arXiv subjects

Daniel Lokshtanov

Publications and source records attributed to Daniel Lokshtanov.

At least 145 records · Page 8Linked to original sources

arXiv2015

Uniform Kernelization Complexity of Hitting Forbidden Minors

The F-Minor-Free Deletion problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. This paper analyzes to what extent provably effective and efficient preprocessing is possible for F-Minor-Free Deletion. Fomin et al. (FOCS 2012) showed that the special case Planar F-Deletion (when F contains at least one planar graph) has a kernel of size f(F) * k^{g(F)} for some functions f and g. The degree g of the polynomial grows very quickly; it is not even known to be computable. Fomin et al. left open whether Planar F-Deletion has kernels whose size is uniformly polynomial, i.e., of the form f(F) * k^c for some universal constant c that does not depend on F. Our results in this paper are twofold. (1) We prove that some Planar F-Deletion problems do not have uniformly polynomial kernels (unless NP is in coNP/poly). In particular, we prove that Treewidth-Eta Deletion does not have a kernel with O(k^{eta/4} - eps) vertices for any eps > 0, unless NP is in coNP/poly. In fact, we even prove the kernelization lower bound for the larger parameter vertex cover number. This resolves an open problem of Cygan et al. (IPEC 2011). It is a natural question whether further restrictions on F lead to uniformly polynomial kernels. However, we prove that even when F contains a path, the degree of the polynomial must, in general, depend on the set F. (2) A canonical F-Minor-Free Deletion problem when F contains a path is Treedepth-eta Deletion: can k vertices be removed to obtain a graph of treedepth at most eta? We prove that Treedepth-eta Deletion admits uniformly polynomial kernels with O(k^6) vertices for every fixed eta. In order to develop the kernelization we prove several new results about the structure of optimal treedepth-decompositions.

cs.DS↗

arXiv2014

Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth

We give a fixed-parameter tractable algorithm that, given a parameter $k$ and two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in $2^{O(k^5\log k)}\cdot n^5$ time that, for a given graph $G$ on $n$ vertices, either concludes that the treewidth of $G$ is at least $k$, or: * finds in an isomorphic-invariant way a graph $\mathfrak{c}(G)$ that is isomorphic to $G$; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes $G$ together with a tree decomposition of $G$ of width $O(k^4)$. Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for $G_1$ and $G_2$ are equal.

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arXiv2014

Parameterized Complexity of Bandwidth on Trees

The bandwidth of a $n$-vertex graph $G$ is the smallest integer $b$ such that there exists a bijective function $f : V(G) \rightarrow \{1,...,n\}$, called a layout of $G$, such that for every edge $uv \in E(G)$, $|f(u) - f(v)| \leq b$. In the {\sc Bandwidth} problem we are given as input a graph $G$ and integer $b$, and asked whether the bandwidth of $G$ is at most $b$. We present two results concerning the parameterized complexity of the {\sc Bandwidth} problem on trees. First we show that an algorithm for {\sc Bandwidth} with running time $f(b)n^{o(b)}$ would violate the Exponential Time Hypothesis, even if the input graphs are restricted to be trees of pathwidth at most two. Our lower bound shows that the classical $2^{O(b)}n^{b+1}$ time algorithm by Saxe [SIAM Journal on Algebraic and Discrete Methods, 1980] is essentially optimal. Our second result is a polynomial time algorithm that given a tree $T$ and integer $b$, either correctly concludes that the bandwidth of $T$ is more than $b$ or finds a layout of $T$ of bandwidth at most $b^{O(b)}$. This is the first parameterized approximation algorithm for the bandwidth of trees.

