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45
Kernel bounds for disjoint cycles and disjoint paths
, 2009
"... In this paper, we give evidence for the problems Disjoint Cycles and Disjoint Paths that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a polynomial in a specified parameter (or, in short: do not have a polynomial kernel); these results ..."
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Cited by 71 (13 self)
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In this paper, we give evidence for the problems Disjoint Cycles and Disjoint Paths that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a polynomial in a specified parameter (or, in short: do not have a polynomial kernel); these results are assuming the validity of certain complexity theoretic assumptions. We build upon recent results by Bodlaender et al. [3] and Fortnow and Santhanam [13], that show that NPcomplete problems that are orcompositional do not have polynomial kernels, unless NP ⊆ coNP/poly. To this machinery, we add a notion of transformation, and thus obtain that Disjoint Cycles and Disjoint Paths do not have polynomial kernels, unless NP ⊆ coNP/poly. We also show that the related Disjoint Cycles Packing problem has a kernel of size O(k log k).
Satisfiability Allows No Nontrivial Sparsification Unless The PolynomialTime Hierarchy Collapses
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 38 (2010)
, 2010
"... Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small ..."
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Cited by 53 (2 self)
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Consider the following twoplayer communication process to decide a language L: The first player holds the entire input x but is polynomially bounded; the second player is computationally unbounded but does not know any part of x; their goal is to cooperatively decide whether x belongs to L at small cost, where the cost measure is the number of bits of communication from the first player to the second player. For any integer d ≥ 3 and positive real ǫ we show that if satisfiability for nvariable dCNF formulas has a protocol of cost O(n d−ǫ) then coNP is in NP/poly, which implies that the polynomialtime hierarchy collapses to its third level. The result even holds when the first player is conondeterministic, and is tight as there exists a trivial protocol for ǫ = 0. Under the hypothesis that coNP is not in NP/poly, our result implies tight lower bounds for parameters of interest in several areas, namely sparsification, kernelization in parameterized complexity, lossy compression, and probabilistically checkable proofs. By reduction, similar results hold for other NPcomplete problems. For the vertex cover problem on nvertex duniform hypergraphs, the above statement holds for any integer d ≥ 2. The case d = 2 implies that no NPhard vertex deletion problem based on a graph property that is inherited by subgraphs can have kernels consisting of O(k 2−ǫ) edges unless coNP is in NP/poly, where k denotes the size of the deletion set. Kernels consisting of O(k 2) edges are known for several problems in the class, including vertex cover, feedback vertex set, and boundeddegree deletion.
Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal
"... The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of ..."
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Cited by 19 (4 self)
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The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4 k kmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed k. It is known that this implies a polynomialtime compression algorithm that turns OCT instances into equivalent instances of size at most O(4 k), a socalled kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in k, has turned into one of the main open questions in the study of kernelization. Despite the impressive progress in the area, including the recent development of lower bound techniques (Bodlaender
Linear kernels and singleexponential algorithms via protrusion decompositions
, 2012
"... A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G−X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been explicitly or ..."
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Cited by 16 (5 self)
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A ttreewidthmodulator of a graph G is a set X ⊆ V (G) such that the treewidth of G−X is at most t − 1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a ttreewidthmodulator. Similar decompositions have already been explicitly or implicitly used for obtaining polynomial kernels [3, 7, 33, 43]. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and is treewidthbounding admits a linear kernel on the class of Htopologicalminorfree graphs, where H is some arbitrary but fixed graph. A parameterized graph problem is called treewidthbounding if all positive instances have a ttreewidthmodulator of size O(k), for some constant t. This result partially extends previous metatheorems on the existence of linear kernels on graphs of bounded genus [7] and Hminorfree graphs [37]. In particular, we show that Chordal Vertex Deletion, Interval Vertex Deletion, Treewidtht Vertex Deletion, and Edge Dominating Set have linear kernels on Htopologicalminorfree graphs.
