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Meirav Zehavi

Publications and source records attributed to Meirav Zehavi.

At least 73 records · Page 4Linked to original sources

arXiv2021

Multivariate Analysis of Scheduling Fair Competitions

A \emph{fair competition}, based on the concept of envy-freeness, is a non-eliminating competition where each contestant (team or individual player) may not play against all other contestants, but the total difficulty for each contestant is the same: the sum of the initial rankings of the opponents for each contestant is the same. Similar to other non-eliminating competitions like the Round-robin competition or the Swiss-system competition, the winner of the fair competition is the contestant who wins the most games. The {\sc Fair Non-Eliminating Tournament} ({\sc Fair-NET}) problem can be used to schedule fair competitions whose infrastructure is known. In the {\sc Fair-NET} problem, we are given an infrastructure of a tournament represented by a graph $G$ and the initial rankings of the contestants represented by a multiset of integers $S$. The objective is to decide whether $G$ is \emph{$S$-fair}, i.e., there exists an assignment of the contestants to the vertices of $G$ such that the sum of the rankings of the neighbors of each contestant in $G$ is the same constant $k\in\mathbb{N}$. We initiate a study of the classical and parameterized complexity of {\sc Fair-NET} with respect to several central structural parameters motivated by real world scenarios, thereby presenting a comprehensive picture of it.

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arXiv2020

On $r$-Simple $k$-Path and Related Problems Parameterized by $k/r$

Abasi et al. (2014) and Gabizon et al. (2015) studied the following problems. In the $r$-Simple $k$-Path problem, given a digraph $G$ on $n$ vertices and integers $r,k$, decide whether $G$ has an $r$-simple $k$-path, which is a walk where every vertex occurs at most $r$ times and the total number of vertex occurrences is $k$. In the $(r,k)$-Monomial Detection problem, given an arithmetic circuit that encodes some polynomial $P$ on $n$ variables and integers $k,r$, decide whether $P$ has a monomial of degree $k$ where the degree of each variable is at most~$r$. In the $p$-Set $(r,q)$-Packing problem, given a universe $V$, positive integers $p,q,r$, and a collection $\cal H$ of sets of size $p$ whose elements belong to $V$, decide whether there exists a subcollection ${\cal H}'$ of $\cal H$ of size $q$ where each element occurs in at most $r$ sets of ${\cal H}'$. Abasi et al. and Gabizon et al. proved that the three problems are single-exponentially fixed-parameter tractable (FPT) when parameterized by $(k/r)\log r$, where $k=pq$ for $p$-Set $(r,q)$-Packing and asked whether the $\log r$ factor in the exponent can be avoided. We consider their question from a wider perspective: are the above problems FPT when parameterized by $k/r$ only? We resolve the wider question by (a) obtaining a $2^{O((k/r)^2\log(k/r))} (n+\log k)^{O(1)}$-time algorithm for $r$-Simple $k$-Path on digraphs and a $2^{O(k/r)} (n+\log k)^{O(1)}$-time algorithm for $r$-Simple $k$-Path on undirected graphs (i.e., for undirected graphs we answer the original question in affirmative), (b) showing that $p$-Set $(r,q)$-Packing is FPT, and (c) proving that $(r,k)$-Monomial Detection is para-NP-hard. For $p$-Set $(r,q)$-Packing, we obtain a polynomial kernel for any fixed $p$, which resolves a question posed by Gabizon et al. regarding the existence of polynomial kernels for problems with relaxed disjointness constraints.

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arXiv2020

Efficient Graph Minors Theory and Parameterized Algorithms for (Planar) Disjoint Paths

