arXiv subjects

Eun Jung Kim

Publications and source records attributed to Eun Jung Kim.

At least 55 records · Page 3Linked to original sources

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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arXiv2017

Complexity and Approximability of Parameterized MAX-CSPs

We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable-constraint incidence graph of the CSP instance. We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k, and we attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPTAS), which computes a $(1-ε)$-approximation in time $f(k,ε)\cdot poly(n)$. 3. There is no FPTAS unless FPT=W[1]. For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results.

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arXiv2016

A polynomial kernel for Block Graph Deletion

In the Block Graph Deletion problem, we are given a graph $G$ on $n$ vertices and a positive integer $k$, and the objective is to check whether it is possible to delete at most $k$ vertices from $G$ to make it a block graph, i.e., a graph in which each block is a clique. In this paper, we obtain a kernel with $\mathcal{O}(k^{6})$ vertices for the Block Graph Deletion problem. This is a first step to investigate polynomial kernels for deletion problems into non-trivial classes of graphs of bounded rank-width, but unbounded tree-width. Our result also implies that Chordal Vertex Deletion admits a polynomial-size kernel on diamond-free graphs. For the kernelization and its analysis, we introduce the notion of `complete degree' of a vertex. We believe that the underlying idea can be potentially applied to other problems. We also prove that the Block Graph Deletion problem can be solved in time $10^{k}\cdot n^{\mathcal{O}(1)}$.

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arXiv2016

An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514--528, 2006]. Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an $n$-vertex graph $G$ and a positive integer $k$, we want to decide whether there is a set of at most $k$ vertices whose removal turns $G$ into a graph of linear rankwidth at most $1$ and find such a vertex set if one exists. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time $f(k)\cdot n^3$ for some function $f$, it is not clear whether this problem allows a running time with a modest exponential function. We first establish that LRW1-Vertex Deletion can be solved in time $8^k\cdot n^{\mathcal{O}(1)}$. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define necklace graphs and investigate their structural properties. Later, we reduce the polynomial factor by refining the trivial branching step based on a cliquewidth expression of a graph, and obtain an algorithm that runs in time $2^{\mathcal{O}(k)}\cdot n^4$. We also prove that the running time cannot be improved to $2^{o(k)}\cdot n^{\mathcal{O}(1)}$ under the Exponential Time Hypothesis assumption. Lastly, we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.

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arXiv2015

Complexity of Grundy coloring and its variants

The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least $k$. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time $O^*(2.443^n)$ and WEAK GRUNDY COLORING in time $O^*(2.716^n)$ on graphs of order $n$. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time $O^*(2^{O(wk)})$ for graphs of treewidth $w$ (where $k$ is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time $O^*(2^{o(w\log w)})$. We also describe an $O^*(2^{2^{O(k)}})$ algorithm for WEAK GRUNDY COLORING, which is therefore $\fpt$ for the parameter $k$. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for GRUNDY COLORING). Although we do not know whether GRUNDY COLORING is in $\fpt$, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with the two other problems, we show that CONNECTED GRUNDY COLORING is $\np$-complete already for $k=7$ colors.

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arXiv2015

Recognizing k-equistable graphs in FPT time

A graph $G = (V,E)$ is called equistable if there exist a positive integer $t$ and a weight function $w : V \to \mathbb{N}$ such that $S \subseteq V$ is a maximal stable set of $G$ if and only if $w(S) = t$. Such a function $w$ is called an equistable function of $G$. For a positive integer $k$, a graph $G = (V,E)$ is said to be $k$-equistable if it admits an equistable function which is bounded by $k$. We prove that the problem of recognizing $k$-equistable graphs is fixed parameter tractable when parameterized by $k$, affirmatively answering a question of Levit et al. In fact, the problem admits an $O(k^5)$-vertex kernel that can be computed in linear time.

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arXiv2014

A Polynomial-time Algorithm for Outerplanar Diameter Improvement

The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an integer $D$, whether it is possible to add edges to $G$ in a way that the resulting graph is outerplanar and has diameter at most $D$. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open.

