arXiv subjects

Daniel Lokshtanov

Publications and source records attributed to Daniel Lokshtanov.

At least 109 records · Page 6Linked to original sources

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.

cs.DS↗

arXiv2017

Algorithms for low-distortion embeddings into arbitrary 1-dimensional spaces

We study the problem of finding a minimum-distortion embedding of the shortest path metric of an unweighted graph into a "simpler" metric $X$. Computing such an embedding (exactly or approximately) is a non-trivial task even when $X$ is the metric induced by a path, or, equivalently, into the real line. In this paper we give approximation and fixed-parameter tractable (FPT) algorithms for minimum-distortion embeddings into the metric of a subdivision of some fixed graph $H$, or, equivalently, into any fixed 1-dimensional simplicial complex. More precisely, we study the following problem: For given graphs $G$, $H$ and integer $c$, is it possible to embed $G$ with distortion $c$ into a graph homeomorphic to $H$? Then embedding into the line is the special case $H=K_2$, and embedding into the cycle is the case $H=K_3$, where $K_k$ denotes the complete graph on $k$ vertices. For this problem we give -an approximation algorithm, which in time $f(H)\cdot \text{poly} (n)$, for some function $f$, either correctly decides that there is no embedding of $G$ with distortion $c$ into any graph homeomorphic to $H$, or finds an embedding with distortion $\text{poly}(c)$; -an exact algorithm, which in time $f'(H, c)\cdot \text{poly} (n)$, for some function $f'$, either correctly decides that there is no embedding of $G$ with distortion $c$ into any graph homeomorphic to $H$, or finds an embedding with distortion $c$. Prior to our work, $\text{poly}(\mathsf{OPT})$-approximation or FPT algorithms were known only for embedding into paths and trees of bounded degrees.

cs.CG↗

arXiv2017

Clustering with Local Restrictions

We study a family of graph clustering problems where each cluster has to satisfy a certain local requirement. Formally, let $μ$ be a function on the subsets of vertices of a graph $G$. In the $(μ,p,q)$-PARTITION problem, the task is to find a partition of the vertices into clusters where each cluster $C$ satisfies the requirements that (1) at most $q$ edges leave $C$ and (2) $μ(C)\le p$. Our first result shows that if $μ$ is an {\em arbitrary} polynomial-time computable monotone function, then $(μ,p,q)$-PARTITION can be solved in time $n^{O(q)}$, i.e., it is polynomial-time solvable {\em for every fixed $q$}. We study in detail three concrete functions $μ$ (the number of vertices in the cluster, number of nonedges in the cluster, maximum number of non-neighbors a vertex has in the cluster), which correspond to natural clustering problems. For these functions, we show that $(μ,p,q)$-PARTITION can be solved in time $2^{O(p)}\cdot n^{O(1)}$ and in time $2^{O(q)}\cdot n^{O(1)}$ on $n$-vertex graphs, i.e., the problem is fixed-parameter tractable parameterized by $p$ or by $q$.

cs.DS↗

arXiv2017

Kernels for (connected) Dominating Set on graphs with Excluded Topological subgraphs

We give the first linear kernels for the (Connected) Dominating Set problems on H-topological minor free graphs. We prove the existence of polynomial time algorithms that, for a given H-topological-minor-free graph G and a positive integer k, output an H-topological-minor-free graph G' on O(k) vertices such that G has a (connected) dominating set of size k iff G' has one. Our results extend the known classes of graphs on which the Dominating Set and Connected Dominating Set problems admit linear kernels. Prior to our work, it was known that these problems admit linear kernels on graphs excluding a fixed apex graph H as a minor. Moreover, for Dominating Set, a kernel of size kc(H), where c(H) is a constant depending on the size of H, follows from a more general result on the kernelization of Dominating Set on graphs of bounded degeneracy. Alon and Gutner explicitly asked whether one can obtain a linear kernel for Dominating Set on H-minor-free graphs. We answer this question in the affirmative and in fact prove a more general result. For Connected Dominating Set no polynomial kernel even on H-minor-free graphs was known prior to our work. On the negative side, it is known that Connected Dominating Set on 2-degenerated graphs does not admit a polynomial kernel unless coNP $\subseteq$ NP/poly. Our kernelization algorithm is based on a non-trivial combination of the following ingredients The structural theorem of Grohe and Marx [STOC 2012] for graphs excluding a fixed graph H as a topological minor; A novel notion of protrusions, different than the one defined in [FOCS 2009]; Our results are based on a generic reduction rule that produces an equivalent instance (in case the input graph is H-minor-free) of the problem, with treewidth $O(\sqrt{k})$. The application of this rule in a divide-and-conquer fashion, together with the new notion of protrusions, gives us the linear kernels.

