arXiv subjects

Dimitrios M. Thilikos

Publications and source records attributed to Dimitrios M. Thilikos.

At least 73 records · Page 4Linked to original sources

arXiv2020

Bidimensionality and Kernels

Bidimensionality Theory was introduced by [E.D. Demaine, F.V. Fomin, M.Hajiaghayi, and D.M. Thilikos. Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs, J. ACM, 52 (2005), pp.866--893] as a tool to obtain sub-exponential time parameterized algorithms on H-minor-free graphs. In [E.D. Demaine and M.Hajiaghayi, Bidimensionality: new connections between FPT algorithms and PTASs, in Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2005, pp.590--601] this theory was extended in order to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this work, we establish a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In particular, we prove that every minor (respectively contraction) bidimensional problem that satisfies a separation property and is expressible in Countable Monadic Second Order Logic (CMSO), admits a linear kernel for classes of graphs that exclude a fixed graph (respectively an apex graph) H as a minor. Our results imply that a multitude of bidimensional problems g graph classes. For most of these problems no polynomial kernels on H-minor-free graphs were known prior to our work.

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arXiv2020

Structure and enumeration of K4-minor-free links and link-diagrams

We study the class L of link-types that admit a K4-minor-free diagram, i.e., they can be projected on the plane so that the resulting graph does not contain any subdivision of K4. We prove that L is the closure of a subclass of torus links under the operation of connected sum. Using this structural result, we enumerate L and subclasses of it, with respect to the minimum number of crossings or edges in a projection of L' in L. Further, we obtain counting formulas and asymptotic estimates for the connected K4-minor-free link-diagrams, minimal K4-minor-free link-diagrams, and K4-minor-free diagrams of the unknot.

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arXiv2019

On the Parameterized Complexity of Graph Modification to First-Order Logic Properties

We consider the problems of deciding whether an input graph can be modified by removing/adding at most k vertices/edges such that the result of the modification satisfies some property definable in first-order logic. We establish a number of sufficient and necessary conditions on the quantification pattern of the first-order formula ϕfor the problem to be fixed-parameter tractable or to admit a polynomial kernel.

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arXiv2019

Minimum Reload Cost Graph Factors

The concept of Reload cost in a graph refers to the cost that occurs while traversing a vertex via two of its incident edges. This cost is uniquely determined by the colors of the two edges. This concept has various applications in transportation networks, communication networks, and energy distribution networks. Various problems using this model are defined and studied in the literature. The problem of finding a spanning tree whose diameter with respect to the reload costs is the smallest possible, the problems of finding a path, trail or walk with minimum total reload cost between two given vertices, problems about finding a proper edge coloring of a graph such that the total reload cost is minimized, the problem of finding a spanning tree such that the sum of the reload costs of all paths between all pairs of vertices is minimized, and the problem of finding a set of cycles of minimum reload cost, that cover all the vertices of a graph, are examples of such problems. % In this work we focus on the last problem. Noting that a cycle cover of a graph is a 2-factor of it, we generalize the problem to that of finding an $r$-factor of minimum reload cost of an edge colored graph. We prove several NP-hardness results for special cases of the problem. Namely, bounded degree graphs, planar graphs, bounded total cost, and bounded number of distinct costs. For the special case of $r=2$, our results imply an improved NP-hardness result. On the positive side, we present a polynomial-time solvable special case which provides a tight boundary between the polynomial and hard cases in terms of $r$ and the maximum degree of the graph. We then investigate the parameterized complexity of the problem, prove W[1]-hardness results and present an FPT algorithm.

