Showing papers by "Yuri Rabinovich published in 2013"

On Multiplicative $\lambda$-Approximations and Some Geometric Applications

Abstract: Let $\mathcal{F}$ be a set system over an underlying finite set $X$, and let $\mu$ be a nonnegative measure over $X$; i.e., for every $S \subseteq X$, $\mu(S)=\sum_{x\in S} \mu(x)$. A measure $\mu^*$ on $X$ is called a multiplicative ${\lambda}$-approximation of $\mu$ on $(\mathcal{F},X)$ if for every $S\in \mathcal{F}$ it holds that $a\mu(S) \leq \mu^*(S) \leq b \mu(S)$, and $b/a = \lambda \geq 1$. The central question raised and partially answered in the present paper is about the existence of meaningful structural properties of $\mathcal{F}$ implying that for any $\mu$ on $X$ there exists an ${{1+\epsilon} \over {1-\epsilon}}$-approximation $\mu^*$ supported on a small subset of $X$. It turns out that the parameter that governs the support size of a multiplicative approximation is the triangular rank of $\mathcal{F}$, ${\rm trk}(\mathcal{F})$. It is defined as the maximal length of a sequence of sets $\{S_i\}_{i=1}^t $ in $\mathcal{F}$ such that for all $1

Upper Bounds on Boolean-Width with Applications to Exact Algorithms

Abstract: Boolean-width is similar to clique-width, rank-width and NLC-width in that all these graph parameters are constantly bounded on the same classes of graphs. In many classes where these parameters are not constantly bounded, boolean-width is distinguished by its much lower value, such as in permutation graphs and interval graphs where boolean-width was shown to be O(logn) [1]. Together with FPT algorithms having runtime O *(c boolw ) for a constant c this helped explain why a variety of problems could be solved in polynomial-time on these graph classes.