cs.DS↗

arXiv2014

Deterministic Truncation of Linear Matroids

Let $M=(E,{\cal I})$ be a matroid. A {\em $k$-truncation} of $M$ is a matroid {$M'=(E,{\cal I}')$} such that for any $A\subseteq E$, $A\in {\cal I}'$ if and only if $|A|\leq k$ and $A\in {\cal I}$. Given a linear representation of $M$ we consider the problem of finding a linear representation of the $k$-truncation of this matroid. This problem can be abstracted out to the following problem on matrices. Let $M$ be a $n\times m$ matrix over a field $\mathbb{F}$. A {\em rank $k$-truncation} of the matrix $M$ is a $k\times m$ matrix $M_k$ (over $\mathbb{F}$ or a related field) such that for every subset $I\subseteq \{1,\ldots,m\}$ of size at most $k$, the set of columns corresponding to $I$ in $M$ has rank $|I|$ if and only of the corresponding set of columns in $M_k$ has rank $|I|$. Finding rank $k$-truncation of matrices is a common way to obtain a linear representation of $k$-truncation of linear matroids, which has many algorithmic applications. A common way to compute a rank $k$-truncation of a $n \times m$ matrix is to multiply the matrix with a random $k\times n$ matrix (with the entries from a field of an appropriate size), yielding a simple randomized algorithm. So a natural question is whether it possible to obtain a rank $k$-truncations of a matrix, {\em deterministically}. In this paper we settle this question for matrices over any finite field or the field of rationals ($\mathbb Q$). We show that given a matrix $M$ over a field $\mathbb{F}$ we can compute a $k$-truncation $M_k$ over the ring $\mathbb{F}[X]$ in deterministic polynomial time.

cs.DS↗

arXiv2014

On Problems as Hard as CNFSAT

The field of exact exponential time algorithms for NP-hard problems has thrived over the last decade. While exhaustive search remains asymptotically the fastest known algorithm for some basic problems, difficult and non-trivial exponential time algorithms have been found for a myriad of problems, including Graph Coloring, Hamiltonian Path, Dominating Set and 3-CNF-Sat. In some instances, improving these algorithms further seems to be out of reach. The CNF-Sat problem is the canonical example of a problem for which the trivial exhaustive search algorithm runs in time O(2^n), where n is the number of variables in the input formula. While there exist non-trivial algorithms for CNF-Sat that run in time o(2^n), no algorithm was able to improve the growth rate 2 to a smaller constant, and hence it is natural to conjecture that 2 is the optimal growth rate. The strong exponential time hypothesis (SETH) by Impagliazzo and Paturi [JCSS 2001] goes a little bit further and asserts that, for every epsilon<1, there is a (large) integer k such that that k-CNF-Sat cannot be computed in time 2^{epsilon n}. In this paper, we show that, for every epsilon < 1, the problems Hitting Set, Set Splitting, and NAE-Sat cannot be computed in time O(2^{epsilon n}) unless SETH fails. Here n is the number of elements or variables in the input. For these problems, we actually get an equivalence to SETH in a certain sense. We conjecture that SETH implies a similar statement for Set Cover, and prove that, under this assumption, the fastest known algorithms for Steinter Tree, Connected Vertex Cover, Set Partitioning, and the pseudo-polynomial time algorithm for Subset Sum cannot be significantly improved. Finally, we justify our assumption about the hardness of Set Cover by showing that the parity of the number of set covers cannot be computed in time O(2^{epsilon n}) for any epsilon<1 unless SETH fails.

cs.DS↗

arXiv2014

Minimum Bisection is fixed parameter tractable

In the classic Minimum Bisection problem we are given as input a graph $G$ and an integer $k$. The task is to determine whether there is a partition of $V(G)$ into two parts $A$ and $B$ such that $||A|-|B|| \leq 1$ and there are at most $k$ edges with one endpoint in $A$ and the other in $B$. In this paper we give an algorithm for Minimum Bisection with running time $O(2^{O(k^{3})}n^3 \log^3 n)$. This is the first fixed parameter tractable algorithm for Minimum Bisection. At the core of our algorithm lies a new decomposition theorem that states that every graph $G$ can be decomposed by small separators into parts where each part is "highly connected" in the following sense: any cut of bounded size can separate only a limited number of vertices from each part of the decomposition. Our techniques generalize to the weighted setting, where we seek for a bisection of minimum weight among solutions that contain at most $k$ edges.

