Meirav Zehavi

Representative Families: A Unified Tradeoff-Based Approach

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.

Mixing Color Coding-Related Techniques

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.

Algorithms for k-Internal Out-Branching

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.

Representative families

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.

Deterministic Parameterized Algorithms for the Graph Motif Problem

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.

Deterministic Algorithms for Matching and Packing Problems Based on Representative Sets

In this work, we study the well-known

Parameterized Algorithms for Module Motif

r

Feedback vertex set inspired kernel for chordal vertex deletion

-Dimensional

Finding, Hitting and Packing Cycles in Subexponential Time on Unit Disk Graphs

k

Copy-Number Evolution Problems: Complexity and Algorithms

-Matching (