arXiv subjects

Fedor V. Fomin

Publications and source records attributed to Fedor V. Fomin.

At least 145 records · Page 8Linked to original sources

arXiv2013

Long Circuits and Large Euler Subgraphs

An undirected graph is Eulerian if it is connected and all its vertices are of even degree. Similarly, a directed graph is Eulerian, if for each vertex its in-degree is equal to its out-degree. It is well known that Eulerian graphs can be recognized in polynomial time while the problems of finding a maximum Eulerian subgraph or a maximum induced Eulerian subgraph are NP-hard. In this paper, we study the parameterized complexity of the following Euler subgraph problems: - Large Euler Subgraph: For a given graph G and integer parameter k, does G contain an induced Eulerian subgraph with at least k vertices? - Long Circuit: For a given graph G and integer parameter k, does G contain an Eulerian subgraph with at least k edges? Our main algorithmic result is that Large Euler Subgraph is fixed parameter tractable (FPT) on undirected graphs. We find this a bit surprising because the problem of finding an induced Eulerian subgraph with exactly k vertices is known to be W[1]-hard. The complexity of the problem changes drastically on directed graphs. On directed graphs we obtained the following complexity dichotomy: Large Euler Subgraph is NP-hard for every fixed k>3 and is solvable in polynomial time for k<=3. For Long Circuit, we prove that the problem is FPT on directed and undirected graphs.

cs.DS↗

arXiv2013

Subexponential fixed-parameter tractability of cluster editing

In the Correlation Clustering, also known as Cluster Editing, we are given an undirected n-vertex graph G and a positive integer k. The task is to decide if G can be transformed into a cluster graph, i.e., a disjoint union of cliques, by changing at most k adjacencies, i.e. by adding/deleting at most k edges. We give a subexponential algorithm that, in time 2^O(sqrt(pk)) + n^O(1) decides whether G can be transformed into a cluster graph with p cliques by changing at most k adjacencies. We complement our algorithmic findings by the following tight lower bounds on the asymptotic behaviour of our algorithm. We show that, unless ETH fails, for any constant 0 < s <= 1, there is p = Theta(k^s) such that there is no algorithm deciding in time 2^o(sqrt(pk)) n^O(1) whether G can be transformed into a cluster graph with p cliques by changing at most k adjacencies.

cs.CC↗

arXiv2013

Subexponential parameterized algorithm for computing the cutwidth of a semi-complete digraph

Cutwidth of a digraph is a width measure introduced by Chudnovsky, Fradkin, and Seymour [4] in connection with development of a structural theory for tournaments, or more generally, for semi-complete digraphs. In this paper we provide an algorithm with running time 2^{O(\sqrt{k log k})} * n^{O(1)} that tests whether the cutwidth of a given n-vertex semi-complete digraph is at most k, improving upon the currently fastest algorithm of the second author [18] that works in 2^{O(k)} * n^2 time. As a byproduct, we obtain a new algorithm for Feedback Arc Set in tournaments (FAST) with running time 2^{c\sqrt{k}} * n^{O(1)}, where c = 2π/ \sqrt(3)*\ln(2) <= 5.24, that is simpler than the algorithms of Feige [9] and of Karpinski and Schudy[16], both also working in 2^{O(\sqrt{k})} * n^{O(1)} time. Our techniques can be applied also to other layout problems on semi-complete digraphs. We show that the Optimal Linear Arrangement problem, a close relative of Feedback Arc Set, can be solved in 2^{O(k^{1/3} \sqrt{\log k})} * n^{O(1)} time, where k is the target cost of the ordering.

cs.DS↗

arXiv2012

Jungles, bundles, and fixed parameter tractability

We give a fixed-parameter tractable (FPT) approximation algorithm computing the path-width of a tournament, and more generally, of a semi-complete digraph. Based on this result, we prove that topological containment and rooted immersion problems are FPT on semi-complete digraphs.

cs.DS↗

arXiv2012

A Polynomial kernel for Proper Interval Vertex Deletion

It is known that the problem of deleting at most k vertices to obtain a proper interval graph (Proper Interval Vertex Deletion) is fixed parameter tractable. However, whether the problem admits a polynomial kernel or not was open. Here, we answers this question in affirmative by obtaining a polynomial kernel for Proper Interval Vertex Deletion. This resolves an open question of van Bevern, Komusiewicz, Moser, and Niedermeier.