In the Disjoint Paths problem, the input consists of an $n$-vertex graph $G$ and a collection of $k$ vertex pairs, $\{(s_i,t_i)\}_{i=1}^k$, and the objective is to determine whether there exists a collection $\{P_i\}_{i=1}^k$ of $k$ pairwise vertex-disjoint paths in $G$ where the end-vertices of $P_i$ are $s_i$ and $t_i$. This problem was shown to admit an $f(k)n^3$-time algorithm by Robertson and Seymour (Graph Minors XIII, The Disjoint Paths Problem, JCTB). In modern terminology, this means that Disjoint Paths is fixed parameter tractable (FPT) with respect to $k$. Remarkably, the above algorithm for Disjoint Paths is a cornerstone of the entire Graph Minors Theory, and conceptually vital to the $g(k)n^3$-time algorithm for Minor Testing (given two undirected graphs, $G$ and $H$ on $n$ and $k$ vertices, respectively, determine whether $G$ contains $H$ as a minor). In this semi-survey, we will first give an exposition of the Graph Minors Theory with emphasis on efficiency from the viewpoint of Parameterized Complexity. Secondly, we will review the state of the art with respect to the Disjoint Paths and Planar Disjoint Paths problems. Lastly, we will discuss the main ideas behind a new algorithm that combines treewidth reduction and an algebraic approach to solve Planar Disjoint Paths in time $2^{k^{O(1)}}n^{O(1)}$ (for undirected graphs).

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arXiv2020

On the (Parameterized) Complexity of Almost Stable Marriage

In the Stable Marriage problem. when the preference lists are complete, all agents of the smaller side can be matched. However, this need not be true when preference lists are incomplete. In most real-life situations, where agents participate in the matching market voluntarily and submit their preferences, it is natural to assume that each agent wants to be matched to someone in his/her preference list as opposed to being unmatched. In light of the Rural Hospital Theorem, we have to relax the "no blocking pair" condition for stable matchings in order to match more agents. In this paper, we study the question of matching more agents with fewest possible blocking edges. In particular, we find a matching whose size exceeds that of stable matching in the graph by at least t and has at most k blocking edges. We study this question in the realm of parameterized complexity with respect to several natural parameters, k,t,d, where d is the maximum length of a preference list. Unfortunately, the problem remains intractable even for the combined parameter k+t+d. Thus, we extend our study to the local search variant of this problem, in which we search for a matching that not only fulfills each of the above conditions but is "closest", in terms of its symmetric difference to the given stable matching, and obtain an FPT algorithm.

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arXiv2020

Computation of Hadwiger Number and Related Contraction Problems: Tight Lower Bounds

We prove that the Hadwiger number of an $n$-vertex graph $G$ (the maximum size of a clique minor in $G$) cannot be computed in time $n^{o(n)}$, unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of $n^{o(n)}$-time algorithms (up to ETH) for a large class of computational problems concerning edge contractions in graphs.

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arXiv2020

Parameterized Study of Steiner Tree on Unit Disk Graphs

We study the Steiner Tree problem on unit disk graphs. Given a $n$ vertex unit disk graph $G$, a subset $R\subseteq V(G)$ of $t$ vertices and a positive integer $k$, the objective is to decide if there exists a tree $T$ in $G$ that spans over all vertices of $R$ and uses at most $k$ vertices from $V\setminus R$. The vertices of $R$ are referred to as terminals and the vertices of $V(G)\setminus R$ as Steiner vertices. First, we show that the problem is NP-Hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in $n^{O(\sqrt{t+k})}$ time. We also show that the Steiner Tree problem on unit disk graphs parameterized by $k$ has an FPT algorithm with running time $2^{O(k)}n^{O(1)}$. In fact, the algorithms are designed for a more general class of graphs, called clique-grid graphs. We mention that the algorithmic results can be made to work for the Steiner Tree on disk graphs with bounded aspect ratio. Finally, we prove that the Steiner Tree on disk graphs parameterized by $k$ is W[1]-hard.

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arXiv2020

The Parameterized Complexity of Guarding Almost Convex Polygons

Art Gallery is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow, ESA'16], the classic variant has an O(log k)-approximation algorithm [Bonnet and Miltzow, SoCG'17], and it may require irrational guards [Abrahamsen et al., SoCG'17]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be \exists R-complete~[Abrahamsen et al., STOC'18]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP. Given the first discovery, the following question was posed by Giannopoulos [Lorentz Center Workshop, 2016]: Is Art Gallery FPT with respect to r, the number of reflex vertices? We focus on the variant where G and C are all the vertices of P, called Vertex-Vertex Art Gallery. We show that Vertex-Vertex Art Gallery is solvable in time r^{O(r^2)}n^{O(1)}. Our approach also extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT. We utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-Vertex Art Gallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions.

