Hardness of Graph Pricing through Generalized Max-Dicut

The Graph Pricing problem is among the fundamental problems whose approximability is not well-understood. While there is a simple combinatorial 1/4-approximation algorithm, the best hardness result remains at 1/2 assuming the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate within a factor better than 1/4 under the UGC, so that the simple combinatorial algorithm might be the best possible. We also prove that for any $\epsilon > 0$, there exists $\delta > 0$ such that the integrality gap of $n^{\delta}$-rounds of the Sherali-Adams hierarchy of linear programming for Graph Pricing is at most 1/2 + $\epsilon$. This work is based on the effort to view the Graph Pricing problem as a Constraint Satisfaction Problem (CSP) simpler than the standard and complicated formulation. We propose the problem called Generalized Max-Dicut($T$), which has a domain size $T + 1$ for every $T \geq 1$. Generalized Max-Dicut(1) is well-known Max-Dicut. There is an approximation-preserving reduction from Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and both our results are achieved through this reduction. Besides its connection to Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own right since in most arity two CSPs studied in the literature, SDP-based algorithms perform better than LP-based or combinatorial algorithms --- for this arity two CSP, a simple combinatorial algorithm does the best.Comment: 28 page

Fast Structuring of Radio Networks for Multi-Message Communications

We introduce collision free layerings as a powerful way to structure radio networks. These layerings can replace hard-to-compute BFS-trees in many contexts while having an efficient randomized distributed construction. We demonstrate their versatility by using them to provide near optimal distributed algorithms for several multi-message communication primitives. Designing efficient communication primitives for radio networks has a rich history that began 25 years ago when Bar-Yehuda et al. introduced fast randomized algorithms for broadcasting and for constructing BFS-trees. Their BFS-tree construction time was $O(D \log^2 n)$ rounds, where $D$ is the network diameter and $n$ is the number of nodes. Since then, the complexity of a broadcast has been resolved to be $T_{BC} = \Theta(D \log \frac{n}{D} + \log^2 n)$ rounds. On the other hand, BFS-trees have been used as a crucial building block for many communication primitives and their construction time remained a bottleneck for these primitives. We introduce collision free layerings that can be used in place of BFS-trees and we give a randomized construction of these layerings that runs in nearly broadcast time, that is, w.h.p. in $T_{Lay} = O(D \log \frac{n}{D} + \log^{2+\epsilon} n)$ rounds for any constant $\epsilon>0$. We then use these layerings to obtain: (1) A randomized algorithm for gathering $k$ messages running w.h.p. in $O(T_{Lay} + k)$ rounds. (2) A randomized $k$-message broadcast algorithm running w.h.p. in $O(T_{Lay} + k \log n)$ rounds. These algorithms are optimal up to the small difference in the additive poly-logarithmic term between $T_{BC}$ and $T_{Lay}$. Moreover, they imply the first optimal $O(n \log n)$ round randomized gossip algorithm

Path loss measurements at 3.5 GHz: A trial test WiMAX based in rural environment

Abstract: This paper addresses the dimensioning of the emerging wireless broadband networks operating in 3.5 GHz band by focusing on the key problem of propagation loss. The characteristics of the path loss in the 3.5 GHz band measured in a rural macro-cellular environment are presented. The existing empirical prediction models are compared with the measured data and a comparative analysis is carried out. The measurements are performed within the experimental activities developed on a WiMAX based platform located in an Italian rural area

Faster deterministic broadcasting in ad hoc radio networks

D'epartement d'informatique, Universit'e du Qu'ebec en Outaouais

On the wake-up problem in radio networks

Abstract. Radio networks model wireless communication when processing units communicate using one wave frequency. This is captured by the property that multiple messages arriving simultaneously to a node interfere with one another and none of them can be read reliably. We present improved solutions to the problem of waking up such a network. This requires activating all nodes in a scenario when some nodes start to be active spontaneously, while every sleeping node needs to be awaken by receiving successfully a message from a neighbor. Our contributions concern the existence and efficient construction of universal radio synchronizers, which are combinatorial structures introduced in [6] as building blocks of efficient wake-up algorithms. First we show by counting that there are (n, g)-universal synchronizers for g(k) = O(k log k log n). Next we show an explicit construction of (n, g)-universal-synchronizers for g(k) = O(k 2 polylog n). By way of applications, we obtain an existential wake-up algorithm which works in time O(n log 2 n) and an explicitly instantiated algorithm that works in time O(n ∆ polylog n), where n is the number of nodes and ∆ is the maximum in-degree in the network. Algorithms for leader-election and synchronization can be developed on top of wake-up ones, as shown in [7], such that they work in time slower by a factor of O(log n) than the underlying wake-up ones.

Efficient k-Shot Broadcasting in Radio Networks

The paper concerns time-efficient k-shot broadcasting in undirected radio networks. In a k-shot broadcasting algorithm, each node in the network is allowed to transmit at most k times. Both known and unknown topology models are considered. For the known topology model, the problem has been studied before by G¸asieniec et al. [14], who established an upper bound of D + O(kn 1/(k−2) log 2 n) and a lower bound of D + Ω((n − D) 1/2k) on the length of k-shot broadcasting schedules for n-node graphs of diameter D. We improve both the upper and the lower bound, providing a randomized algorithm for constructing a k-shot broadcasting schedule of length D + O(kn 1/2k log 2+1/k n) on undirected graphs, and a lower bound of D + Ω(k · (n − D) 1/2k), which almost closes the gap between these bounds. For the unknown topology model, we provide the first k-shot broadcasting algorithm. Assuming that each node knows only the network size n (or a linear upper bound on it), our randomized k-shot broadcasting algorithm completes broadcasting in O((D +min{D · k, log n}) · n 1/(k−1) log n) rounds with high probability. Moreover, we present an Θ(log n)-shot broadcasting algorithm that completes broadcasting in at most O(D log n +log 2 n) rounds with high probability. This algorithm matches the broadcasting time of the algorithm of Bar-Yehuda et al. [3], which assumes no limitation on the maximum number of transmissions per node

Channel interference in ieee 802.11b systems

Abstract — There are many different channels defined in the IEEE 802.11 standard. However, the performance of WiFi networks still greatly suffers from the interference between users, even if they are using different channels. In this paper, we conduct some theoretical analysis of the interference between two channels, which is further verified by experiments. We show that there is indeed serious interference between two nonoverlapping channels if they are close to each other. I