Meirav Zehavi

Hadas Shachnai; Meirav Zehavi

Let \(M=(E,{\cal I})\) be a matroid, and let \({\cal S}\) be a family of subsets of size p of E. A subfamily \(\widehat{\cal S}\subseteq{\cal S}\) represents \({\cal S}\) if for every pair of sets \(X\in{\cal S}\) and \(Y\subseteq E\setminus\ X\) such that \(X\cup Y\in{\cal I}\), there is a set \(\widehat{X}\in\widehat{\cal S}\) disjoint from Y such that \(\widehat{X}\cup Y\in{\cal I}\). Fomin et al. (Proc. ACM-SIAM Symposium on Discrete Algorithms, 2014) introduced a powerful technique for fast computation of representative families for uniform matroids. In this paper, we show that this technique leads to a unified approach for substantially improving the running times of parameterized algorithms for some classic problems. This includes, among others, k -Partial Cover, k -Internal Out-Branching, and Long Directed Cycle. Our approach exploits an interesting tradeoff between running time and the size of the representative families.

Meirav Zehavi

Narrow sieves, representative sets and divide-and-color are three breakthrough techniques related to color coding, which led to the design of extremely fast parameterized algorithms. We present a novel family of strategies for applying mixtures of them. This includes: (a) a mix of representative sets and narrow sieves; (b) a faster computation of representative sets under certain separateness conditions, mixed with divide-and-color and a new technique, called “balanced cutting”; (c) two mixtures of representative sets and a new technique, called “unbalanced cutting”. We demonstrate our strategies by obtaining, among other results, significantly faster algorithms for k-Internal Out-Branching and Weighted 3-Set k-Packing, and a general framework for speeding-up the previous best deterministic algorithms for k-Path, k-Tree, r-Dimensional k-Matching, Graph Motif and Partial Cover.

Meirav Zehavi

The k-Internal Out-Branching (k-IOB) problem asks if a given directed graph has an out-branching (i.e., a spanning tree with exactly one node of in-degree 0) with at least k internal nodes. The k-Internal Spanning Tree (k-IST) problem is a special case of k-IOB, which asks if a given undirected graph has a spanning tree with at least k internal nodes. We present an O *(4 k ) time randomized algorithm for k-IOB, which improves the O * running times of the best known algorithms for both k-IOB and k-IST. Moreover, for graphs of bounded degree Δ, we present an \(O^*(2^{(2-\frac{\Delta+1}{\Delta(\Delta-1)})k})\) time randomized algorithm for k-IOB. Both our algorithms use polynomial space.

Hadas Shachnai; Meirav Zehavi

Given a matroid M = ( E , I ) , and a family S of p-subsets of E, a subfamily S ? ? S represents S if for any X ? S and Y ? E ? X satisfying X ? Y ? I , there is a set X ? ? S ? disjoint from Y, where X ? ? Y ? I . We show that a powerful technique for computing representative families, introduced by Fomin et al. (2014) 5, leads to a unified approach for substantially improving the running times of parameterized algorithms for some classic problems. This includes k-Partial Cover, k-Internal Out-Branching, and Long Directed Cycle, among others. Our approach exploits an interesting tradeoff between running time and the representative family size. A unified tradeoff-based approach for computing representative families.Faster FPT algorithms for problems previously solved by using representative families.Fastest FPT algorithm for the k-Partial Cover problem.Faster deterministic FPT algorithm for the k-Internal Out-Branching problem.

Ron Y. Pinter; Hadas Shachnai; Meirav Zehavi

We study the classic Graph Motif problem: given a graph G = (V,E) with a set of colors for each node, and a multiset M of colors, we seek a subtree T ⊆ G, and a coloring of the nodes in T, such that T carries exactly (also with respect to multiplicity) the colors in M. Graph Motif plays a central role in the study of pattern matching problems, primarily motivated from the analysis of complex biological networks.

Prachi Goyal; Neeldhara Misra; Fahad Panolan; Meirav Zehavi

In this work, we study the well-known

Meirav Zehavi

r

Akanksha Agrawal; Daniel Lokshtanov; Pranabendu Misra; Saket Saurabh; Meirav Zehavi

-Dimensional

Fedor V. Fomin; Daniel Lokshtanov; Fahad Panolan; Saket Saurabh; Meirav Zehavi

k

Mohammed El-Kebir; Benjamin J. Raphael; Ron Shamir; Roded Sharan; Simone Zaccaria; Meirav Zehavi; Ron Zeira

-Matching (