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arXiv2014

The Complexity of Repairing, Adjusting, and Aggregating of Extensions in Abstract Argumentation

We study the computational complexity of problems that arise in abstract argumentation in the context of dynamic argumentation, minimal change, and aggregation. In particular, we consider the following problems where always an argumentation framework F and a small positive integer k are given. - The Repair problem asks whether a given set of arguments can be modified into an extension by at most k elementary changes (i.e., the extension is of distance k from the given set). - The Adjust problem asks whether a given extension can be modified by at most k elementary changes into an extension that contains a specified argument. - The Center problem asks whether, given two extensions of distance k, whether there is a "center" extension that is a distance at most (k-1) from both given extensions. We study these problems in the framework of parameterized complexity, and take the distance k as the parameter. Our results covers several different semantics, including admissible, complete, preferred, semi-stable and stable semantics.

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arXiv2013

Betweenness Parameterized Above Tight Lower Bound

We study ordinal embedding relaxations in the realm of parameterized complexity. We prove the existence of a quadratic kernel for the {\sc Betweenness} problem parameterized above its tight lower bound, which is stated as follows. For a set $V$ of variables and set $\mathcal C$ of constraints "$v_i$ \mbox{is between} $v_j$ \mbox{and} $v_k$", decide whether there is a bijection from $V$ to the set $\{1,\ldots,|V|\}$ satisfying at least $|\mathcal C|/3 + κ$ of the constraints in $\mathcal C$. Our result solves an open problem attributed to Benny Chor in Niedermeier's monograph "Invitation to Fixed-Parameter Algorithms." The betweenness problem is of interest in molecular biology. An approach developed in this paper can be used to determine parameterized complexity of a number of other optimization problems on permutations parameterized above or below tight bounds.

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arXiv2012

Don't Be Strict in Local Search!

Local Search is one of the fundamental approaches to combinatorial optimization and it is used throughout AI. Several local search algorithms are based on searching the k-exchange neighborhood. This is the set of solutions that can be obtained from the current solution by exchanging at most k elements. As a rule of thumb, the larger k is, the better are the chances of finding an improved solution. However, for inputs of size n, a naïve brute-force search of the k-exchange neighborhood requires n to the power of O(k) time, which is not practical even for very small values of k. Fellows et al. (IJCAI 2009) studied whether this brute-force search is avoidable and gave positive and negative answers for several combinatorial problems. They used the notion of local search in a strict sense. That is, an improved solution needs to be found in the k-exchange neighborhood even if a global optimum can be found efficiently. In this paper we consider a natural relaxation of local search, called permissive local search (Marx and Schlotter, IWPEC 2009) and investigate whether it enhances the domain of tractable inputs. We exemplify this approach on a fundamental combinatorial problem, Vertex Cover. More precisely, we show that for a class of inputs, finding an optimum is hard, strict local search is hard, but permissive local search is tractable. We carry out this investigation in the framework of parameterized complexity.

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arXiv2012

Linear kernels and single-exponential algorithms via protrusion decompositions

A \emph{$t$-treewidth-modulator} of a graph $G$ is a set $X \subseteq V(G)$ such that the treewidth of $G-X$ is at most some constant $t-1$. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs $G$ that come equipped with a $t$-treewidth-modulator. This decomposition, called a \emph{protrusion decomposition}, is the cornerstone in obtaining the following two main results. We first show that any parameterized graph problem (with parameter $k$) that has \emph{finite integer index} and is \emph{treewidth-bounding} admits a linear kernel on $H$-topological-minor-free graphs, where $H$ is some arbitrary but fixed graph. A parameterized graph problem is called treewidth-bounding if all positive instances have a $t$-treewidth-modulator of size $O(k)$, for some constant $t$. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus [Bodlaender et al., FOCS 2009] and $H$-minor-free graphs [Fomin et al., SODA 2010]. Our second application concerns the Planar-$\mathcal{F}$-Deletion problem. Let $\mathcal{F}$ be a fixed finite family of graphs containing at least one planar graph. Given an $n$-vertex graph $G$ and a non-negative integer $k$, Planar-$\mathcal{F}$-Deletion asks whether $G$ has a set $X\subseteq V(G)$ such that $|X|\leq k$ and $G-X$ is $H$-minor-free for every $H\in \mathcal{F}$. Very recently, an algorithm for Planar-$\mathcal{F}$-Deletion with running time $2^{O(k)} n \log^2 n$ (such an algorithm is called \emph{single-exponential}) has been presented in [Fomin et al., FOCS 2012] under the condition that every graph in $\mathcal{F}$ is connected. Using our algorithm to construct protrusion decompositions as a building block, we get rid of this connectivity constraint and present an algorithm for the general Planar-$\mathcal{F}$-Deletion problem running in time $2^{O(k)} n^2$.