cs.DS↗

arXiv2017

Balanced Judicious Partition is Fixed-Parameter Tractable

The family of judicious partitioning problems, introduced by Bollobás and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollobás and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT).

cs.DS↗

arXiv2017

Covering vectors by spaces: Regular matroids

Seymour's decomposition theorem for regular matroids is a fundamental result with a number of combinatorial and algorithmic applications. In this work we demonstrate how this theorem can be used in the design of parameterized algorithms on regular matroids. We consider the problem of covering a set of vectors of a given finite dimensional linear space (vector space) by a subspace generated by a set of vectors of minimum size. Specifically, in the Space Cover problem, we are given a matrix M and a subset of its columns T; the task is to find a minimum set F of columns of M disjoint with T such that that the linear span of F contains all vectors of T. For graphic matroids this problem is essentially Stainer Forest and for cographic matroids this is a generalization of Multiway Cut. Our main result is the algorithm with running time 2^{O(k)}||M|| ^{O(1)} solving Space Cover in the case when M is a totally unimodular matrix over rationals, where k is the size of F. In other words, we show that on regular matroids the problem is fixed-parameter tractable parameterized by the rank of the covering subspace.

cs.DS↗

arXiv2017

Polylogarithmic Approximation Algorithms for Weighted-$\mathcal{F}$-Deletion Problems

For a family of graphs $\cal F$, the canonical Weighted $\cal F$ Vertex Deletion problem is defined as follows: given an $n$-vertex undirected graph $G$ and a weight function $w: V(G)\rightarrow\mathbb{R}$, find a minimum weight subset $S\subseteq V(G)$ such that $G-S$ belongs to $\cal F$. We devise a recursive scheme to obtain $O(\log^{O(1)}n)$-approximation algorithms for such problems, building upon the classic technique of finding balanced separators in a graph. Roughly speaking, our scheme applies to problems where an optimum solution $S$, together with a well-structured set $X$, form a balanced separator of $G$. We obtain the first $O(\log^{O(1)}n)$-approximation algorithms for the following problems. * We give an $O(\log^2n)$-factor approximation algorithm for Weighted Chordal Vertex Deletion (WCVD), the vertex deletion problem to the family of chordal graphs. On the way, we also obtain a constant factor approximation algorithm for Multicut on chordal graphs. * We give an $O(\log^3n)$-factor approximation algorithm for Weighted Distance Hereditary Vertex Deletion (WDHVD). This is the vertex deletion problem to the family of distance hereditary graphs, or equivalently, the family of graphs of rankwidth 1. Our methods also allow us to obtain in a clean fashion a $O(\log^{1.5}n)$-approximation algorithm for the Weighted $\cal F$ Vertex Deletion problem when $\cal F$ is a minor closed family excluding at least one planar graph. For the unweighted version of the problem constant factor approximation algorithms are were known~[Fomin et al., FOCS~2012], while for the weighted version considered here an $O(\log n \log\log n)$-approximation algorithm follows from~[Bansal et al., SODA~2017]. We believe that our recursive scheme can be applied to obtain $O(\log^{O(1)}n)$-approximation algorithms for many other problems as well.

