arXiv subjects

Fedor V. Fomin

Publications and source records attributed to Fedor V. Fomin.

At least 109 records · Page 6Linked to original sources

arXiv2018

Partial complementation of graphs

A partial complement of the graph $G$ is a graph obtained from $G$ by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph $G$ and graph class $\mathcal{G}$, is there a partial complement of $G$ which is in $\mathcal{G}$? We show that this problem can be solved in polynomial time for various choices of the graphs class $\mathcal{G}$, such as bipartite, degenerate, or cographs. We complement these results by proving that the problem is NP-complete when $\mathcal{G}$ is the class of $r$-regular graphs.

cs.CC↗

arXiv2018

Parameterized Low-Rank Binary Matrix Approximation

We provide a number of algorithmic results for the following family of problems: For a given binary m\times n matrix A and integer k, decide whether there is a "simple" binary matrix B which differs from A in at most k entries. For an integer r, the "simplicity" of B is characterized as follows. - Binary r-Means: Matrix B has at most r different columns. This problem is known to be NP-complete already for r=2. We show that the problem is solvable in time 2^{O(k\log k)}\cdot(nm)^{O(1)} and thus is fixed-parameter tractable parameterized by k. We prove that the problem admits a polynomial kernel when parameterized by r and k but it has no polynomial kernel when parameterized by k only unless NP\subseteq coNP/poly. We also complement these result by showing that when being parameterized by r and k, the problem admits an algorithm of running time 2^{O(r\cdot \sqrt{k\log{(k+r)}})}(nm)^{O(1)}, which is subexponential in k for r\in O(k^{1/2 -ε}) for any ε>0. - Low GF(2)-Rank Approximation: Matrix B is of GF(2)-rank at most r. This problem is known to be NP-complete already for r=1. It also known to be W[1]-hard when parameterized by k. Interestingly, when parameterized by r and k, the problem is not only fixed-parameter tractable, but it is solvable in time 2^{O(r^{ 3/2}\cdot \sqrt{k\log{k}})}(nm)^{O(1)}, which is subexponential in k. - Low Boolean-Rank Approximation: Matrix B is of Boolean rank at most r. The problem is known to be NP-complete for k=0 as well as for r=1. We show that it is solvable in subexponential in k time 2^{O(r2^r\cdot \sqrt{k\log k})}(nm)^{O(1)}.

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

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

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

Structured Connectivity Augmentation

We initiate the algorithmic study of the following "structured augmentation" question: is it possible to increase the connectivity of a given graph G by superposing it with another given graph H? More precisely, graph F is the superposition of G and H with respect to injective mapping ϕ: V(H)->V(G) if every edge uv of F is either an edge of G, or ϕ^{-1}(u)ϕ^{-1}(v) is an edge of H. We consider the following optimization problem. Given graphs G,H, and a weight function ωassigning non-negative weights to pairs of vertices of V(G), the task is to find φof minimum weight ω(ϕ)=\sum_{xy\in E(H)}ω(ϕ(x)φ(y)) such that the edge connectivity of the superposition F of G and H with respect to ϕis higher than the edge connectivity of G. Our main result is the following "dichotomy" complexity classification. We say that a class of graphs C has bounded vertex-cover number, if there is a constant t depending on C only such that the vertex-cover number of every graph from C does not exceed t. We show that for every class of graphs C with bounded vertex-cover number, the problems of superposing into a connected graph F and to 2-edge connected graph F, are solvable in polynomial time when H\in C. On the other hand, for any hereditary class C with unbounded vertex-cover number, both problems are NP-hard when H\in C. For the unweighted variants of structured augmentation problems, i.e. the problems where the task is to identify whether there is a superposition of graphs of required connectivity, we provide necessary and sufficient combinatorial conditions on the existence of such superpositions. These conditions imply polynomial time algorithms solving the unweighted variants of the problems.

cs.DS↗

arXiv2017

Finding Detours is Fixed-parameter Tractable

We consider the following natural "above guarantee" parameterization of the classical Longest Path problem: For given vertices s and t of a graph G, and an integer k, the problem Longest Detour asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k. Furthermore, we study the related problem Exact Detour that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k, and a deterministic algorithm with running time about 6.745^k, showing that this problem is FPT as well. Our algorithms for Exact Detour apply to both undirected and directed graphs.

