For 3 ≤ q < Q we consider the ApproxColoring(q,Q) problem of deciding for a given graph G whether χ(G) ≤ q or χ(G) ≥ Q. It was show in [DMR09] that the problem ApproxColoring(q,Q) is NP-hard for q = 3, 4 and arbitrary large constant Q under variants of the Unique Games Conjecture. In this paper we give a tighter analysis of the reduction of [DMR09] from Unique Games to the ApproxColoring proble...

We show that an encryption scheme cannot have a simple decryption function and be homomorphic at the same time, even with added noise. Specifically, if a scheme can homomorphically evaluate the majority function, then its decryption cannot be weakly-learnable (in particular, linear), even if the probability of decryption error is high. (In contrast, without homomorphism, such schemes do exist a...

The absolute loss is the absolute difference between the desired and predicted outcome. This paper demonstrates worst-case upper bounds on the absolute loss for the Perceptron learning algorithm and the Exponentiated Update learning algorithm, which is related to the Weighted Majority algorithm. The bounds characterize the behavior of the algorithms over any sequence of trials, where each trial...

The case-study presented in this paper is aimed at assessing the dependability of a Programmable Logic Controller (PLC) devoted to safety functions. This case study has been brought to our attention by a national environmental agency and has been partially abstracted and anonymized to protect proprietary information. The PLC consists of a triplicated channel with a (2 : 3) majority voting logic...

A general theory is developed for constructing the asymptotically shallowest networks and the asymptotically smallest networks (with respect to formula size) for the carry save addition of n numbers using any given basic carry save adder as a building block. Using these optimal carry save addition networks the shallowest known multiplication circuits and the shortest formulae for the majority f...

Perhaps one of the most natural ways to aggregate binary preferences from individual agents to a group of agents is simple majority rule, which prescribes that one alternative is socially preferred to another whenever a majority of agents prefers the former to the latter. Majority rule intuitively appeals to democratic principles, is easy to understand and—most importantly—satisfies some attrac...

This paper gives the first separation between the power of formulas and circuits of equal depth in the AC[⊕] basis (unbounded fan-in AND, OR, NOT and MOD2 gates). We show, for all d(n) ≤ O( logn log logn ), that there exist polynomial-size depth-d circuits that are not equivalent to depth-d formulas of size n (moreover, this is optimal in that n cannot be improved to n). This result is obtained...

Least absolute deviation (LAD) regression is an important tool used in numerous applications throughout science and engineering, mainly due to the intrinsic robust characteristics of LAD. In this paper, we show that the optimization needed to solve the LAD regression problem can be viewed as a sequence of maximum likelihood estimates (MLE) of location. The derived algorithm reduces to an iterat...

This paper considers the problem of minimizing the ordered weighted average (or ordered median) function of finitely many rational functions over compact semi-algebraic sets. Ordered weighted averages of rational functions are not, in general, neither rational functions nor the supremum of rational functions so that current results available for the minimization of rational functions cannot be ...

Aprofileπ = (x1, . . . , xk), of length k, in a finite connected graphG is a sequence of vertices ofG, with repetitions allowed. Amedian x ofπ is a vertex forwhich the sumof the distances from x to the vertices in the profile is minimum. The median function finds the set of all medians of a profile. Medians are important in location theory and consensus theory. A median graph is a graph for whi...