arXiv subjects

Danny Hermelin

Publications and source records attributed to Danny Hermelin.

49 records · Page 3Linked to original sources

arXiv2015

Parameterized Complexity of Critical Node Cuts

We consider the following natural graph cut problem called Critical Node Cut (CNC): Given a graph $G$ on $n$ vertices, and two positive integers $k$ and $x$, determine whether $G$ has a set of $k$ vertices whose removal leaves $G$ with at most $x$ connected pairs of vertices. We analyze this problem in the framework of parameterized complexity. That is, we are interested in whether or not this problem is solvable in $f(κ) \cdot n^{O(1)}$ time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters $κ$. We consider four such parameters: - The size $k$ of the required cut. - The upper bound $x$ on the number of remaining connected pairs. - The lower bound $y$ on the number of connected pairs to be removed. - The treewidth $w$ of $G$. We determine whether or not CNC is fixed-parameter tractable for each of these parameters. We determine this also for all possible aggregations of these four parameters, apart from $w+k$. Moreover, we also determine whether or not CNC admits a polynomial kernel for all these parameterizations. That is, whether or not there is an algorithm that reduces each instance of CNC in polynomial time to an equivalent instance of size $κ^{O(1)}$, where $κ$ is the given parameter.

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arXiv2015

Parameterized Complexity Dichotomy for Steiner Multicut

The Steiner Multicut problem asks, given an undirected graph G, terminals sets T1,...,Tt $\subseteq$ V(G) of size at most p, and an integer k, whether there is a set S of at most k edges or nodes s.t. of each set Ti at least one pair of terminals is in different connected components of G \ S. This problem generalizes several graph cut problems, in particular the Multicut problem (the case p = 2), which is fixed-parameter tractable for the parameter k [Marx and Razgon, Bousquet et al., STOC 2011]. We provide a dichotomy of the parameterized complexity of Steiner Multicut. That is, for any combination of k, t, p, and the treewidth tw(G) as constant, parameter, or unbounded, and for all versions of the problem (edge deletion and node deletion with and without deletable terminals), we prove either that the problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or even (para-)NP-complete). We highlight that: - The edge deletion version of Steiner Multicut is fixed-parameter tractable for the parameter k+t on general graphs (but has no polynomial kernel, even on trees). We present two proofs: one using the randomized contractions technique of Chitnis et al, and one relying on new structural lemmas that decompose the Steiner cut into important separators and minimal s-t cuts. - In contrast, both node deletion versions of Steiner Multicut are W[1]-hard for the parameter k+t on general graphs. - All versions of Steiner Multicut are W[1]-hard for the parameter k, even when p=3 and the graph is a tree plus one node. Hence, the results of Marx and Razgon, and Bousquet et al. do not generalize to Steiner Multicut. Since we allow k, t, p, and tw(G) to be any constants, our characterization includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to constant or unbounded).

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arXiv2014

On the Average-case Complexity of Parameterized Clique

The k-Clique problem is a fundamental combinatorial problem that plays a prominent role in classical as well as in parameterized complexity theory. It is among the most well-known NP-complete and W[1]-complete problems. Moreover, its average-case complexity analysis has created a long thread of research already since the 1970s. Here, we continue this line of research by studying the dependence of the average-case complexity of the k-Clique problem on the parameter k. To this end, we define two natural parameterized analogs of efficient average-case algorithms. We then show that k-Clique admits both analogues for Erdős-Rényi random graphs of arbitrary density. We also show that k-Clique is unlikely to admit neither of these analogs for some specific computable input distribution.

