In this section we demonstrate that our upper bounds are not far from optimal, by proving lower bounds for exact and approximate backup placement. We begin with a lower bound for an exact (optimal load) solution.

The quadratic dependence on n in (1) can be removed when ω corresponds to a snowflake, leading to sharp bounds in this case. The appropriate variant of Lemma 9 is as follows.

Given a set $F$ of $n$ positive functions over a ground set $X$, we consider the problem of computing $x^*$ that minimizes the expression $\sum_{f\in F}f(x)$, over $x\in X$. A typical application is \emph{shape fitting}, where we wish to approximate…

The concept of decomposition in computer science and engineering is considered a fundamental component of computational thinking and is prevalent in design of algorithms, software construction, hardware design, and more. We propose a simple and natu…

In this section, we describe a deterministic algorithm (given as Algorithm 4) whose output obeys (roughly) the same properties guaranteed by Proposition 3.10 for Algorithm 2, and thus, using it in Algorithm 3 instead of Algorithm 2 does not affect t…

We considered a setting in which a group of voters wishes to elect the top K candidates out of a given list of candidates, according to some voting rule. We designed secure multiparty protocols for that purpose; to the best of our knowledge, ours is…

The main open question is whether the k-Connectivity Augmentation problem admits a constant ratio approximation algorithm also for large values of k, say k=n−√n. In this context we mention four papers. In [10] is is shown that for values of k close …

We thank Nitish Korula, Vahab S. Mirrokni and Morteza Zadimoghaddam for sharing with us the full version of their paper [24].