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We study the problem of reaching agreement in a synchronous distributed system by $n$ autonomous parties, when the communication links from/to faulty parties can omit messages. The faulty parties are selected and controlled by an adaptive, full-information, computationally unbounded adversary. We design a randomized algorithm that works in $O(\sqrt{n}\log^2 n)$ rounds and sends $O(n^2\log^3 n)$ communication bits, where the number of faulty parties is $\Theta(n)$. Our result is simultaneously tight for both these measures within polylogarithmic factors: due to the $\Omega(n^2)$ lower bound on communication by Abraham et al. (PODC'19) and $\Omega(\sqrt{n/\log n})$ lower bound on the number of rounds by Bar-Joseph and Ben-Or (PODC'98). We also quantify how much randomness is necessary and sufficient to reduce time complexity to a certain value, while keeping the communication complexity (nearly) optimal. We prove that no MC algorithm can work in less than $\Omega(\frac{n^2}{\max\{R,n\}\log n})$ rounds if it uses less than $O(R)$ calls to a random source, assuming a constant fraction of faulty parties. This can be contrasted with a long line of work on consensus against an {\em adversary limited to polynomial computation time}, thus unable to break cryptographic primitives, culminating in a work by Ghinea et al. (EUROCRYPT'22), where an optimal $O(r)$-round solution with probability $1-(cr)^{-r}$ is given. Our lower bound strictly separates these two regimes, by excluding such results if the adversary is computationally unbounded. On the upper bound side, we show that for $R\in\tilde{O}(n^{3/2})$ there exists an algorithm solving consensus in $\tilde{O}(\frac{n^2}{R})$ rounds with high probability, where tilde notation hides a polylogarithmic factor. The communication complexity of the algorithm does not depend on the amount of randomness $R$ and stays optimal within polylogarithmic factor.

In this paper, we consider proving inapproximability in terms of OPT and thus we base the foundations of fixed parameter inapproximability.

With more than four billion usage of cellular phones worldwide, mobile advertising has become an attractive alternative to online advertisements. In this paper, we propose a new targeted advertising policy for Wireless Service Providers (WSPs) via SMS or MMS- namely {\em AdCell}. In our model, a WSP charges the advertisers for showing their ads. Each advertiser has a valuation for specific types of customers in various times and locations and has a limit on the maximum available budget. Each query is in the form of time and location and is associated with one individual customer. In order to achieve a non-intrusive delivery, only a limited number of ads can be sent to each customer. Recently, new services have been introduced that offer location-based advertising over cellular network that fit in our model (e.g., ShopAlerts by AT&T) . We consider both online and offline version of the AdCell problem and develop approximation algorithms with constant competitive ratio. For the online version, we assume that the appearances of the queries follow a stochastic distribution and thus consider a Bayesian setting. Furthermore, queries may come from different distributions on different times. This model generalizes several previous advertising models such as online secretary problem \cite{HKP04}, online bipartite matching \cite{KVV90,FMMM09} and AdWords \cite{saberi05}. ...