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arXiv2014

Binary Jumbled Pattern Matching via All-Pairs Shortest Paths

In binary jumbled pattern matching we wish to preprocess a binary string $S$ in order to answer queries $(i,j)$ which ask for a substring of $S$ that is of size $i$ and has exactly $j$ 1-bits. The problem naturally generalizes to node-labeled trees and graphs by replacing "substring" with "connected subgraph". In this paper, we give an ${n^2}/{2^{Ω(\log n/\log \log n)^{1/2}}}$ time solution for both strings and trees. This odd-looking time complexity improves the state of the art $O(n^2/\log^2 n)$ solutions by more than any poly-logarithmic factor. It originates from the recent seminal algorithm of Williams for min-plus matrix multiplication. We obtain the result by giving a black box reduction from trees to strings. This is then combined with a reduction from strings to min-plus matrix multiplications.

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arXiv2014

Binary Jumbled Pattern Matching on Trees and Tree-Like Structures

Binary jumbled pattern matching asks to preprocess a binary string $S$ in order to answer queries $(i,j)$ which ask for a substring of $S$ that is of length $i$ and has exactly $j$ 1-bits. This problem naturally generalizes to vertex-labeled trees and graphs by replacing "substring" with "connected subgraph". In this paper, we give an $O(n^2 / \log^2 n)$-time solution for trees, matching the currently best bound for (the simpler problem of) strings. We also give an $\Oh{g^{2 / 3} n^{4 / 3}/(\log n)^{4/3}}$-time solution for strings that are compressed by a grammar of size $g$. This solution improves the known bounds when the string is compressible under many popular compression schemes. Finally, we prove that the problem is fixed-parameter tractable with respect to the treewidth $w$ of the graph, thus improving the previous best $n^{O(w)}$ algorithm [ICALP'07].

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arXiv2014

Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

The Induced Graph Matching problem asks to find k disjoint induced subgraphs isomorphic to a given graph H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced Graph Matching is fixed-parameter tractable in k on claw-free graphs when H is a fixed connected graph, and even admits a polynomial kernel when H is a complete graph. Both results rely on a new, strong, and generic algorithmic structure theorem for claw-free graphs. Complementing the above positive results, we prove W[1]-hardness of Induced Graph Matching on graphs excluding K_1,4 as an induced subgraph, for any fixed complete graph H. In particular, we show that Independent Set is W[1]-hard on K_1,4-free graphs. Finally, we consider the complexity of Induced Graph Matching on a large subclass of claw-free graphs, namely on proper circular-arc graphs. We show that the problem is either polynomial-time solvable or NP-complete, depending on the connectivity of H and the structure of G.

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arXiv2013

Tight Kernel Bounds for Problems on Graphs with Small Degeneracy

In this paper we consider kernelization for problems on d-degenerate graphs, i.e. graphs such that any subgraph contains a vertex of degree at most $d$. This graph class generalizes many classes of graphs for which effective kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and H-topological-minor free graphs. We show that for several natural problems on d-degenerate graphs the best known kernelization upper bounds are essentially tight.

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arXiv2011

Hierarchies of Inefficient Kernelizability

The framework of Bodlaender et al. (ICALP 2008) and Fortnow and Santhanam (STOC 2008) allows us to exclude the existence of polynomial kernels for a range of problems under reasonable complexity-theoretical assumptions. However, there are also some issues that are not addressed by this framework, including the existence of Turing kernels such as the "kernelization" of Leaf Out Branching(k) into a disjunction over n instances of size poly(k). Observing that Turing kernels are preserved by polynomial parametric transformations, we define a kernelization hardness hierarchy, akin to the M- and W-hierarchy of ordinary parameterized complexity, by the PPT-closure of problems that seem likely to be fundamentally hard for efficient Turing kernelization. We find that several previously considered problems are complete for our fundamental hardness class, including Min Ones d-SAT(k), Binary NDTM Halting(k), Connected Vertex Cover(k), and Clique(k log n), the clique problem parameterized by k log n.

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arXiv2010

Parameterized Two-Player Nash Equilibrium

We study the computation of Nash equilibria in a two-player normal form game from the perspective of parameterized complexity. Recent results proved hardness for a number of variants, when parameterized by the support size. We complement those results, by identifying three cases in which the problem becomes fixed-parameter tractable. These cases occur in the previously studied settings of sparse games and unbalanced games as well as in the newly considered case of locally bounded treewidth games that generalizes both these two cases.

