Average-case complexity

About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.

Worst- and average-case complexity of LLL lattice reduction in MIMO wireless systems

Abstract: Lattice reduction by means of the LLL algorithm has been previously suggested as a powerful preprocessing tool that allows to improve the performance of suboptimal detectors and to reduce the complexity of optimal MIMO detectors. The complexity of the LLL algorithm is often cited as polynomial in the dimension of the lattice. In this paper we argue that this statement is not correct when made in the MIMO context. Specifically, we demonstrate that in typical communication scenarios the worst-case complexity of the LLL algorithm is not even finite. For i.i.d. Rayleigh fading channels, we further prove that the average LLL complexity is polynomial and that the probability for an atypically large number of LLL iterations decays exponentially.

On rank vs. communication complexity

Abstract: This paper concerns the open problem of Lovasz and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We first give an example exhibiting the largest gap known. We then prove two related theorems.

Lower Bounds in Communication Complexity

Abstract: In the 30 years since its inception, communication complexity has become a vital area of theoretical computer science. The applicability of communication complexity to other areas, including circuit and formula complexity, VLSI design, proof complexity, and streaming algorithms, has meant that it has attracted a lot of interest. Lower Bounds in Communication Complexity focuses on showing lower bounds on the communication complexity of explicit functions. It treats different variants of communication complexity, including randomized, quantum, and multiparty models. Many tools have been developed for this purpose from a diverse set of fields including linear algebra, Fourier analysis, and information theory. As is often the case in complexity theory, demonstrating a lower bound is usually the more difficult task. Lower Bounds in Communication Complexity describes a three-step approach for the development and application of these techniques. This approach can be applied in much the same way for different models, be they randomized, quantum, or multiparty. Lower Bounds in Communication Complexity is an ideal primer for anyone with an interest in this current and popular topic.

On the complexity of supervisory control design in the RW framework

Abstract: The time complexity of supervisory control design for a general class of problems is studied. It is shown to be very unlikely that a polynomial-time algorithm can be found when either (1) the plant is composed of m components running concurrently or (2) the set of legal behaviors is given by the intersection of n legal specifications. That is to say, in general, there is no way to avoid constructing a state space which has size exponential in m+n. It is suggested that, rather than discouraging future work in the field, this result should point researchers to more fruitful directions, namely, studying special cases of the problem, where certain structural properties possessed by the plant or specification lend themselves to more efficient algorithms for designing supervisory controls. As no background on the subject of computational complexity is assumed, we have tried to explain all the borrowed material in basic terms, so that this paper may serve as a tutorial for a system engineer not familiar with the subject.