cs.DS↗

arXiv2014

On the Ordered List Subgraph Embedding Problems

In the (parameterized) Ordered List Subgraph Embedding problem (p-OLSE) we are given two graphs $G$ and $H$, each with a linear order defined on its vertices, a function $L$ that associates with every vertex in $G$ a list of vertices in $H$, and a parameter $k$. The question is to decide if we can embed (one-to-one) a subgraph $S$ of $G$ of $k$ vertices into $H$ such that: (1) every vertex of $S$ is mapped to a vertex from its associated list, (2) the linear orders inherited by $S$ and its image under the embedding are respected, and (3) if there is an edge between two vertices in $S$ then there is an edge between their images. If we require the subgraph $S$ to be embedded as an induced subgraph, we obtain the Ordered List Induced Subgraph Embedding problem (p-OLISE). The p-OLSE and p-OLISE problems model various problems in Bioinformatics related to structural comparison/alignment of proteins. We investigate the complexity of p-OLSE and p-OLISE with respect to the following structural parameters: the width $Δ_L$ of the function $L$ (size of the largest list), and the maximum degree $Δ_H$ of $H$ and $Δ_G$ of $G$. In terms of the structural parameters under consideration, we draw a complete complexity landscape of p-OLSE and p-OLISE (and their optimization versions) with respect to the computational frameworks of classical complexity, parameterized complexity, and approximation.

cs.DS↗

arXiv2014

Representative Sets of Product Families

A subfamily ${\cal F}'$ of a set family ${\cal F}$ is said to $q$-{\em represent} ${\cal F}$ if for every $A \in {\cal F}$ and $B$ of size $q$ such that $A \cap B = \emptyset$ there exists a set $A' \in {\cal F}'$ such that $A' \cap B = \emptyset$. In this paper, we consider the efficient computation of $q$-representative sets for {\em product} families ${\cal F}$. A family ${\cal F}$ is a product family if there exist families ${\cal A}$ and ${\cal B}$ such that ${\cal F} = \{A \cup B~:~A \in {\cal A}, B \in {\cal B}, A \cap B = \emptyset\}$. Our main technical contribution is an algorithm which given ${\cal A}$, ${\cal B}$ and $q$ computes a $q$-representative family ${\cal F}'$ of ${\cal F}$. The running time of our algorithm is sublinear in $|{\cal F}|$ for many choices of ${\cal A}$, ${\cal B}$ and $q$ which occur naturally in several dynamic programming algorithms. We also give an algorithm for the computation of $q$-representative sets for product families ${\cal F}$ in the more general setting where $q$-representation also involves independence in a matroid in addition to disjointness. This algorithm considerably outperforms the naive approach where one first computes ${\cal F}$ from ${\cal A}$ and ${\cal B}$, and then computes the $q$-representative family ${\cal F}'$ from ${\cal F}$. We give two applications of our new algorithms for computing $q$-representative sets for product families. The first is a $3.8408^{k}n^{O(1)}$ deterministic algorithm for the Multilinear Monomial Detection ($k$-MlD) problem. The second is a significant improvement of deterministic dynamic programming algorithms for "connectivity problems" on graphs of bounded treewidth.

cs.DS↗

arXiv2013

Tree Deletion Set has a Polynomial Kernel (but no OPT^O(1) approximation)

In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G-S is a tree. The problem is NP-complete and even NP-hard to approximate within any factor of OPT^c for any constant c. In this paper we give a O(k^4) size kernel for the Tree Deletion Set problem. To the best of our knowledge our result is the first counterexample to the "conventional wisdom" that kernelization algorithms automatically provide approximation algorithms with approximation ratio close to the size of the kernel. An appealing feature of our kernelization algorithm is a new algebraic reduction rule that we use to handle the instances on which Tree Deletion Set is hard to approximate.

cs.DS↗

arXiv2013

(Meta) Kernelization

In a parameterized problem, every instance I comes with a positive integer k. The problem is said to admit a polynomial kernel if, in polynomial time, one can reduce the size of the instance I to a polynomial in k, while preserving the answer. In this work we give two meta-theorems on kernelzation. The first theorem says that all problems expressible in Counting Monadic Second Order Logic and satisfying a coverability property admit a polynomial kernel on graphs of bounded genus. Our second result is that all problems that have finite integer index and satisfy a weaker coverability property admit a linear kernel on graphs of bounded genus. These theorems unify and extend all previously known kernelization results for planar graph problems.