cs.DS↗

arXiv2017

Feedback Vertex Set Inspired Kernel for Chordal Vertex Deletion

Given a graph $G$ and a parameter $k$, the Chordal Vertex Deletion (CVD) problem asks whether there exists a subset $U\subseteq V(G)$ of size at most $k$ that hits all induced cycles of size at least 4. The existence of a polynomial kernel for CVD was a well-known open problem in the field of Parameterized Complexity. Recently, Jansen and Pilipczuk resolved this question affirmatively by designing a polynomial kernel for CVD of size $O(k^{161}\log^{58}k)$, and asked whether one can design a kernel of size $O(k^{10})$. While we do not completely resolve this question, we design a significantly smaller kernel of size $O(k^{12}\log^{10}k)$, inspired by the $O(k^2)$-size kernel for Feedback Vertex Set. Furthermore, we introduce the notion of the independence degree of a vertex, which is our main conceptual contribution.

cs.DS↗

arXiv2017

The complexity of independent set reconfiguration on bipartite graphs

We settle the complexity of the Independent Set Reconfiguration problem on bipartite graphs under all three commonly studied reconfiguration models. We show that under the token jumping or token addition/removal model the problem is NP-complete. For the token sliding model, we show that the problem remains PSPACE-complete.

cs.CC↗

arXiv2017

Packing Cycles Faster Than Erdős-Pósa

The Cycle Packing problem asks whether a given undirected graph $G=(V,E)$ contains $k$ vertex-disjoint cycles. Since the publication of the classic Erdős-Pósa theorem in 1965, this problem received significant scientific attention in the fields of Graph Theory and Algorithm Design. In particular, this problem is one of the first problems studied in the framework of Parameterized Complexity. The non-uniform fixed-parameter tractability of Cycle Packing follows from the Robertson-Seymour theorem, a fact already observed by Fellows and Langston in the 1980s. In 1994, Bodlaender showed that Cycle Packing can be solved in time $2^{\mathcal{O}(k^2)}\cdot |V|$ using exponential space. In case a solution exists, Bodlaender's algorithm also outputs a solution (in the same time). It has later become common knowledge that Cycle Packing admits a $2^{\mathcal{O}(k\log^2k)}\cdot |V|$-time (deterministic) algorithm using exponential space, which is a consequence of the Erdős-Pósa theorem. Nowadays, the design of this algorithm is given as an exercise in textbooks on Parameterized Complexity. Yet, no algorithm that runs in time $2^{o(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, has been found. In light of this, it seems natural to ask whether the $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$ bound is essentially optimal. In this paper, we answer this question negatively by developing a $2^{\mathcal{O}(\frac{k\log^2k}{\log\log k})}\cdot |V|$-time (deterministic) algorithm for Cycle Packing. In case a solution exists, our algorithm also outputs a solution (in the same time). Moreover, apart from beating the bound $2^{\mathcal{O}(k\log^2k)}\cdot |V|^{\mathcal{O}(1)}$, our algorithm runs in time linear in $|V|$, and its space complexity is polynomial in the input size.

cs.DS↗

arXiv2017

Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a $d$-degenerate graph $G$ and an integer $k$, outputs an independent set $Y$, such that for every independent set $X$ in $G$ of size at most $k$, the probability that $X$ is a subset of $Y$ is at least $\left({(d+1)k \choose k} \cdot k(d+1)\right)^{-1}$.The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph $G$, a set $T = \{\{s_1, t_1\}, \{s_2, t_2\}, \ldots, \{s_\ell, t_\ell\}\}$ of terminal pairs and an integer $k$, returns an induced subgraph $G^\star$ of $G$ that maintains all the inclusion minimal multicuts of $G$ of size at most $k$, and does not contain any $(k+2)$-vertex connected set of size $2^{{\cal O}(k)}$. In particular, $G^\star$ excludes a clique of size $2^{{\cal O}(k)}$ as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for Stable $s$-$t$ Separator, Stable Odd Cycle Transversal and Stable Multicut on general graphs, and for Stable Directed Feedback Vertex Set on $d$-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our algorithms can be derandomized at the cost of a small overhead in the running time.