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arXiv2020

ETH-Tight Algorithms for Long Path and Cycle on Unit Disk Graphs

We present an algorithm for the extensively studied Long Path and Long Cycle problems on unit disk graphs that runs in time $2^{O(\sqrt{k})}(n+m)$. Under the Exponential Time Hypothesis, Long Path and Long Cycle on unit disk graphs cannot be solved in time $2^{o(\sqrt{k})}(n+m)^{O(1)}$ [de Berg et al., STOC 2018], hence our algorithm is optimal. Besides the $2^{O(\sqrt{k})}(n+m)^{O(1)}$-time algorithm for the (arguably) much simpler Vertex Cover problem by de Berg et al. [STOC 2018] (which easily follows from the existence of a $2k$-vertex kernel for the problem), this is the only known ETH-optimal fixed-parameter tractable algorithm on UDGs. Previously, Long Path and Long Cycle on unit disk graphs were only known to be solvable in time $2^{O(\sqrt{k}\log k)}(n+m)$. This algorithm involved the introduction of a new type of a tree decomposition, entailing the design of a very tedious dynamic programming procedure. Our algorithm is substantially simpler: we completely avoid the use of this new type of tree decomposition. Instead, we use a marking procedure to reduce the problem to (a weighted version of) itself on a standard tree decomposition of width $O(\sqrt{k})$.

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arXiv2020

Hitting Topological Minors is FPT

In the Topological Minor Deletion (TM-Deletion) problem input consists of an undirected graph $G$, a family of undirected graphs ${\cal F}$ and an integer $k$. The task is to determine whether $G$ contains a set of vertices $S$ of size at most $k$, such that the graph $G\setminus S$ obtained from $G$ by removing the vertices of $S$, contains no graph from ${\cal F}$ as a topological minor. We give an algorithm for TM-Deletionwith running time $f(h^\star,k)\cdot |V(G)|^{4}$. Here $h^\star$ is the maximum size of a graph in ${\cal F}$ and $f$ is a computable function of $h^\star$ and $k$. This is the first fixed parameter tractable algorithm (FPT) for the problem. In fact, even for the restricted case of planar inputs the first FPT algorithm was found only recently by Golovach et al. [SODA 2020]. For this case we improve upon the algorithm of Golovach et al. [SODA 2020] by designing an FPT algorithm with explicit dependence on $k$ and $h^\star$.

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arXiv2019

A New Paradigm for Identifying Reconciliation-Scenario Altering Mutations Conferring Environmental Adaptation

An important goal in microbial computational genomics is to identify crucial events in the evolution of a gene that severely alter the duplication, loss and mobilization patterns of the gene within the genomes in which it disseminates. In this paper, we formalize this microbiological goal as a new pattern-matching problem in the domain of Gene tree and Species tree reconciliation, denoted "Reconciliation-Scenario Altering Mutation (RSAM) Discovery". We propose an $O(m\cdot n\cdot k)$ time algorithm to solve this new problem, where $m$ and $n$ are the number of vertices of the input Gene tree and Species tree, respectively, and $k$ is a user-specified parameter that bounds from above the number of optimal solutions of interest. The algorithm first constructs a hypergraph representing the $k$ highest scoring reconciliation scenarios between the given Gene tree and Species tree, and then interrogates this hypergraph for subtrees matching a pre-specified RSAM Pattern. Our algorithm is optimal in the sense that the number of hypernodes in the hypergraph can be lower bounded by $Ω(m\cdot n\cdot k)$. We implement the new algorithm as a tool, called RSAM-finder, and demonstrate its application to -the identification of RSAMs in toxins and drug resistance elements across a dataset spanning hundreds of species.

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arXiv2019

A Brief Note on Single Source Fault Tolerant Reachability

Let $G$ be a directed graph with $n$ vertices and $m$ edges, and let $s \in V(G)$ be a designated source vertex. We consider the problem of single source reachability (SSR) from $s$ in presence of failures of edges (or vertices). Formally, a spanning subgraph $H$ of $G$ is a {\em $k$-Fault Tolerant Reachability Subgraph ($k$-FTRS)} if it has the following property. For any set $F$ of at most $k$ edges (or vertices) in $G$, and for any vertex $v\in V(G)$, the vertex $v$ is reachable from $s$ in $G-F$ if and only if it is reachable from $s$ in $H - F$. Baswana et.al. [STOC 2016, SICOMP 2018] showed that in the setting above, for any positive integer $k$, we can compute a $k$-FTRS with $2^k n$ edges. In this paper, we give a much simpler algorithm for computing a $k$-FTRS, and observe that it extends to higher connectivity as well. Our results follow from a simple application of \emph{important separators}, a well known technique in Parameterized Complexity.