cs.DM↗

arXiv2013

A O(c^k n) 5-Approximation Algorithm for Treewidth

We give an algorithm that for an input n-vertex graph G and integer k>0, in time 2^[O(k)]n either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k+4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time single-exponential in k and linear in n. Treewidth based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is single-exponential in the treewidth and linear in the input size.

cs.DS↗

arXiv2012

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

We study the parameterized complexity of the directed variant of the classical {\sc Steiner Tree} problem on various classes of directed sparse graphs. While the parameterized complexity of {\sc Steiner Tree} parameterized by the number of terminals is well understood, not much is known about the parameterization by the number of non-terminals in the solution tree. All that is known for this parameterization is that both the directed and the undirected versions are W[2]-hard on general graphs, and hence unlikely to be fixed parameter tractable FPT. The undirected {\sc Steiner Tree} problem becomes FPT when restricted to sparse classes of graphs such as planar graphs, but the techniques used to show this result break down on directed planar graphs. In this article we precisely chart the tractability border for {\sc Directed Steiner Tree} (DST) on sparse graphs parameterized by the number of non-terminals in the solution tree. Specifically, we show that the problem is fixed parameter tractable on graphs excluding a topological minor, but becomes W[2]-hard on graphs of degeneracy 2. On the other hand we show that if the subgraph induced by the terminals is required to be acyclic then the problem becomes FPT on graphs of bounded degeneracy. We further show that our algorithm achieves the best possible running time dependence on the solution size and degeneracy of the input graph, under standard complexity theoretic assumptions. Using the ideas developed for DST, we also obtain improved algorithms for {\sc Dominating Set} on sparse undirected graphs. These algorithms are asymptotically optimal.

cs.DS↗

arXiv2012

SeeSite: Efficiently Finding Co-occurring Splice Sites and Exon Splicing Enhancers

The problem of identifying splice sites consists of two sub-problems: finding their boundaries, and characterizing their sequence markers. Other splicing elements---including, enhancers and silencers---that occur in the intronic and exonic regions play an important role in splicing activity. Existing methods for detecting splicing elements are limited to finding either splice sites or enhancers and silencers, even though these elements are well-known to co-occur. We introduce SeeSite, an efficient and accurate tool for detecting splice sites and their complementary exon splicing enhancers (ESEs). SeeSite has three stages: graph construction, finding dense subgraphs, and recovering splice sites and ESEs along with their consensus. The third step involves solving Consensus Sequence with Outliers, an NP-complete string clustering problem. We prove that our algorithm for this problem outputs near-optimal solutions in polynomial time. Using SeeSite we demonstrate that ESEs are preferentially associated with weaker splice sites, and splice sites of a certain canonical form co-occur with specific ESEs.

q-bio.QM↗

arXiv2012

Faster Parameterized Algorithms using Linear Programming

We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an $O^*((2.618)^k)$ algorithm for the problem. Here $k$ is the excess of the vertex cover size over the LP optimum, and we write $O^*(f(k))$ for a time complexity of the form $O(f(k)n^{O(1)})$, where $f (k)$ grows exponentially with $k$. We proceed to show that a more sophisticated branching algorithm achieves a runtime of $O^*(2.3146^k)$. Following this, using known and new reductions, we give $O^*(2.3146^k)$ algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion and Almost 2-SAT, and an $O^*(1.5214^k)$ algorithm for Konig Vertex Deletion, Vertex Cover Param by OCT and Vertex Cover Param by KVD. These algorithms significantly improve the best known bounds for these problems. The most notable improvement is the new bound for Odd Cycle Transversal - this is the first algorithm which beats the dependence on $k$ of the seminal $O^*(3^k)$ algorithm of Reed, Smith and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most $2k - c \log k$ vertices. Our kernel is simpler than previously known kernels achieving the same size bound.