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arXiv2012

A single-exponential FPT algorithm for the $K_4$-minor cover problem

Given an input graph G and an integer k, the parameterized K_4-minor cover problem asks whether there is a set S of at most k vertices whose deletion results in a K_4-minor-free graph, or equivalently in a graph of treewidth at most 2. This problem is inspired by two well-studied parameterized vertex deletion problems, Vertex Cover and Feedback Vertex Set, which can also be expressed as Treewidth-t Vertex Deletion problems: t=0 for Vertex Cover and t=1 for Feedback Vertex Set. While a single-exponential FPT algorithm has been known for a long time for \textsc{Vertex Cover}, such an algorithm for Feedback Vertex Set was devised comparatively recently. While it is known to be unlikely that Treewidth-t Vertex Deletion can be solved in time c^{o(k)}.n^{O(1)}, it was open whether the K_4-minor cover problem could be solved in single-exponential FPT time, i.e. in c^k.n^{O(1)} time. This paper answers this question in the affirmative.

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arXiv2012

On Exact Algorithms for Permutation CSP

In the Permutation Constraint Satisfaction Problem (Permutation CSP) we are given a set of variables $V$ and a set of constraints C, in which constraints are tuples of elements of V. The goal is to find a total ordering of the variables, $π : V \rightarrow [1,...,|V|]$, which satisfies as many constraints as possible. A constraint $(v_1,v_2,...,v_k)$ is satisfied by an ordering $π$ when $π(v_1)<π(v_2)<...<π(v_k)$. An instance has arity $k$ if all the constraints involve at most $k$ elements. This problem expresses a variety of permutation problems including {\sc Feedback Arc Set} and {\sc Betweenness} problems. A naive algorithm, listing all the $n!$ permutations, requires $2^{O(n\log{n})}$ time. Interestingly, {\sc Permutation CSP} for arity 2 or 3 can be solved by Held-Karp type algorithms in time $O^*(2^n)$, but no algorithm is known for arity at least 4 with running time significantly better than $2^{O(n\log{n})}$. In this paper we resolve the gap by showing that {\sc Arity 4 Permutation CSP} cannot be solved in time $2^{o(n\log{n})}$ unless ETH fails.

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arXiv2011

Improved Parameterized Algorithms for Constraint Satisfaction

For many constraint satisfaction problems, the algorithm which chooses a random assignment achieves the best possible approximation ratio. For instance, a simple random assignment for {\sc Max-E3-Sat} allows 7/8-approximation and for every $\eps >0$ there is no polynomial-time ($7/8+\eps$)-approximation unless P=NP. Another example is the {\sc Permutation CSP} of bounded arity. Given the expected fraction $ρ$ of the constraints satisfied by a random assignment (i.e. permutation), there is no $(ρ+\eps)$-approximation algorithm for every $\eps >0$, assuming the Unique Games Conjecture (UGC). In this work, we consider the following parameterization of constraint satisfaction problems. Given a set of $m$ constraints of constant arity, can we satisfy at least $ρm +k$ constraint, where $ρ$ is the expected fraction of constraints satisfied by a random assignment? {\sc Constraint Satisfaction Problems above Average} have been posed in different forms in the literature \cite{Niedermeier2006,MahajanRamanSikdar09}. We present a faster parameterized algorithm for deciding whether $m/2+k/2$ equations can be simultaneously satisfied over ${\mathbb F}_2$. As a consequence, we obtain $O(k)$-variable bikernels for {\sc boolean CSPs} of arity $c$ for every fixed $c$, and for {\sc permutation CSPs} of arity 3. This implies linear bikernels for many problems under the "above average" parameterization, such as {\sc Max-$c$-Sat}, {\sc Set-Splitting}, {\sc Betweenness} and {\sc Max Acyclic Subgraph}. As a result, all the parameterized problems we consider in this paper admit $2^{O(k)}$-time algorithms. We also obtain non-trivial hybrid algorithms for every Max $c$-CSP: for every instance $I$, we can either approximate $I$ beyond the random assignment threshold in polynomial time, or we can find an optimal solution to $I$ in subexponential time.