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arXiv2019

The Parameterized Complexity of Motion Planning for Snake-Like Robots

We study the parameterized complexity of a variant of the classic video game Snake that models real-world problems of motion planning. Given a snake-like robot with an initial position and a final position in an environment (modeled by a graph), our objective is to determine whether the robot can reach the final position from the initial position without intersecting itself. Naturally, this problem models a wide-variety of scenarios, ranging from the transportation of linked wagons towed by a locomotor at an airport or a supermarket to the movement of a group of agents that travel in an `ant-like' fashion and the construction of trains in amusement parks. Unfortunately, already on grid graphs, this problem is PSPACE-complete [Biasi and Ophelders, 2016]. Nevertheless, we prove that even on general graphs, the problem is solvable in time $k^{\mathcal{O}(k)}|I|^{\mathcal{O}(1)}$ where $k$ is the size of the snake, and $|I|$ is the input size. In particular, this shows that the problem is fixed-parameter tractable (FPT). Towards this, we show how to employ color-coding to sparsify the configuration graph of the problem to have size $k^{\mathcal{O}(k)}|I|^{\mathcal{O}(1)}$ rather than $|I|^{\mathcal{O}(k)}$. We believe that our approach will find other applications in motion planning. Additionally, we show that the problem is unlikely to admit a polynomial kernel even on grid graphs, but it admits a treewidth-reduction procedure. To the best of our knowledge, the study of the parameterized complexity of motion planning problems (where the intermediate configurations of the motion are of importance) has so far been largely overlooked. Thus, our work is pioneering in this regard.

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arXiv2019

Decomposition of Map Graphs with Applications

Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or its treewidth is $O(\sqrt{k})$. However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new decomposition lemma for map graphs. Informally, our lemma states the following. For any map graph $G$, there exists a collection $(U_1,\ldots,U_t)$ of cliques of $G$ with the following property: $G$ either contains a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or it admits a tree decomposition where every bag is the union of $O(\sqrt{k})$ of the cliques in the above collection. The new lemma appears to be a handy tool in the design of subexponential parameterized algorithms on map graphs. We demonstrate its usability by designing algorithms on map graphs with running time $2^{O({\sqrt{k}\log{k}})} \cdot n^{O(1)}$ for the Connected Planar $\cal F$-Deletion problem (that encompasses problems such as Feedback Vertex Set and Vertex Cover). Obtaining subexponential algorithms for Longest Cycle/Path and Cycle Packing is more challenging. We have to construct tree decompositions with more powerful properties and to prove sublinear bounds on the number of ways an optimum solution could "cross" bags in these decompositions. For Longest Cycle/Path, these are the first subexponential-time parameterized algorithms on map graphs. For Feedback Vertex Set and Cycle Packing, we improve upon known $2^{O({k^{0.75}\log{k}})} \cdot n^{O(1)}$-time algorithms on map graphs.

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arXiv2019

Covering Vectors by Spaces in Perturbed Graphic Matroids and Their Duals

Perturbed graphic matroids are binary matroids that can be obtained from a graphic matroid by adding a noise of small rank. More precisely, r-rank perturbed graphic matroid M is a binary matroid that can be represented in the form I +P, where I is the incidence matrix of some graph and P is a binary matrix of rank at most r. Such matroids naturally appear in a number of theoretical and applied settings. The main motivation behind our work is an attempt to understand which parameterized algorithms for various problems on graphs could be lifted to perturbed graphic matroids. We study the parameterized complexity of a natural generalization (for matroids) of the following fundamental problems on graphs: Steiner Tree and Multiway Cut. In this generalization, called the Space Cover problem, we are given a binary matroid M with a ground set E, a set of terminals T\subseteq E, and a non-negative integer k. The task is to decide whether T can be spanned by a subset of E\setminus T of size at most k. We prove that on graphic matroid perturbations, for every fixed r, Space Cover is fixed-parameter tractable parameterized by k. On the other hand, the problem becomes W[1]-hard when parameterized by r+k+|T| and it is NP-complete for r\leq 2 and |T|\leq 2. On cographic matroids, that are the duals of graphic matroids, Space Cover generalizes another fundamental and well-studied problem, namely Multiway Cut. We show that on the duals of perturbed graphic matroids the Space Cover problem is fixed-parameter tractable parameterized by r+k.