cs.DS↗

arXiv2011

Outlier Detection for DNA Fragment Assembly

Given $n$ length-$\ell$ strings $S =\{s_1, ..., s_n\}$ over a constant size alphabet $Σ$ together with parameters $d$ and $k$, the objective in the {\em Consensus String with Outliers} problem is to find a subset $S^*$ of $S$ of size $n-k$ and a string $s$ such that $\sum_{s_i \in S^*} d(s_i, s) \leq d$. Here $d(x, y)$ denotes the Hamming distance between the two strings $x$ and $y$. We prove 1. a variant of {\em Consensus String with Outliers} where the number of outliers $k$ is fixed and the objective is to minimize the total distance $\sum_{s_i \in S^*} d(s_i, s)$ admits a simple PTAS. (ii) Under the natural assumption that the number of outliers $k$ is small, the PTAS for the distance minimization version of {\em Consensus String with Outliers} performs well. In particular, as long as $k\leq cn$ for a fixed constant $c < 1$, the algorithm provides a $(1+ε)$-approximate solution in time $f(1/ε)(n\ell)^{O(1)}$ and thus, is an EPTAS. 2. In order to improve the PTAS for {\em Consensus String with Outliers} to an EPTAS, the assumption that $k$ is small is necessary. Specifically, when $k$ is allowed to be arbitrary the {\em Consensus String with Outliers} problem does not admit an EPTAS unless FPT=W[1]. This hardness result holds even for binary alphabets. 3. The decision version of {\em Consensus String with Outliers} is fixed parameter tractable when parameterized by $\frac{d}{n-k}$. and thus, also when parameterized by just $d$. To the best of our knowledge, {\em Consensus String with Outliers} is the first problem that admits a PTAS, and is fixed parameter tractable when parameterized by the value of the objective function but does not admit an EPTAS under plausible complexity assumptions.

cs.DS↗

arXiv2011

Bidimensionality and Geometric Graphs

In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d.

cs.DS↗

arXiv2011

Bidimensionality and EPTAS

Bidimensionality theory is a powerful framework for the development of metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al related the theory to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. Two of the most widely used approaches to obtain PTASs on planar graphs are the Lipton-Tarjan separator based approach, and Baker's approach. Demaine and Hajiaghayi strengthened both approaches using bidimensionality and obtained EPTASs for a multitude of problems. We unify the two strenghtened approaches to combine the best of both worlds. At the heart of our framework is a decomposition lemma which states that for "most" bidimensional problems, there is a polynomial time algorithm which given an H-minor-free graph G as input and an e > 0 outputs a vertex set X of size e * OPT such that the treewidth of G n X is f(e). Here, OPT is the objective function value of the problem in question and f is a function depending only on e. This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including cycle packing, vertex-h-packing, maximum leaf spanning tree, and partial r-dominating set no EPTASs on planar graphs were previously known.

cs.DS↗

arXiv2011

Contracting Graphs to Paths and Trees

Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. Interestingly, the study of edge contraction problems of this type from a parameterized perspective has so far been left largely unexplored. We consider two basic edge contraction problems, which we call Path-Contractibility and Tree-Contractibility. Both problems take an undirected graph $G$ and an integer $k$ as input, and the task is to determine whether we can obtain a path or an acyclic graph, respectively, by contracting at most $k$ edges of $G$. Our main contribution is an algorithm with running time $4^{k+O(\log^2 k)} + n^{O(1)}$ for Path-Contractibility and an algorithm with running time $4.88^k n^{O(1)}$ for Tree-Contractibility, based on a novel application of the color coding technique of Alon, Yuster and Zwick. Furthermore, we show that Path-Contractibility has a kernel with at most $5k+3$ vertices, while Tree-Contractibility does not have a polynomial kernel unless coNP $\subseteq$ NP/poly. We find the latter result surprising, because of the strong connection between Tree-Contractibility and Feedback Vertex Set, which is known to have a vertex kernel with size $O(k^2)$.

cs.DS↗