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arXiv2018

Data-compression for Parametrized Counting Problems on Sparse graphs

We study the concept of \emph{compactor}, which may be seen as a counting-analogue of kernelization in counting parameterized complexity. For a function $F:Σ^*\to \Bbb{N}$ and a parameterization $κ: Σ^*\to \Bbb{N}$, a compactor $({\sf P},{\sf M})$ consists of a polynomial-time computable function ${\sf P}$, called \emph{condenser}, and a computable function ${\sf M}$, called \emph{extractor}, such that $F={\sf M}\circ {\sf P}$, and the condensing ${\sf P}(x)$ of $x$ has length at most $s(κ(x))$, for any input $x\in Σ^*.$ If $s$ is a polynomial function, then the compactor is said to be of polynomial-size. Although the study on counting-analogue of kernelization is not unprecedented, it has received little attention so far. We study a family of vertex-certified counting problems on graphs that are MSOL-expressible; that is, for an MSOL-formula $ϕ$ with one free set variable to be interpreted as a vertex subset, we want to count all $A\subseteq V(G)$ where $|A|=k$ and $(G,A)\models ϕ.$ In this paper, we prove that every vertex-certified counting problems on graphs that is \emph{MSOL-expressible} and \emph{treewidth modulable}, when parameterized by $k$, admits a polynomial-size compactor on $H$-topological-minor-free graphs with condensing time $O(k^2n^2)$ and decoding time $2^{O(k)}.$ This implies the existence of an {\sf FPT}-algorithm of running time $O(n^2k^2)+2^{O(k)}.$ All aforementioned complexities are under the Uniform Cost Measure (UCM) model where numbers can be stored in constant space and arithmetic operations can be done in constant time.

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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.

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arXiv2018

Clustering to Given Connectivities

We define a general variant of the graph clustering problem where the criterion of density for the clusters is (high) connectivity. In {\sc Clustering to Given Connectivities}, we are given an $n$-vertex graph $G$, an integer $k$, and a sequence $Λ=\langle λ_{1},\ldots,λ_{t}\rangle$ of positive integers and we ask whether it is possible to remove at most $k$ edges from $G$ such that the resulting connected components are {\sl exactly} $t$ and their corresponding edge connectivities are lower-bounded by the numbers in $Λ$. We prove that this problem, parameterized by $k$, is fixed parameter tractable i.e., can be solved by an $f(k)\cdot n^{O(1)}$-step algorithm, for some function $f$ that depends only on the parameter $k$. Our algorithm uses the recursive understanding technique that is especially adapted so to deal with the fact that, in out setting, we do not impose any restriction to the connectivity demands in $Λ$.

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arXiv2018

Acyclic Edge Coloring through the Lovász Local Lemma

We give a probabilistic analysis of a Moser-type algorithm for the Lovász Local Lemma (LLL), adjusted to search for acyclic edge colorings of a graph. We thus improve the best known upper bound to acyclic chromatic index, also obtained by analyzing a similar algorithm, but through the entropic method (basically counting argument). Specifically we show that a graph with maximum degree $Δ$ has an acyclic proper edge coloring with at most $\lceil 3.74(Δ-1)\rceil+1 $ colors, whereas, previously, the best bound was $4(Δ-1)$. The main contribution of this work is that it comprises a probabilistic analysis of a Moser-type algorithm applied to events pertaining to dependent variables.

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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.

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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.

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arXiv2017

A Linear Kernel for Planar Red-Blue Dominating Set

In the Red-Blue Dominating Set problem, we are given a bipartite graph $G = (V_B \cup V_R,E)$ and an integer $k$, and asked whether $G$ has a subset $D \subseteq V_B$ of at most $k$ "blue" vertices such that each "red" vertex from $V_R$ is adjacent to a vertex in $D$. We provide the first explicit linear kernel for this problem on planar graphs, of size at most $43k$.

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arXiv2017

Parameterized complexity of finding a spanning tree with minimum reload cost diameter