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arXiv2019

Going Far From Degeneracy

An undirected graph G is d-degenerate if every subgraph of G has a vertex of degree at most d. By the classical theorem of Erdős and Gallai from 1959, every graph of degeneracy d>1 contains a cycle of length at least d+1. The proof of Erdős and Gallai is constructive and can be turned into a polynomial time algorithm constructing a cycle of length at least d+1. But can we decide in polynomial time whether a graph contains a cycle of length at least d+2? An easy reduction from Hamiltonian Cycle provides a negative answer to this question: deciding whether a graph has a cycle of length at least d+2 is NP-complete. Surprisingly, the complexity of the problem changes drastically when the input graph is 2-connected. In this case we prove that deciding whether G contains a cycle of length at least d+k can be done in time 2^{O(k)}|V(G)|^{O(1)}. In other words, deciding whether a 2-connected n-vertex G contains a cycle of length at least d+log n can be done in polynomial time. Similar algorithmic results hold for long paths in graphs. We observe that deciding whether a graph has a path of length at least d+1 is NP-complete. However, we prove that if graph G is connected, then deciding whether G contains a path of length at least d+k can be done in time 2^{O(k)}n^{O(1)}. We complement these results by showing that the choice of degeneracy as the `above guarantee parameterization' is optimal in the following sense: For any ε>0 it is NP-complete to decide whether a connected (2-connected) graph of degeneracy d has a path (cycle) of length at least (1+ε)d.

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arXiv2018

Popular Matching in Roommates Setting is NP-hard

An input to the Popular Matching problem, in the roommates setting, consists of a graph $G$ and each vertex ranks its neighbors in strict order, known as its preference. In the Popular Matching problem the objective is to test whether there exists a matching $M^\star$ such that there is no matching $M$ where more people are happier with $M$ than with $M^\star$. In this paper we settle the computational complexity of the Popular Matching problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly, explicitly asked over the last decade.

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arXiv2018

The Parameterized Complexity of Packing Arc-Disjoint Cycles in Tournaments

Given a directed graph $D$ on $n$ vertices and a positive integer $k$, the Arc-Disjoint Cycle Packing problem is to determine whether $D$ has $k$ arc-disjoint cycles. This problem is known to be W[1]-hard in general directed graphs. In this paper, we initiate a systematic study on the parameterized complexity of the problem restricted to tournaments. We show that the problem is fixed-parameter tractable and admits a polynomial kernel when parameterized by the solution size $k$. In particular, we show that it can be solved in $2^{\mathcal{O}(k \log k)} n^{\mathcal{O}(1)}$ time and has a kernel with $\mathcal{O}(k)$ vertices. The primary ingredient in both these results is a min-max theorem that states that every tournament either contains $k$ arc-disjoint triangles or has a feedback arc set of size at most $6k$. Our belief is that this combinatorial result is of independent interest and could be useful in other problems related to cycles in tournaments.

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arXiv2018

Reducing CMSO Model Checking to Highly Connected Graphs

Given a Counting Monadic Second Order (CMSO) sentence $ψ$, the CMSO$[ψ]$ problem is defined as follows. The input to CMSO$[ψ]$ is a graph $G$, and the objective is to determine whether $G\models ψ$. Our main theorem states that for every CMSO sentence $ψ$, if CMSO$[ψ]$ is solvable in polynomial time on "globally highly connected graphs", then CMSO$[ψ]$ is solvable in polynomial time (on general graphs). We demonstrate the utility of our theorem in the design of parameterized algorithms. Specifically we show that technical problem-specific ingredients of a powerful method for designing parameterized algorithms, recursive understanding, can be replaced by a black-box invocation of our main theorem. We also show that our theorem can be easily deployed to show fixed parameterized tractability of a wide range of problems, where the input is a graph $G$ and the task is to find a connected induced subgraph of $G$ such that "few" vertices in this subgraph have neighbors outside the subgraph, and additionally the subgraph has a CMSO-definable property.

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