cs.DS↗

arXiv2012

A Linear Vertex Kernel for Maximum Internal Spanning Tree

We present a polynomial time algorithm that for any graph G and integer k >= 0, either finds a spanning tree with at least k internal vertices, or outputs a new graph G' on at most 3k vertices and an integer k' such that G has a spanning tree with at least k internal vertices if and only if G' has a spanning tree with at least k' internal vertices. In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k, has a 3k-vertex kernel. Our result is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that "a hypergraph H contains a hypertree if and only if H is partition connected."

cs.DS↗

arXiv2011

k-Gap Interval Graphs

We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n+k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k=0), we parameterize graph problems by k, and find FPT algorithms for several problems, including Feedback Vertex Set, Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal. The Coloring problem turns out to be W[1]-hard and we design an XP algorithm for the recognition problem.

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

Parameterized Complexity of Firefighting Revisited

The Firefighter problem is to place firefighters on the vertices of a graph to prevent a fire with known starting point from lighting up the entire graph. In each time step, a firefighter may be permanently placed on an unburned vertex and the fire spreads to its neighborhood in the graph in so far no firefighters are protecting those vertices. The goal is to let as few vertices burn as possible. This problem is known to be NP-complete, even when restricted to bipartite graphs or to trees of maximum degree three. Initial study showed the Firefighter problem to be fixed-parameter tractable on trees in various parameterizations. We complete these results by showing that the problem is in FPT on general graphs when parameterized by the number of burned vertices, but has no polynomial kernel on trees, resolving an open problem. Conversely, we show that the problem is W[1]-hard when parameterized by the number of unburned vertices, even on bipartite graphs. For both parameterizations, we additionally give refined algorithms on trees, improving on the running times of the known algorithms.

cs.DM↗

arXiv2011

Subexponential Parameterized Algorithm for Minimum Fill-in

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. Kaplan, Shamir, and Tarjan [FOCS 1994] have shown that the problem is solvable in time O(2^(O(k)) + k2 * nm) on graphs with n vertices and m edges and thus is fixed parameter tractable. Here, we give the first subexponential parameterized algorithm solving Minimum Fill-in in time O(2^(O(\sqrt{k} log k)) + k2 * nm). This substantially lower the complexity of the problem. Techniques developed for Minimum Fill-in can be used to obtain subexponential parameterized algorithms for several related problems including Minimum Chain Completion, Chordal Graph Sandwich, and Triangulating Colored Graph.

cs.DS↗

arXiv2010

Hitting forbidden minors: Approximation and Kernelization

We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leves, as an induced subgraph, where $t$ is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph $θ_c$ as a minor for a fixed integer $c$. The graph $θ_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.

cs.DS↗

arXiv2010

Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc $k$-Leaf Out-Branching}, which is to find an oriented spanning tree with at least $k$ leaves, we obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} n+ n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed graph $H$ as a minor. For the special case when the input directed graph is planar, the running time can be improved to $2^{O(\sqrt{k})}n + n^{O(1)}$. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc $k$-Internal Out-Branching}, which is to find an oriented spanning tree with at least $k$ internal vertices. We obtain an algorithm solving the problem in time $2^{O(\sqrt{k} \log k)} + n^{O(1)}$ on directed graphs whose underlying undirected graph excludes some fixed apex graph $H$ as a minor. Finally, we observe that for any $ε>0$, the {\sc $k$-Directed Path} problem is solvable in time $O((1+ε)^k n^{f(ε)})$, where $f$ is some function of $\ve$. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs.

