arXiv subjects

O-joung Kwon

Publications and source records attributed to O-joung Kwon.

At least 55 records · Page 3Linked to original sources

arXiv2019

Measuring what Matters: A Hybrid Approach to Dynamic Programming with Treewidth

We develop a framework for applying treewidth-based dynamic programming on graphs with "hybrid structure", i.e., with parts that may not have small treewidth but instead possess other structural properties. Informally, this is achieved by defining a refinement of treewidth which only considers parts of the graph that do not belong to a pre-specified tractable graph class. Our approach allows us to not only generalize existing fixed-parameter algorithms exploiting treewidth, but also fixed-parameter algorithms which use the size of a modulator as their parameter. As the flagship application of our framework, we obtain a parameter that combines treewidth and rank-width to obtain fixed-parameter algorithms for Chromatic Number, Hamiltonian Cycle, and Max-Cut.

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arXiv2019

On low rank-width colorings

We introduce the concept of low rank-width colorings, generalising the notion of low tree-depth colorings introduced by Nešetřil and Ossona de Mendez in [Grad and classes with bounded expansion I. Decompositions. EJC, 2008]. We say that a class $\mathcal{C}$ of graphs admits low rank-width colourings if there exist functions $N\colon \mathbb{N}\rightarrow\mathbb{N}$ and $Q\colon \mathbb{N}\rightarrow\mathbb{N}$ such that for all $p\in \mathbb{N}$, every graph $G\in \mathcal{C}$ can be vertex colored with at most $N(p)$ colors such that the union of any $i\leq p$ color classes induces a subgraph of rank-width at most $Q(i)$. Graph classes admitting low rank-width colorings strictly generalize graph classes admitting low tree-depth colorings and graph classes of bounded rank-width. We prove that for every graph class $\mathcal{C}$ of bounded expansion and every positive integer $r$, the class $\{G^r\colon G\in \mathcal{C}\}$ of $r$th powers of graphs from $\mathcal{C}$, as well as the classes of unit interval graphs and bipartite permutation graphs admit low rank-width colorings. All of these classes have unbounded rank-width and do not admit low tree-depth colorings. We also show that the classes of interval graphs and permutation graphs do not admit low rank-width colorings. As interesting side properties, we prove that every graph class admitting low rank-width colorings has the Erdős-Hajnal property and is $χ$-bounded.

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arXiv2019

An optimal XP algorithm for Hamiltonian cycle on graphs of bounded clique-width

In this paper, we prove that, given a clique-width $k$-expression of an $n$-vertex graph, \textsc{Hamiltonian Cycle} can be solved in time $n^{\mathcal{O}(k)}$. This improves the naive algorithm that runs in time $n^{\mathcal{O}(k^2)}$ by Espelage et al. (WG 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time $n^{o(k)}$ (SIAM. J. Computing 2014). We present a technique of representative sets using two-edge colored multigraphs on $k$ vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on $k$ vertices with at most $n$ edges.

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arXiv2019

Classes of graphs with no long cycle as a vertex-minor are polynomially $χ$-bounded

A class $\mathcal G$ of graphs is $χ$-bounded if there is a function $f$ such that for every graph $G\in \mathcal G$ and every induced subgraph $H$ of $G$, $χ(H)\le f(ω(H))$. In addition, we say that $\mathcal G$ is polynomially $χ$-bounded if $f$ can be taken as a polynomial function. We prove that for every integer $n\ge3$, there exists a polynomial $f$ such that $χ(G)\le f(ω(G))$ for all graphs with no vertex-minor isomorphic to the cycle graph $C_n$. To prove this, we show that if $\mathcal G$ is polynomially $χ$-bounded, then so is the closure of $\mathcal G$ under taking the $1$-join operation.

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arXiv2019

Erdös-Pósa property of minor-models with prescribed vertex sets

A minor-model of a graph $H$ in a graph $G$ is a subgraph of $G$ that can be contracted to $H$. We prove that for a positive integer $\ell$ and a non-empty planar graph $H$ with at least $\ell-1$ connected components, there exists a function $f_{H, \ell}:\mathbb{N}\rightarrow \mathbb{R}$ satisfying the property that every graph $G$ with a family of vertex subsets $Z_1, \ldots, Z_m$ contains either $k$ pairwise vertex-disjoint minor-models of $H$ each intersecting at least $\ell$ sets among prescribed vertex sets, or a vertex subset of size at most $f_{H, \ell}(k)$ that meets all such minor-models of $H$. This function $f_{H, \ell}$ is independent with the number $m$ of given sets, and thus, our result generalizes Mader's $\mathcal S$-path Theorem, by applying $\ell=2$ and $H$ to be the one-vertex graph. We prove that such a function $f_{H, \ell}$ does not exist if $H$ consists of at most $\ell-2$ connected components.