Conondeterminism in compositions: A kernelization lower bound for a Ramseytype problem
, 2012
"... Until recently, techniques for obtaining lower bounds for kernelization were one of the most sought after tools in the field of parameterized complexity. Now, after a strong influx of techniques, we are in the fortunate situation of having tools available that are even stronger than what has been re ..."
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Cited by 10 (2 self)
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Until recently, techniques for obtaining lower bounds for kernelization were one of the most sought after tools in the field of parameterized complexity. Now, after a strong influx of techniques, we are in the fortunate situation of having tools available that are even stronger than what has been required in their applications so far. Based on a result of Fortnow and Santhanam (STOC 2008, JCSS 2011), Bodlaender et al. (ICALP 2008, JCSS 2009) showed that, unless NP ⊆ coNP/poly, the existence of a deterministic polynomialtime composition algorithm, i.e., an algorithm which outputs an instance of bounded parameter value which is yes if and only if one of t input instances is yes, rules out the existence of polynomial kernels for a problem. Dell and van Melkebeek (STOC 2010) continued this line
Preprocessing of Min Ones Problems: A Dichotomy
"... Min Ones Constraint Satisfaction Problems, i.e., the task of finding a satisfying assignment with at most k true variables (Min Ones SAT(Γ)), can express a number of interesting and natural problems. We study the preprocessing properties of this class of problems with respect to k, using the notion ..."
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Cited by 8 (3 self)
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Min Ones Constraint Satisfaction Problems, i.e., the task of finding a satisfying assignment with at most k true variables (Min Ones SAT(Γ)), can express a number of interesting and natural problems. We study the preprocessing properties of this class of problems with respect to k, using the notion of kernelization to capture the viability of preprocessing. We give a dichotomy of Min Ones SAT(Γ) problems into admitting or not admitting a kernelization with size guarantee polynomial in k, based on the constraint language Γ. We introduce a property of boolean relations called mergeability that can be easily checked for any Γ. When all relations in Γ are mergeable, then we show a polynomial kernelization for Min Ones SAT(Γ). Otherwise, any Γ containing a nonmergeable relation and such that Min Ones SAT(Γ) is NPcomplete permits us to prove that Min Ones SAT(Γ) does not admit a polynomial kernelization unless NP ⊆ coNP/poly, by a reduction from a particular parameterization of Exact Hitting Set.
Contracting graphs to paths and trees
"... Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a param ..."
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Cited by 8 (4 self)
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Vertex deletion and edge deletion problems play a central role in Parameterized Complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G and an integer k, and the task is to determine whether we can obtain an acyclic graph or a path, respectively, by a sequence of at most k edge contractions in G. We present an algorithm with running time 4.98 k n O(1) for Tree Contraction, based on a variant of the color coding technique of Alon, Yuster and Zwick, and an algorithm with running time 2 k+o(k) + n O(1) for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k + 3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising, because of the connection between Tree Contraction and Feedback Vertex Set, which is known to have a kernel with O(k²) vertices.
On Tractable Cases of Target Set Selection
"... We study the NPcomplete TARGET SET SELECTION (TSS) problem occurring in social network analysis. Complementing results on its approximability and extending results for its restriction to trees and bounded treewidth graphs, we classify the influence of the parameters “diameter”, “cluster edge delet ..."
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Cited by 7 (3 self)
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We study the NPcomplete TARGET SET SELECTION (TSS) problem occurring in social network analysis. Complementing results on its approximability and extending results for its restriction to trees and bounded treewidth graphs, we classify the influence of the parameters “diameter”, “cluster edge deletion number”, “vertex cover number”, and “feedback edge set number ” of the underlying graph on the problem’s complexity, revealing both tractable and intractable cases. For instance, even for diametertwo split graphs TSS remains very hard. TSS can be efficiently solved on graphs with small feedback edge set number and also turns out to be fixedparameter tractable when parameterized by the vertex cover number, both results contrasting known parameterized intractability results for the parameter treewidth. While these tractability results are relevant for sparse networks, we also show efficient fixedparameter algorithms for the parameter cluster edge deletion number, yielding tractability for certain dense networks.