cs.DS↗

arXiv2009

Finding Induced Subgraphs via Minimal Triangulations

Potential maximal cliques and minimal separators are combinatorial objects which were introduced and studied in the realm of minimal triangulations problems including Minimum Fill-in and Treewidth. We discover unexpected applications of these notions to the field of moderate exponential algorithms. In particular, we show that given an n-vertex graph G together with its set of potential maximal cliques Pi_G, and an integer t, it is possible in time |Pi_G| * n^(O(t)) to find a maximum induced subgraph of treewidth t in G; and for a given graph F of treewidth t, to decide if G contains an induced subgraph isomorphic to F. Combined with an improved algorithm enumerating all potential maximal cliques in time O(1.734601^n), this yields that both problems are solvable in time 1.734601^n * n^(O(t)).

cs.DS↗

arXiv2009

Faster Algorithms for Finding and Counting Subgraphs

In this paper we study a natural generalization of both {\sc $k$-Path} and {\sc $k$-Tree} problems, namely, the {\sc Subgraph Isomorphism} problem. In the {\sc Subgraph Isomorphism} problem we are given two graphs $F$ and $G$ on $k$ and $n$ vertices respectively as an input, and the question is whether there exists a subgraph of $G$ isomorphic to $F$. We show that if the treewidth of $F$ is at most $t$, then there is a randomized algorithm for the {\sc Subgraph Isomorphism} problem running in time $\cO^*(2^k n^{2t})$. To do so, we associate a new multivariate {Homomorphism polynomial} of degree at most $k$ with the {\sc Subgraph Isomorphism} problem and construct an arithmetic circuit of size at most $n^{\cO(t)}$ for this polynomial. Using this polynomial, we also give a deterministic algorithm to count the number of homomorphisms from $F$ to $G$ that takes $n^{\cO(t)}$ time and uses polynomial space. For the counting version of the {\sc Subgraph Isomorphism} problem, where the objective is to count the number of distinct subgraphs of $G$ that are isomorphic to $F$, we give a deterministic algorithm running in time and space $\cO^*({n \choose k/2}n^{2p})$ or ${n\choose k/2}n^{\cO(t \log k)}$. We also give an algorithm running in time $\cO^{*}(2^{k}{n \choose k/2}n^{5p})$ and taking space polynomial in $n$. Here $p$ and $t$ denote the pathwidth and the treewidth of $F$, respectively. Thus our work not only improves on known results on {\sc Subgraph Isomorphism} but it also extends and generalize most of the known results on {\sc $k$-Path} and {\sc $k$-Tree}.

cs.DS↗

arXiv2009

Kernels for Feedback Arc Set In Tournaments

A tournament T=(V,A) is a directed graph in which there is exactly one arc between every pair of distinct vertices. Given a digraph on n vertices and an integer parameter k, the Feedback Arc Set problem asks whether the given digraph has a set of k arcs whose removal results in an acyclic digraph. The Feedback Arc Set problem restricted to tournaments is known as the k-Feedback Arc Set in Tournaments (k-FAST) problem. In this paper we obtain a linear vertex kernel for k-FAST. That is, we give a polynomial time algorithm which given an input instance T to k-FAST obtains an equivalent instance T' on O(k) vertices. In fact, given any fixed e>0, the kernelized instance has at most (2+e)k vertices. Our result improves the previous known bound of O(k^2) on the kernel size for k-FAST. Our kernelization algorithm solves the problem on a subclass of tournaments in polynomial time and uses a known polynomial time approximation scheme for k-FAST.

cs.DS↗

arXiv2009

Rank-width and Tree-width of H-minor-free Graphs

We prove that for any fixed r>=2, the tree-width of graphs not containing K_r as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as K_r-minor free graphs and graphs of bounded genus.

math.CO↗

arXiv2009

Algorithm for Finding $k$-Vertex Out-trees and its Application to $k$-Internal Out-branching Problem

An out-tree $T$ is an oriented tree with only one vertex of in-degree zero. A vertex $x$ of $T$ is internal if its out-degree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a given out-tree with $k$ vertices. The algorithms are of runtime $O^*(5.704^k)$ and $O^*(5.704^{k(1+o(1))})$, respectively. We apply the deterministic algorithm to obtain a deterministic algorithm of runtime $O^*(c^k)$, where $c$ is a constant, for deciding whether an input digraph contains a spanning out-tree with at least $k$ internal vertices. This answers in affirmative a question of Gutin, Razgon and Kim (Proc. AAIM'08).

cs.DS↗