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arXiv2010

Unified Compression-Based Acceleration of Edit-Distance Computation

The edit distance problem is a classical fundamental problem in computer science in general, and in combinatorial pattern matching in particular. The standard dynamic programming solution for this problem computes the edit-distance between a pair of strings of total length O(N) in O(N^2) time. To this date, this quadratic upper-bound has never been substantially improved for general strings. However, there are known techniques for breaking this bound in case the strings are known to compress well under a particular compression scheme. The basic idea is to first compress the strings, and then to compute the edit distance between the compressed strings. As it turns out, practically all known o(N^2) edit-distance algorithms work, in some sense, under the same paradigm described above. It is therefore natural to ask whether there is a single edit-distance algorithm that works for strings which are compressed under any compression scheme. A rephrasing of this question is to ask whether a single algorithm can exploit the compressibility properties of strings under any compression method, even if each string is compressed using a different compression. In this paper we set out to answer this question by using straight line programs. These provide a generic platform for representing many popular compression schemes including the LZ-family, Run-Length Encoding, Byte-Pair Encoding, and dictionary methods. For two strings of total length N having straight-line program representations of total size n, we present an algorithm running in O(nN log(N/n)) time for computing the edit-distance of these two strings under any rational scoring function, and an O(n^{2/3}N^{4/3}) time algorithm for arbitrary scoring functions. Our new result, while providing a signi cant speed up for highly compressible strings, does not surpass the quadratic time bound even in the worst case scenario.

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arXiv2010

Mod/Resc Parsimony Inference

We address in this paper a new computational biology problem that aims at understanding a mechanism that could potentially be used to genetically manipulate natural insect populations infected by inherited, intra-cellular parasitic bacteria. In this problem, that we denote by \textsc{Mod/Resc Parsimony Inference}, we are given a boolean matrix and the goal is to find two other boolean matrices with a minimum number of columns such that an appropriately defined operation on these matrices gives back the input. We show that this is formally equivalent to the \textsc{Bipartite Biclique Edge Cover} problem and derive some complexity results for our problem using this equivalence. We provide a new, fixed-parameter tractability approach for solving both that slightly improves upon a previously published algorithm for the \textsc{Bipartite Biclique Edge Cover}. Finally, we present experimental results where we applied some of our techniques to a real-life data set.

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arXiv2010

Minimum Vertex Cover in Rectangle Graphs

We consider the Vertex Cover problem in intersection graphs of axis-parallel rectangles on the plane. We present two algorithms: The first is an EPTAS for non-crossing rectangle families, rectangle families $\calR$ where $R_1 \setminus R_2$ is connected for every pair of rectangles $R_1,R_2 \in \calR$. This algorithm extends to intersection graphs of pseudo-disks. The second algorithm achieves a factor of $(1.5 + \varepsilon)$ in general rectangle families, for any fixed $\varepsilon > 0$, and works also for the weighted variant of the problem. Both algorithms exploit the plane properties of axis-parallel rectangles in a non-trivial way.

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arXiv2009

A Unified Algorithm for Accelerating Edit-Distance Computation via Text-Compression

We present a unified framework for accelerating edit-distance computation between two compressible strings using straight-line programs. For two strings of total length $N$ having straight-line program representations of total size $n$, we provide an algorithm running in $O(n^{1.4}N^{1.2})$ time for computing the edit-distance of these two strings under any rational scoring function, and an $O(n^{1.34}N^{1.34})$ time algorithm for arbitrary scoring functions. This improves on a recent algorithm of Tiskin that runs in $O(nN^{1.5})$ time, and works only for rational scoring functions. Also, in the last part of the paper, we show how the classical four-russians technique can be incorporated into our SLP edit-distance scheme, giving us a simple $Ω(\lg N)$ speed-up in the case of arbitrary scoring functions, for any pair of strings.

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