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arXiv2019

Generalized feedback vertex set problems on bounded-treewidth graphs: chordality is the key to single-exponential parameterized algorithms

It has long been known that Feedback Vertex Set can be solved in time $2^{\mathcal{O}(w\log w)}n^{\mathcal{O}(1)}$ on $n$-vertex graphs of treewidth $w$, but it was only recently that this running time was improved to $2^{\mathcal{O}(w)}n^{\mathcal{O}(1)}$, that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class $\mathcal{P}$ of graphs, the Bounded $\mathcal{P}$-Block Vertex Deletion problem asks, given a graph~$G$ on $n$ vertices and positive integers~$k$ and~$d$, whether $G$ contains a set~$S$ of at most $k$ vertices such that each block of $G-S$ has at most $d$ vertices and is in $\mathcal{P}$. Assuming that $\mathcal{P}$ is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of $d$: if $\mathcal{P}$ consists only of chordal graphs, then the problem can be solved in time $2^{\mathcal{O}(wd^2)} n^{\mathcal{O}(1)}$, and if $\mathcal{P}$ contains a graph with an induced cycle of length $\ell\ge 4$, then the problem is not solvable in time $2^{o(w\log w)} n^{\mathcal{O}(1)}$ even for fixed $d=\ell$, unless the ETH fails. We also study a similar problem, called Bounded $\mathcal{P}$-Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if $d$ is part of the input and $\mathcal{P}$ contains all chordal graphs, then it cannot be solved in time $f(w)n^{o(w)}$ for some function $f$, unless the ETH fails.

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arXiv2018

Packing and covering induced subdivisions

A class $\mathcal{F}$ of graphs has the induced Erdős-Pósa property if there exists a function $f$ such that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ pairwise vertex-disjoint induced subgraphs that belong to $\mathcal{F}$, or a vertex set of size at most $f(k)$ hitting all induced copies of graphs in $\mathcal{F}$. Kim and Kwon (SODA'18) showed that for a cycle $C_{\ell}$ of length $\ell$, the class of $C_{\ell}$-subdivisions has the induced Erdős-Pósa property if and only if $\ell\le 4$. In this paper, we investigate whether or not the class of $H$-subdivisions has the induced Erdős-Pósa property for other graphs $H$. We completely settle the case when $H$ is a forest or a complete bipartite graph. Regarding the general case, we identify necessary conditions on $H$ for the class of $H$-subdivisions to have the induced Erdős-Pósa property. For this, we provide three basic constructions that are useful to prove that the class of the subdivisions of a graph does not have the induced Erdős-Pósa property. Among remaining graphs, we prove that if $H$ is either the diamond, the $1$-pan, or the $2$-pan, then the class of $H$-subdivisions has the induced Erdős-Pósa property.

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arXiv2018

Generalized distance domination problems and their complexity on graphs of bounded mim-width

We generalize the family of $(σ, ρ)$-problems and locally checkable vertex partition problems to their distance versions, which naturally captures well-known problems such as distance-$r$ dominating set and distance-$r$ independent set. We show that these distance problems are XP parameterized by the structural parameter mim-width, and hence polynomial on graph classes where mim-width is bounded and quickly computable, such as $k$-trapezoid graphs, Dilworth $k$-graphs, (circular) permutation graphs, interval graphs and their complements, convex graphs and their complements, $k$-polygon graphs, circular arc graphs, complements of $d$-degenerate graphs, and $H$-graphs if given an $H$-representation. To supplement these findings, we show that many classes of (distance) $(σ, ρ)$-problems are W[1]-hard parameterized by mim-width + solution size.

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arXiv2018

Chi-boundedness of graph classes excluding wheel vertex-minors

A class of graphs is $χ$-bounded if there exists a function $f:\mathbb N\rightarrow \mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less than or equal to $f(q)$. We denote by $W_n$ the wheel graph on $n+1$ vertices. We show that the class of graphs having no vertex-minor isomorphic to $W_n$ is $χ$-bounded. This generalizes several previous results; $χ$-boundedness for circle graphs, for graphs having no $W_5$ vertex-minors, and for graphs having no fan vertex-minors.