We study the minimum diameter spanning tree problem under the reload cost model (DIAMETER-TREE for short) introduced by Wirth and Steffan (2001). In this problem, given an undirected edge-colored graph $G$, reload costs on a path arise at a node where the path uses consecutive edges of different colors. The objective is to find a spanning tree of $G$ of minimum diameter with respect to the reload costs. We initiate a systematic study of the parameterized complexity of the DIAMETER-TREE problem by considering the following parameters: the cost of a solution, and the treewidth and the maximum degree $Δ$ of the input graph. We prove that DIAMETER-TREE is para-NP-hard for any combination of two of these three parameters, and that it is FPT parameterized by the three of them. We also prove that the problem can be solved in polynomial time on cactus graphs. This result is somehow surprising since we prove DIAMETER-TREE to be NP-hard on graphs of treewidth two, which is best possible as the problem can be trivially solved on forests. When the reload costs satisfy the triangle inequality, Wirth and Steffan (2001) proved that the problem can be solved in polynomial time on graphs with $Δ= 3$, and Galbiati (2008) proved that it is NP-hard if $Δ= 4$. Our results show, in particular, that without the requirement of the triangle inequality, the problem is NP-hard if $Δ= 3$, which is also best possible. Finally, in the case where the reload costs are polynomially bounded by the size of the input graph, we prove that DIAMETER-TREE is in XP and W[1]-hard parameterized by the treewidth plus $Δ$.

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arXiv2017

Cutwidth: obstructions and algorithmic aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most $k$ are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most $k$. We prove that every minimal immersion obstruction for cutwidth at most $k$ has size at most $2^{O(k^3\log k)}$. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $2^{O(k^2\log k)}\cdot n$, where $k$ is the optimum width and $n$ is the number of vertices. While being slower by a $\log k$-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth II: Algorithms for partial $w$-trees of bounded degree, J. Algorithms, 56(1):25--49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.

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arXiv2017

An $O(\log OPT)$-approximation for covering and packing minor models of $θ_r$

Given two graphs $G$ and $H$, we define $\textsf{v-cover}_{H}(G)$ (resp. $\textsf{e-cover}_{H}(G)$) as the minimum number of vertices (resp. edges) whose removal from $G$ produces a graph without any minor isomorphic to ${H}$. Also $\textsf{v-pack}_{H}(G)$ (resp. $\textsf{v-pack}_{H}(G)$) is the maximum number of vertex- (resp. edge-) disjoint subgraphs of $G$ that contain a minor isomaorphic to $H$. We denote by $θ_r$ the graph with two vertices and $r$ parallel edges between them. When $H=θ_r$, the parameters $\textsf{v-cover}_{H}$, $\textsf{e-cover}_{H}$, $\textsf{v-pack}_{H}$, and $\textsf{v-pack}_{H}$ are NP-hard to compute (for sufficiently big values of $r$). Drawing upon combinatorial results in [Minors in graphs of large $θ_r$-girth, Chatzidimitriou et al., arXiv:1510.03041], we give an algorithmic proof that if $\textsf{v-pack}_{θ_r}(G)\leq k$, then $\textsf{v-cover}_{θ_r}(G) = O(k\log k)$, and similarly for $\textsf{v-pack}_{θ_r}$ and $\textsf{e-cover}_{θ_r}$. In other words, the class of graphs containing ${θ_r}$ as a minor has the vertex/edge Erdős-Pósa property, for every positive integer $r$. Using the algorithmic machinery of our proofs, we introduce a unified approach for the design of an $O(\log {\rm OPT})$-approximation algorithm for $\textsf{v-pack}_{θ_r}$, $\textsf{v-cover}_{θ_r}$, $\textsf{v-pack}_{θ_r}$, and $\textsf{e-cover}_{θ_r}$ that runs in $O(n\cdot \log(n)\cdot m)$ steps. Also, we derive several new Erdős-Pósa-type results from the techniques that we introduce.

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arXiv2017

Minors in graphs of large $θ_r$-girth

For every $r \in \mathbb{N}$, let $θ_r$ denote the graph with two vertices and $r$ parallel edges. The $θ_r$-girth of a graph $G$ is the minimum number of edges of a subgraph of $G$ that can be contracted to $θ_r$. This notion generalizes the usual concept of girth which corresponds to the case $r=2$. In [Minors in graphs of large girth, Random Structures & Algorithms, 22(2):213--225, 2003], Kühn and Osthus showed that graphs of sufficiently large minimum degree contain clique-minors whose order is an exponential function of their girth. We extend this result for the case of $θ_{r}$-girth and we show that the minimum degree can be replaced by some connectivity measurement. As an application of our results, we prove that, for every fixed $r$, graphs excluding as a minor the disjoint union of $k$ $θ_{r}$'s have treewidth $O(k\cdot \log k)$.