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arXiv2011

Solving MAX-r-SAT Above a Tight Lower Bound

We present an exact algorithm that decides, for every fixed $r \geq 2$ in time $O(m) + 2^{O(k^2)}$ whether a given multiset of $m$ clauses of size $r$ admits a truth assignment that satisfies at least $((2^r-1)m+k)/2^r$ clauses. Thus \textsc{Max-$r$-Sat} is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound $(1-2^{-r})m$. This solves an open problem of Mahajan et al. (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with $O(k^2)$ variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size $O(k^2)$, then there is a truth assignment satisfying the required number of clauses. We introduce a new notion of bikernelization from a parameterized problem to another one and apply it to prove that the above-mentioned parameterized \textsc{Max-$r$-Sat} admits a polynomial-size kernel. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of \textsc{Max-2-Sat} with $m$ clauses has at least $3k$ variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least $(3m+k)/4$ clauses. We also outline how the fixed-parameter tractability and polynomial-size kernel results on \textsc{Max-$r$-Sat} can be extended to more general families of Boolean Constraint Satisfaction Problems.

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arXiv2011

Parameterized Complexity Results for General Factors in Bipartite Graphs with an Application to Constraint Programming

The NP-hard general factor problem asks, given a graph and for each vertex a list of integers, whether the graph has a spanning subgraph where each vertex has a degree that belongs to its assigned list. The problem remains NP-hard even if the given graph is bipartite with partition U+V, and each vertex in U is assigned the list {1}; this subproblem appears in the context of constraint programming as the consistency problem for the extended global cardinality constraint. We show that this subproblem is fixed-parameter tractable when parameterized by the size of the second partite set V. More generally, we show that the general factor problem for bipartite graphs, parameterized by |V|, is fixed-parameter tractable as long as all vertices in U are assigned lists of length 1, but becomes W[1]-hard if vertices in U are assigned lists of length at most 2. We establish fixed-parameter tractability by reducing the problem instance to a bounded number of acyclic instances, each of which can be solved in polynomial time by dynamic programming.

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arXiv2011

Algorithms and Complexity Results for Persuasive Argumentation

The study of arguments as abstract entities and their interaction as introduced by Dung (Artificial Intelligence 177, 1995) has become one of the most active research branches within Artificial Intelligence and Reasoning. A main issue for abstract argumentation systems is the selection of acceptable sets of arguments. Value-based argumentation, as introduced by Bench-Capon (J. Logic Comput. 13, 2003), extends Dung's framework. It takes into account the relative strength of arguments with respect to some ranking representing an audience: an argument is subjectively accepted if it is accepted with respect to some audience, it is objectively accepted if it is accepted with respect to all audiences. Deciding whether an argument is subjectively or objectively accepted, respectively, are computationally intractable problems. In fact, the problems remain intractable under structural restrictions that render the main computational problems for non-value-based argumentation systems tractable. In this paper we identify nontrivial classes of value-based argumentation systems for which the acceptance problems are polynomial-time tractable. The classes are defined by means of structural restrictions in terms of the underlying graphical structure of the value-based system. Furthermore we show that the acceptance problems are intractable for two classes of value-based systems that where conjectured to be tractable by Dunne (Artificial Intelligence 171, 2007).

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arXiv2009

Vertex Cover Problem Parameterized Above and Below Tight Bounds

We study the well-known Vertex Cover problem parameterized above and below tight bounds. We show that two of the parameterizations (both were suggested by Mahajan, Raman and Sikdar, J. Computer and System Sciences, 75(2):137--153, 2009) are fixed-parameter tractable and two other parameterizations are W[1]-hard (one of them is, in fact, W[2]-hard).

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