficient (technology determined) factor. Considerable progress has been made in recent years in elucidating essential algorithmic properties of the SINR model, involving various aspects of networks such as connectivity scheduling [1], [2], maximum feasible set of links [3], [4], [5], SINR diagrams [6], distributed scheduling [7], [8], distributed broadcast [9], [10], and others. Early work on the scheduling problem includes [11], [12], [13]. Gupta and Kumar [14] proposed the geometric version of SINR and initiated average-case analysis of network capacity known as scaling laws. NP-completeness has been shown for scheduling with different forms of power control: none [15], limited [16], and unbounded [17]. Moscibroda and Wattenhofer [1] initiated worst-case analysis in the SINR model.

Although the standard analytic assumption that signal decays polynomially with the distance traveled is far from realistic [18], [19], it has been shown that results obtained with that assumption can be translated to the setting of arbitrary measured signal decay [20], [21], as well as to Rayleigh fading model – a stochastic fading model [22].

The setting of transmission powers of the links – power control – is an integral part of scheduling, as the same set of links can be feasible with one power assignment and infeasible with another. The scheduling problem has been considered both for arbitrary power control and for fixed oblivious power assignments, where the power chosen for a link is local and depends only on the link itself. This paper addresses algorithms dealing with both scenarios.

Finding constant-factor approximation to the scheduling problem has proved challenging. However, there are several approaches giving logarithmic approximation.

I. Greedy first-fit algorithms. The links are processed in a non-decreasing order of length (the transmitter-receiver distance) and are assigned to the first slot where they have low mutual interference with already scheduled links [23], [4], [3], [24]. An approximation ratio of $O(\log n)$ can be proved for such algorithms, by first arguing that a similar first-fit algorithm provides constant factor approximation to the capacity problem, where the goal is to select maximal size feasible subset, then extending this result to the scheduling problem via a generic argument. It has also been shown that such algorithms perform well for some randomly generated network instances [25].

II. Randomized Contention Resolution. The links act in rounds $r=1,2,\dots$, where in each round r, every link transmits independently with probability $p_r$ or stays silent, for a sequence of probability values $p_r$, $r=1,2,\dots$, and each link is assigned to the time slot r if its transmission succeeded in round r [8], [26]. This is another approach giving a $O(\log n)$-approximation (for fixed power assignments) with high probability, under certain choices of probabilities.

III. Partitioning into Length Classes. The strategy is to divide the links into groups of nearly equal length and schedule each group separately. Following this approach, numerous $O(\log \mathrm{\Delta })$-approximation results have been argued [15], [27], [28], where Δ is the ratio between the longest and shortest link length.

The only known algorithms to achieve non-trivial approximation for scheduling in terms of the number of links n are from the first two families. The only known constant-factor approximation algorithms for scheduling are obtained in the case of the linear power scheme [29], [30]. In recent papers, we have developed a conflict-graph based approach that gives $O({\log }^{⁎}\mathrm{\Delta })$ and $O(\log \log \mathrm{\Delta })$-approximation algorithms for scheduling with power control and fixed power assignments, respectively, but here too, as in the third family of algorithms above, the approximation factor is only $O(n)$ in terms of n [31], [32]. The algorithms from [31], [32] are somewhat similar to first-fit algorithms considered here, but they process the links in the opposite order. This rather unintuitive approach gives better approximation in terms of Δ but no non-trivial approximation in terms of n.

The optimum number of slots for scheduling has also been approximated through interference measures [8], [26], [33]. However, no measure is known to give better than $O(\log n)$ approximation. It is also not evident how to efficiently compute such measures. Many variants of the scheduling problem are known to be NP-hard but we are not aware of any significant inapproximability result.

While for the third family of algorithms it is easy to find examples attaining the approximation ratio $\mathrm{\Omega }(\log \mathrm{\Delta })$, we are not aware of constructions showing that the approximation ratio of $O(\log n)$ of the first and second families cannot be improved. For randomized contention resolution algorithms (the second family), an additive $\mathrm{\Omega }(\log n)$ approximation bound has been proven in [26], and it was asked whether another analysis of their algorithm could give approximation ratio smaller than $O(\log n)$. In fact, for one particular power assignment (the linear power scheme, defined below), it is known that such an algorithm provides additive $O({\log }^{2}n)$ approximation.

Our results  We show that first-fit algorithms (including the algorithms for both fixed power assignments and optimized powers) achieve no better than $\mathrm{\Omega }(\log n)$ (in terms of only n) or $\mathrm{\Omega }\left(\frac{\log \mathrm{\Delta }}{\log \log \mathrm{\Delta }}\right)$ (in terms of only Δ) approximation, and randomized contention resolution algorithms achieve no better than $\mathrm{\Omega }\left(\frac{\log n}{\log \log n}\right)$ or $\mathrm{\Omega }\left(\frac{\log \mathrm{\Delta }}{\log \log \mathrm{\Delta }}\right)$ approximation, for any sequence of probabilities $p_r$, $r=1,2,\dots$ These results hold even in one-dimensional networks, where the nodes are embedded in the real line. Our constructions are obtained by modeling sets of links after certain graphs that are hard instances for analogous algorithms for graph coloring.

Note, however, that our constructions do not work for the uniform power assignment (where all links use equal power) and the linear power assignment (defined below). For the latter, constant factor approximation algorithms are known [29], [30]. The case of the uniform power assignment is still open.

These results suggest that new methods are needed for better than logarithmic approximation of scheduling.