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arXiv2016

Recent techniques and results on the Erdős-Pósa property

Several min-max relations in graph theory can be expressed in the framework of the Erdős-Pósa property. Typically, this property reveals a connection between packing and covering problems on graphs. We describe some recent techniques for proving this property that are related to tree-like decompositions. We also provide an unified presentation of the current state of the art on this topic.

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arXiv2016

Explicit linear kernels for packing problems

During the last years, several algorithmic meta-theorems have appeared (Bodlaender et al. [FOCS 2009], Fomin et al. [SODA 2010], Kim et al. [ICALP 2013]) guaranteeing the existence of linear kernels on sparse graphs for problems satisfying some generic conditions. The drawback of such general results is that it is usually not clear how to derive from them constructive kernels with reasonably low explicit constants. To fill this gap, we recently presented [STACS 2014] a framework to obtain explicit linear kernels for some families of problems whose solutions can be certified by a subset of vertices. In this article we enhance our framework to deal with packing problems, that is, problems whose solutions can be certified by collections of subgraphs of the input graph satisfying certain properties. ${\mathcal F}$-Packing is a typical example: for a family ${\mathcal F}$ of connected graphs that we assume to contain at least one planar graph, the task is to decide whether a graph $G$ contains $k$ vertex-disjoint subgraphs such that each of them contains a graph in ${\mathcal F}$ as a minor. We provide explicit linear kernels on sparse graphs for the following two orthogonal generalizations of ${\mathcal F}$-Packing: for an integer $\ell \geq 1$, one aims at finding either minor-models that are pairwise at distance at least $\ell$ in $G$ ($\ell$-${\mathcal F}$-Packing), or such that each vertex in $G$ belongs to at most $\ell$ minors-models (${\mathcal F}$-Packing with $\ell$-Membership). Finally, we also provide linear kernels for the versions of these problems where one wants to pack subgraphs instead of minors.

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arXiv2016

Linear kernels for edge deletion problems to immersion-closed graph classes

Suppose $\mathcal{F}$ is a finite family of graphs. We consider the following meta-problem, called $\mathcal{F}$-Immersion Deletion: given a graph $G$ and integer $k$, decide whether the deletion of at most $k$ edges of $G$ can result in a graph that does not contain any graph from $\mathcal{F}$ as an immersion. This problem is a close relative of the $\mathcal{F}$-Minor Deletion problem studied by Fomin et al. [FOCS 2012], where one deletes vertices in order to remove all minor models of graphs from $\mathcal{F}$. We prove that whenever all graphs from $\mathcal{F}$ are connected and at least one graph of $\mathcal{F}$ is planar and subcubic, then the $\mathcal{F}$-Immersion Deletion problem admits: a constant-factor approximation algorithm running in time $O(m^3 \cdot n^3 \cdot \log m)$; a linear kernel that can be computed in time $O(m^4 \cdot n^3 \cdot \log m)$; and a $O(2^{O(k)} + m^4 \cdot n^3 \cdot \log m)$-time fixed-parameter algorithm, where $n,m$ count the vertices and edges of the input graph. These results mirror the findings of Fomin et al. [FOCS 2012], who obtained a similar set of algorithmic results for $\mathcal{F}$-Minor Deletion, under the assumption that at least one graph from $\mathcal{F}$ is planar. An important difference is that we are able to obtain a linear kernel for $\mathcal{F}$-Immersion Deletion, while the exponent of the kernel of Fomin et al. for $\mathcal{F}$-Minor Deletion depends heavily on the family $\mathcal{F}$. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ICALP 2015]. This reveals that the kernelization complexity of $\mathcal{F}$-Immersion Deletion is quite different than that of $\mathcal{F}$-Minor Deletion.

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