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3

Here's an example that suggests that the stronger claim in the earlier version (Lemma 5.3) is false. I've only given a cursory look at the papers, so please check this carefully to make sure I am understanding correctly, thanks. Consider a cluster $X$ consisting of a rooted star: a root $r$ and $n-1$ nodes $v_1,v_2,\ldots, v_{n-1}$ such that $d(r, v_i) = 1$...

4

Why is it sometimes easier to achieve $\epsilon$-additive accuracy than $\epsilon$-multiplicative? Consider a problem where the objective value OPT is guaranteed to lie in a constant non-negative real interval such as $[0, c]$. For example, the paper you mention is about computing approximation solutions to two-player zero-sum matrix games in which the ...

1

To simplify life, let $\mathcal V = [n] := \{1,2,\ldots,n\}$. For $A \subseteq [n]$, define $h(A):=\sum_{i \in A}c_i$. Note that $h$ defines a modular (i.e additive) set function. Now, suppose there exists $\gamma \in [0, 1]$ such that $f$ is weakly $\gamma$-submodular, i.e such that  \sum_{i \in B\setminus A}f(A \cup \{i\}) - f(A) \ge \gamma (f(A \cup B)...

1

Dependent randomized rounding is a broad tool that is used in many approximation algorithms. Lot depends on the problem structure, objective and constraints and there is no single unified answer/framework. Some papers like the ones you mention identify generic settings where one has some nice probabilistic tools and provide some applications. I list a paper ...

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