cs.DS↗

arXiv2017

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

We give algorithms with running time $2^{O({\sqrt{k}\log{k}})} \cdot n^{O(1)}$ for the following problems. Given an $n$-vertex unit disk graph $G$ and an integer $k$, decide whether $G$ contains (1) a path on exactly/at least $k$ vertices, (2) a cycle on exactly $k$ vertices, (3) a cycle on at least $k$ vertices, (4) a feedback vertex set of size at most $k$, and (5) a set of $k$ pairwise vertex-disjoint cycles. For the first three problems, no subexponential time parameterized algorithms were previously known. For the remaining two problems, our algorithms significantly outperform the previously best known parameterized algorithms that run in time $2^{O(k^{0.75}\log{k})} \cdot n^{O(1)}$. Our algorithms are based on a new kind of tree decompositions of unit disk graphs where the separators can have size up to $k^{O(1)}$ and there exists a solution that crosses every separator at most $O(\sqrt{k})$ times. The running times of our algorithms are optimal up to the $\log{k}$ factor in the exponent, assuming the Exponential Time Hypothesis.

cs.DS↗

arXiv2017

Parameterized Complexity and Approximability of Directed Odd Cycle Transversal

A directed odd cycle transversal of a directed graph (digraph) $D$ is a vertex set $S$ that intersects every odd directed cycle of $D$. In the Directed Odd Cycle Transversal (DOCT) problem, the input consists of a digraph $D$ and an integer $k$. The objective is to determine whether there exists a directed odd cycle transversal of $D$ of size at most $k$. In this paper, we settle the parameterized complexity of DOCT when parameterized by the solution size $k$ by showing that DOCT does not admit an algorithm with running time $f(k)n^{O(1)}$ unless FPT = W[1]. On the positive side, we give a factor $2$ fixed parameter tractable (FPT) approximation algorithm for the problem. More precisely, our algorithm takes as input $D$ and $k$, runs in time $2^{O(k^2)}n^{O(1)}$, and either concludes that $D$ does not have a directed odd cycle transversal of size at most $k$, or produces a solution of size at most $2k$. Finally, we provide evidence that there exists $ε> 0$ such that DOCT does not admit a factor $(1+ε)$ FPT-approximation algorithm.

cs.DS↗

arXiv2017

Parameterized Integer Quadratic Programming: Variables and Coefficients

In the Integer Quadratic Programming problem input is an n*n integer matrix Q, an m*n integer matrix A and an m-dimensional integer vector b. The task is to find a vector x in Z^n, minimizing x^TQx, subject to Ax <= b. We give a fixed parameter tractable algorithm for Integer Quadratic Programming parameterized by n+a. Here a is the largest absolute value of an entry of Q and A. As an application of our main result we show that Optimal Linear Arrangement is fixed parameter tractable parameterized by the size of the smallest vertex cover of the input graph. This resolves an open problem from the recent monograph by Downey and Fellows.

cs.DS↗

arXiv2017

The Half-integral Erdös-Pósa Property for Non-null Cycles

A Group Labeled Graph is a pair $(G,Λ)$ where $G$ is an oriented graph and $Λ$ is a mapping from the arcs of $G$ to elements of a group. A (not necessarily directed) cycle $C$ is called non-null if for any cyclic ordering of the arcs in $C$, the group element obtained by `adding' the labels on forward arcs and `subtracting' the labels on reverse arcs is not the identity element of the group. Non-null cycles in group labeled graphs generalize several well-known graph structures, including odd cycles. In this paper, we prove that non-null cycles on Group Labeled Graphs have the half-integral Erdös-Pósa property. That is, there is a function $f:{\mathbb N}\to {\mathbb N}$ such that for any $k\in {\mathbb N}$, any group labeled graph $(G,Λ)$ has a set of $k$ non-null cycles such that each vertex of $G$ appears in at most two of these cycles or there is a set of at most $f(k)$ vertices that intersects every non-null cycle. Since it is known that non-null cycles do not have the integeral Erdös-Pósa property in general, a half-integral Erdös-Pósa result is the best one could hope for.