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↗

arXiv2016

Spanning Circuits in Regular Matroids

We consider the fundamental Matroid Theory problem of finding a circuit in a matroid spanning a set T of given terminal elements. For graphic matroids this corresponds to the problem of finding a simple cycle passing through a set of given terminal edges in a graph. The algorithmic study of the problem on regular matroids, a superclass of graphic matroids, was initiated by Gavenčiak, Král', and Oum [ICALP'12], who proved that the case of the problem with |T|=2 is fixed-parameter tractable (FPT) when parameterized by the length of the circuit. We extend the result of Gavenčiak, Král', and Oum by showing that for regular matroids - the Minimum Spanning Circuit problem, deciding whether there is a circuit with at most \ell elements containing T, is FPT parameterized by k=\ell-|T|; - the Spanning Circuit problem, deciding whether there is a circuit containing T, is FPT parameterized by |T|. We note that extending our algorithmic findings to binary matroids, a superclass of regular matroids, is highly unlikely: Minimum Spanning Circuit parameterized by \ell is W[1]-hard on binary matroids even when |T|=1. We also show a limit to how far our results can be strengthened by considering a smaller parameter. More precisely, we prove that Minimum Spanning Circuit parameterized by |T| is W[1]-hard even on cographic matroids, a proper subclass of regular matroids.

cs.DS↗

arXiv2016

Subexponential parameterized algorithms for planar and apex-minor-free graphs via low treewidth pattern covering

We prove the following theorem. Given a planar graph $G$ and an integer $k$, it is possible in polynomial time to randomly sample a subset $A$ of vertices of $G$ with the following properties: (i) $A$ induces a subgraph of $G$ of treewidth $\mathcal{O}(\sqrt{k}\log k)$, and (ii) for every connected subgraph $H$ of $G$ on at most $k$ vertices, the probability that $A$ covers the whole vertex set of $H$ is at least $(2^{\mathcal{O}(\sqrt{k}\log^2 k)}\cdot n^{\mathcal{O}(1)})^{-1}$, where $n$ is the number of vertices of $G$. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound $2^{\mathcal{O}(\sqrt{k} \log^2 k)} n^{\mathcal{O}(1)}$. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph, examples of such problems include Directed $k$-Path, Weighted $k$-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface.

cs.DS↗

arXiv2016

Vertex Cover Structural Parameterization Revisited

A pseudoforest is a graph whose connected components have at most one cycle. Let X be a pseudoforest modulator of graph G, i. e. a vertex subset of G such that G-X is a pseudoforest. We show that Vertex Cover admits a polynomial kernel being parameterized by the size of the pseudoforest modulator. In other words, we provide a polynomial time algorithm that for an input graph G and integer k, outputs a graph G' and integer k', such that G' has O(|X|12) vertices and G has a vertex cover of size k if and only if G' has vertex cover of size k'. We complement our findings by proving that there is no polynomial kernel for Vertex Cover parameterized by the size of a modulator to a mock forest (a graph where no cycles share a vertex) unless NP is a subset of coNP/poly. In particular, this also rules out polynomial kernels when parameterized by the size of a modulator to outerplanar and cactus graphs.

cs.DS↗

arXiv2016

Efficient Computation of Representative Sets with Applications in Parameterized and Exact Algorithms

We give two algorithms computing representative families of linear and uniform matroids and demonstrate how to use representative families for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include - LONGEST DIRECTED CYCLE - MINIMUM EQUIVALENT GRAPH (MEG) - Algorithms on graphs of bounded treewidth -k-PATH, k-TREE, and more generally, k-SUBGRAPH ISOMORPHISM, where the k-vertex pattern graph is of constant treewidth.

cs.DS↗

arXiv2016

Tight Lower Bounds on Graph Embedding Problems

We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of $|V(H)|^{o(|V(H)|)}$-time algorithm deciding if graph $G$ is a subgraph of $H$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems. Moreover, as a consequence of our reductions conditional lower bounds follow for other related problems such as Locally Injective Homomorphism, Graph Minors, Topological Graph Minors, Minimum Distortion Embedding and Quadratic Assignment Problem.

cs.DS↗

arXiv2016

Metric Dimension of Bounded Tree-length Graphs

The notion of resolving sets in a graph was introduced by Slater (1975) and Harary and Melter (1976) as a way of uniquely identifying every vertex in a graph. A set of vertices in a graph is a resolving set if for any pair of vertices x and y there is a vertex in the set which has distinct distances to x and y. A smallest resolving set in a graph is called a metric basis and its size, the metric dimension of the graph. The problem of computing the metric dimension of a graph is a well-known NP-hard problem and while it was known to be polynomial time solvable on trees, it is only recently that efforts have been made to understand its computational complexity on various restricted graph classes. In recent work, Foucaud et al. (2015) showed that this problem is NP-complete even on interval graphs. They complemented this result by also showing that it is fixed-parameter tractable (FPT) parameterized by the metric dimension of the graph. In this work, we show that this FPT result can in fact be extended to all graphs of bounded tree-length. This includes well-known classes like chordal graphs, AT-free graphs and permutation graphs. We also show that this problem is FPT parameterized by the modular-width of the input graph.