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arXiv2018

A unified polynomial-time algorithm for Feedback Vertex Set on graphs of bounded mim-width

We give a first polynomial-time algorithm for (Weighted) Feedback Vertex Set on graphs of bounded maximum induced matching width (mim-width). Explicitly, given a branch decomposition of mim-width $w$, we give an $n^{\mathcal{O}(w)}$-time algorithm that solves Feedback Vertex Set. This provides a unified algorithm for many well-known classes, such as Interval graphs and Permutation graphs, and furthermore, it gives the first polynomial-time algorithms for other classes of bounded mim-width, such as Circular Permutation and Circular $k$-Trapezoid graphs for fixed $k$. In all these classes the decomposition is computable in polynomial time, as shown by Belmonte and Vatshelle [Theor. Comput. Sci. 2013]. We show that powers of graphs of tree-width $w - 1$ or path-width $w$ and powers of graphs of clique-width $w$ have mim-width at most $w$. These results extensively provide new classes of bounded mim-width. We prove a slight strengthening of the first statement which implies that, surprisingly, Leaf Power graphs which are of importance in the field of phylogenetic studies have mim-width at most $1$. Given a tree decomposition of width $w-1$, a path decomposition of width $w$, or a clique-width $w$-expression of a graph, one can for any value of $k$ find a mim-width decomposition of its $k$-power in polynomial time, and apply our algorithm to solve Feedback Vertex Set on the $k$-power in time $n^{\mathcal{O}(w)}$. In contrast to Feedback Vertex Set, we show that Hamiltonian Cycle is NP-complete even on graphs of linear mim-width $1$, which further hints at the expressive power of the mim-width parameter.

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arXiv2017

A note on the complexity of Feedback Vertex Set parameterized by mim-width

We complement the recent algorithmic result that Feedback Vertex Set is XP-time solvable parameterized by the mim-width of a given branch decomposition of the input graph [3] by showing that the problem is W[1]-hard in this parameterization. The hardness holds even for linear mim-width, as well as for H-graphs, where the parameter is the number of edges in H. To obtain this result, we adapt a reduction due to Fomin, Golovach and Raymond [2], following the same line of reasoning but adding a new gadget.

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arXiv2017

Polynomial-time algorithms for the Longest Induced Path and Induced Disjoint Paths problems on graphs of bounded mim-width

We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give $n^{\mathcal{O}(w)}$-time algorithms on graphs of mim-width at most $w$, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and $H$-Induced Topological Minor for fixed $H$. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs, Convex graphs, $k$-Trapezoid, Circular $k$-Trapezoid, $k$-Polygon, Dilworth-$k$ and Co-$k$-Degenerate graphs for fixed $k$.

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arXiv2017

Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions

In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomial-time algorithm, Algorithmica 78(1):342--377, 2017], we presented a characterization of the linear rank-width of distance-hereditary graphs, from which we derived an algorithm to compute it in polynomial time. In this paper, we investigate structural properties of distance-hereditary graphs based on this characterization. First, we prove that for a fixed tree $T$, every distance-hereditary graph of sufficiently large linear rank-width contains a vertex-minor isomorphic to $T$. We extend this property to bigger graph classes, namely, classes of graphs whose prime induced subgraphs have bounded linear rank-width. Here, prime graphs are graphs containing no splits. We conjecture that for every tree $T$, every graph of sufficiently large linear rank-width contains a vertex-minor isomorphic to $T$. Our result implies that it is sufficient to prove this conjecture for prime graphs. For a class $Φ$ of graphs closed under taking vertex-minors, a graph $G$ is called a vertex-minor obstruction for $Φ$ if $G\notin Φ$ but all of its proper vertex-minors are contained in $Φ$. Secondly, we provide, for each $k\ge 2$, a set of distance-hereditary graphs that contains all distance-hereditary vertex-minor obstructions for graphs of linear rank-width at most $k$. Also, we give a simpler way to obtain the known vertex-minor obstructions for graphs of linear rank-width at most $1$.

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arXiv2017

A single-exponential fixed-parameter algorithm for Distance-Hereditary Vertex Deletion

Vertex deletion problems ask whether it is possible to delete at most $k$ vertices from a graph so that the resulting graph belongs to a specified graph class. Over the past years, the parameterized complexity of vertex deletion to a plethora of graph classes has been systematically researched. Here we present the first single-exponential fixed-parameter tractable algorithm for vertex deletion to distance-hereditary graphs, a well-studied graph class which is particularly important in the context of vertex deletion due to its connection to the graph parameter rank-width. We complement our result with matching asymptotic lower bounds based on the exponential time hypothesis. As an application of our algorithm, we show that a vertex deletion set to distance-hereditary graphs can be used as a parameter which allows single-exponential fixed-parameter tractable algorithms for classical NP-hard problems.