cs.DM↗

arXiv2016

Lossy Kernelization

In this paper we propose a new framework for analyzing the performance of preprocessing algorithms. Our framework builds on the notion of kernelization from parameterized complexity. However, as opposed to the original notion of kernelization, our definitions combine well with approximation algorithms and heuristics. The key new definition is that of a polynomial size $α$-approximate kernel. Loosely speaking, a polynomial size $α$-approximate kernel is a polynomial time pre-processing algorithm that takes as input an instance $(I,k)$ to a parameterized problem, and outputs another instance $(I',k')$ to the same problem, such that $|I'|+k' \leq k^{O(1)}$. Additionally, for every $c \geq 1$, a $c$-approximate solution $s'$ to the pre-processed instance $(I',k')$ can be turned in polynomial time into a $(c \cdot α)$-approximate solution $s$ to the original instance $(I,k)$. Our main technical contribution are $α$-approximate kernels of polynomial size for three problems, namely Connected Vertex Cover, Disjoint Cycle Packing and Disjoint Factors. These problems are known not to admit any polynomial size kernels unless $NP \subseteq coNP/poly$. Our approximate kernels simultaneously beat both the lower bounds on the (normal) kernel size, and the hardness of approximation lower bounds for all three problems. On the negative side we prove that Longest Path parameterized by the length of the path and Set Cover parameterized by the universe size do not admit even an $α$-approximate kernel of polynomial size, for any $α\geq 1$, unless $NP \subseteq coNP/poly$. In order to prove this lower bound we need to combine in a non-trivial way the techniques used for showing kernelization lower bounds with the methods for showing hardness of approximation

cs.DS↗

arXiv2016

Below all subsets for Minimal Connected Dominating Set

A vertex subset $S$ in a graph $G$ is a dominating set if every vertex not contained in $S$ has a neighbor in $S$. A dominating set $S$ is a connected dominating set if the subgraph $G[S]$ induced by $S$ is connected. A connected dominating set $S$ is a minimal connected dominating set if no proper subset of $S$ is also a connected dominating set. We prove that there exists a constant $\varepsilon > 10^{-50}$ such that every graph $G$ on $n$ vertices has at most $O(2^{(1-\varepsilon)n})$ minimal connected dominating sets. For the same $\varepsilon$ we also give an algorithm with running time $2^{(1-\varepsilon)n}\cdot n^{O(1)}$ to enumerate all minimal connected dominating sets in an input graph $G$.

cs.DS↗

arXiv2016

A Linear Time Parameterized Algorithm for Directed Feedback Vertex Set

In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph $D$ on $n$ vertices and $m$ edges, and an integer $k$. The objective is to determine whether there exists a set of at most $k$ vertices intersecting every directed cycle of $D$. Whether or not DFVS admits a fixed parameter tractable (FPT) algorithm was considered the most important open problem in parameterized complexity until Chen, Liu, Lu, O'Sullivan and Razgon [JACM 2008] answered the question in the affirmative. They gave an algorithm for the problem with running time $O(k!4^kk^4nm)$. Since then, no faster algorithm for the problem has been found. In this paper, we give an algorithm for DFVS with running time $O(k!4^kk^5(n+m))$. Our algorithm is the first algorithm for DFVS with linear dependence on input size. Furthermore, the asymptotic dependence of the running time of our algorithm on the parameter $k$ matches up to a factor $k$ the algorithm of Chen, Liu, Lu, O'Sullivan and Razgon. On the way to designing our algorithm for DFVS, we give a general methodology to shave off a factor of $n$ from iterative-compression based algorithms for a few other well-studied covering problems in parameterized complexity. We demonstrate the applicability of this technique by speeding up by a factor of $n$, the current best FPT algorithms for Multicut [STOC 2011, SICOMP 2014] and Directed Subset Feedback Vertex Set [ICALP 2012, TALG 2014].

cs.DS↗