cs.DS↗

arXiv2015

Exact Algorithms via Monotone Local Search

We give a new general approach for designing exact exponential-time algorithms for subset problems. In a subset problem the input implicitly describes a family of sets over a universe of size n and the task is to determine whether the family contains at least one set. Our approach is based on "monotone local search", where the goal is to extend a partial solution to a solution by adding as few elements as possible. More formally, in the extension problem we are also given as input a subset X of the universe and an integer k. The task is to determine whether one can add at most k elements to X to obtain a set in the (implicitly defined) family. Our main result is that an O*(c^k) time algorithm for the extension problem immediately yields a randomized algorithm for finding a solution of any size with running time O*((2-1/c)^n). In many cases, the extension problem can be reduced to simply finding a solution of size at most k. Furthermore, efficient algorithms for finding small solutions have been extensively studied in the field of parameterized algorithms. Directly applying these algorithms, our theorem yields in one stroke significant improvements over the best known exponential-time algorithms for several well-studied problems, including d-Hitting Set, Feedback Vertex Set, Node Unique Label Cover, and Weighted d-SAT. Our results demonstrate an interesting and very concrete connection between parameterized algorithms and exact exponential-time algorithms. We also show how to derandomize our algorithms at the cost of a subexponential multiplicative factor in the running time. Our derandomization is based on an efficient construction of a new pseudo-random object that might be of independent interest. Finally, we extend our methods to establish new combinatorial upper bounds and develop enumeration algorithms.

cs.DS↗

arXiv2015

Fully polynomial-time parameterized computations for graphs and matrices of low treewidth

We investigate the complexity of several fundamental polynomial-time solvable problems on graphs and on matrices, when the given instance has low treewidth; in the case of matrices, we consider the treewidth of the graph formed by non-zero entries. In each of the considered cases, the best known algorithms working on general graphs run in polynomial time, however the exponent of the polynomial is large. Therefore, our main goal is to construct algorithms with running time of the form $\textrm{poly}(k)\cdot n$ or $\textrm{poly}(k)\cdot n\log n$, where $k$ is the width of the tree decomposition given on the input. Such procedures would outperform the best known algorithms for the considered problems already for moderate values of the treewidth, like $O(n^{1/c})$ for some small constant $c$. Our results include: -- an algorithm for computing the determinant and the rank of an $n\times n$ matrix using $O(k^3\cdot n)$ time and arithmetic operations; -- an algorithm for solving a system of linear equations using $O(k^3\cdot n)$ time and arithmetic operations; -- an $O(k^3\cdot n\log n)$-time randomized algorithm for finding the cardinality of a maximum matching in a graph; -- an $O(k^4\cdot n\log^2 n)$-time randomized algorithm for constructing a maximum matching in a graph; -- an $O(k^2\cdot n\log n)$-time algorithm for finding a maximum vertex flow in a directed graph. Moreover, we give an approximation algorithm for treewidth with time complexity suited to the running times as above. Namely, the algorithm, when given a graph $G$ and integer $k$, runs in time $O(k^7\cdot n\log n)$ and either correctly reports that the treewidth of $G$ is larger than $k$, or constructs a tree decomposition of $G$ of width $O(k^2)$.

cs.DS↗

arXiv2015

Kernelization and Sparseness: the case of Dominating Set

We prove that for every positive integer $r$ and for every graph class $\mathcal G$ of bounded expansion, the $r$-Dominating Set problem admits a linear kernel on graphs from $\mathcal G$. Moreover, when $\mathcal G$ is only assumed to be nowhere dense, then we give an almost linear kernel on $\mathcal G$ for the classic Dominating Set problem, i.e., for the case $r=1$. These results generalize a line of previous research on finding linear kernels for Dominating Set and $r$-Dominating Set. However, the approach taken in this work, which is based on the theory of sparse graphs, is radically different and conceptually much simpler than the previous approaches. We complement our findings by showing that for the closely related Connected Dominating Set problem, the existence of such kernelization algorithms is unlikely, even though the problem is known to admit a linear kernel on $H$-topological-minor-free graphs. Also, we prove that for any somewhere dense class $\mathcal G$, there is some $r$ for which $r$-Dominating Set is W[$2$]-hard on $\mathcal G$. Thus, our results fall short of proving a sharp dichotomy for the parameterized complexity of $r$-Dominating Set on subgraph-monotone graph classes: we conjecture that the border of tractability lies exactly between nowhere dense and somewhere dense graph classes.

cs.DS↗

arXiv2015

Tight Bounds for Subgraph Isomorphism and Graph Homomorphism

We prove that unless Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. Combined with the reduction of Cygan, Pachocki, and Socała, our result rules out (subject to ETH) a possibility of $|V(G)|^{o(|V(G)|)}$-time algorithm deciding if graph $H$ is a subgraph of $G$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems.

cs.DS↗