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arXiv2017

A width parameter useful for chordal and co-comparability graphs

We investigate new graph classes of bounded mim-width, strictly extending interval graphs and permutation graphs. The graphs $K_t \boxminus K_t$ and $K_t \boxminus S_t$ are graphs obtained from the disjoint union of two cliques of size $t$, and one clique of size $t$ and one independent set of size $t$ respectively, by adding a perfect matching. We prove that : (1) interval graphs are $(K_3\boxminus S_3)$-free chordal graphs; and $(K_t\boxminus S_t)$-free chordal graphs have mim-width at most $t-1$, (2) permutation graphs are $(K_3\boxminus K_3)$-free co-comparability graphs; and $(K_t\boxminus K_t)$-free co-comparability graphs have mim-width at most $t-1$, (3) chordal graphs and co-comparability graphs have unbounded mim-width in general. We obtain several algorithmic consequences; for instance, while Minimum Dominating Set is NP-complete on chordal graphs, it can be solved in time $n^{\mathcal{O}(t)}$ on $(K_t\boxminus S_t)$-free chordal graphs. The third statement strengthens a result of Belmonte and Vatshelle stating that either those classes do not have constant mim-width or a decomposition with constant mim-width cannot be computed in polynomial time unless $P=NP$. We generalize these ideas to bigger graph classes. We introduce a new width parameter sim-width, of stronger modelling power than mim-width, by making a small change in the definition of mim-width. We prove that chordal graphs and co-comparability graphs have sim-width at most 1. We investigate a way to bound mim-width for graphs of bounded sim-width by excluding $K_t\boxminus K_t$ and $K_t\boxminus S_t$ as induced minors or induced subgraphs, and give algorithmic consequences. Lastly, we show that circle graphs have unbounded sim-width, and thus also unbounded mim-width.

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arXiv2017

A Polynomial Kernel for Distance-Hereditary Vertex Deletion

A graph is distance-hereditary if for any pair of vertices, their distance in every connected induced subgraph containing both vertices is the same as their distance in the original graph. The Distance-Hereditary Vertex Deletion problem asks, given a graph $G$ on $n$ vertices and an integer $k$, whether there is a set $S$ of at most $k$ vertices in $G$ such that $G-S$ is distance-hereditary. This problem is important due to its connection to the graph parameter rank-width that distance-hereditary graphs are exactly graphs of rank-width at most $1$. Eiben, Ganian, and Kwon (MFCS' 16) proved that Distance-Hereditary Vertex Deletion can be solved in time $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$, and asked whether it admits a polynomial kernelization. We show that this problem admits a polynomial kernel, answering this question positively. For this, we use a similar idea for obtaining an approximate solution for Chordal Vertex Deletion due to Jansen and Pilipczuk (SODA' 17) to obtain an approximate solution with $\mathcal{O}(k^3\log n)$ vertices when the problem is a YES-instance, and we exploit the structure of split decompositions of distance-hereditary graphs to reduce the total size.

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arXiv2016

Neighborhood complexity and kernelization for nowhere dense classes of graphs

We prove that whenever $G$ is a graph from a nowhere dense graph class $\mathcal{C}$, and $A$ is a subset of vertices of $G$, then the number of subsets of $A$ that are realized as intersections of $A$ with $r$-neighborhoods of vertices of $G$ is at most $f(r,ε)\cdot |A|^{1+ε}$, where $r$ is any positive integer, $ε$ is any positive real, and $f$ is a function that depends only on the class $\mathcal{C}$. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by Reidl et al. As an algorithmic application of the above result, we show that for every fixed $r$, the parameterized Distance-$r$ Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by Drange et al., and shows that the limit of parameterized tractability of Distance-$r$ Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.

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arXiv2016

Parameterized vertex deletion problems for hereditary graph classes with a block property

For a class of graphs $\mathcal{P}$, the Bounded $\mathcal{P}$-Block Vertex Deletion problem asks, given a graph $G$ on $n$ vertices and positive integers $k$ and $d$, whether there is a set $S$ of at most $k$ vertices such that each block of $G-S$ has at most $d$ vertices and is in $\mathcal{P}$. We show that when $\mathcal{P}$ satisfies a natural hereditary property and is recognizable in polynomial time, Bounded $\mathcal{P}$-Block Vertex Deletion can be solved in time $2^{O(k \log d)}n^{O(1)}$. When $\mathcal{P}$ contains all split graphs, we show that this running time is essentially optimal unless the Exponential Time Hypothesis fails. On the other hand, if $\mathcal{P}$ consists of only complete graphs, or only cycle graphs and $K_2$, then Bounded $\mathcal{P}$-Block Vertex Deletion admits a $c^{k}n^{O(1)}$-time algorithm for some constant $c$ independent of $d$. We also show that Bounded $\mathcal{P}$-Block Vertex Deletion admits a kernel with $O(k^2 d^7)$ vertices.

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