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19
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 76 (16 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 56 (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.
Incompressibility through Colors and IDs
"... In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown t ..."
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Cited by 46 (5 self)
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In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the nonexistence of polynomial kernels for specific problems has been developed by Bodlaender et al. [6] and Fortnow and Santhanam [15]. With few exceptions, all known kernelization lower bounds result have been obtained by directly applying this framework. In this paper we show how to combine these results with combinatorial reductions which use colors and IDs in order to prove kernelization lower bounds for a variety of basic problems. Below we give a summary of our main results. All our results are under the assumption that the polynomial hierarchy does not collapse to the third level. • We show that the Steiner Tree problem parameterized by the number of terminals and solution size, and the Connected Vertex Cover and Capacitated Vertex Cover problems do not admit a polynomial kernel. The two latter results are surprising because the closely related Vertex Cover problem admits a kernel of size 2k.
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 11 (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
FPT Algorithms and Kernels for the Directed kLeaf Problem
, 2008
"... A subgraph T of a digraph D is an outbranching if T is an oriented spanning tree with only one vertex of indegree zero (called the root). The vertices of T of outdegree zero are leaves. In the Directed kLeaf Problem, we are given a digraph D and an integral parameter k, and we are to decide whet ..."
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Cited by 9 (1 self)
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A subgraph T of a digraph D is an outbranching if T is an oriented spanning tree with only one vertex of indegree zero (called the root). The vertices of T of outdegree zero are leaves. In the Directed kLeaf Problem, we are given a digraph D and an integral parameter k, and we are to decide whether D has an outbranching with at least k leaves. Recently, Kneis et al. (2008) obtained an algorithm for the problem of running time 4 k · n O(1). We describe a new algorithm for the problem of running time 3.72 k · n O(1). In Rooted Directed kLeaf Problem, apart from D and k, we are given a vertex r of D and we are to decide whether D has an outbranching rooted at r with at least k leaves. Very recently, Fernau et al. (2008) found an O(k 3)size kernel for Rooted Directed kLeaf. In this paper, we obtain an O(k) kernel for Rooted Directed kLeaf restricted to acyclic digraphs.
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.
Kernelization  Preprocessing with A Guarantee
"... Data reduction techniques are widely applied to deal with computationally hard problems in real world applications. It has been a longstanding challenge to formally express the efficiency and accuracy of these “preprocessing” procedures. The framework of parameterized complexity turns out to be ..."
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Cited by 8 (1 self)
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Data reduction techniques are widely applied to deal with computationally hard problems in real world applications. It has been a longstanding challenge to formally express the efficiency and accuracy of these “preprocessing” procedures. The framework of parameterized complexity turns out to be particularly suitable for a mathematical analysis of preprocessing heuristics. A kernelization algorithm is a preprocessing algorithm which simplifies the instances given as input in polynomial time, and the extent of simplification desired is quantified with the help of the additional parameter. We give an overview of some of the early work in the area and also survey newer techniques that have emerged in the design and analysis of kernelization algorithms. We also outline the framework of Bodlaender et al. [9] and Fortnow and Santhanam [38] for showing kernelization lower bounds under reasonable assumptions from classical complexity theory, and highlight some of the recent results that strengthen and generalize this framework.
Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs
"... In 2000 Alber et al. [SWAT 2000] obtained the first parameterized subexponential algorithm on undirected planar graphs by showing that kDOMINATING SET is solvable in time 2 O( √ k) ..."
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Cited by 7 (6 self)
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In 2000 Alber et al. [SWAT 2000] obtained the first parameterized subexponential algorithm on undirected planar graphs by showing that kDOMINATING SET is solvable in time 2 O( √ k)
Kernel(s) for Problems With No Kernel: On OutTrees With Many Leaves
, 2011
"... The kLEAF OUTBRANCHING problem is to find an outbranching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the kLEA ..."
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Cited by 5 (1 self)
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The kLEAF OUTBRANCHING problem is to find an outbranching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the kLEAFOUTBRANCHING problem. We give the first polynomial kernel for ROOTED kLEAFOUTBRANCHING, a variant of kLEAFOUTBRANCHING where the root of the tree searched for is also a part of the input. Our kernel with O(k 3) vertices is obtained using extremal combinatorics. For the kLEAFOUTBRANCHING problem, we show that no polynomialsized kernel is possible unless coNP is in NP/poly. However, our positive results for ROOTED kLEAFOUTBRANCHING immediately imply that the seemingly intractable kLEAFOUTBRANCHING problem admits a data reduction to n independent polynomialsized kernels. These two results, tractability and intractability side by side, are the first ones separating Karp kernelization from Turing kernelization. This answers affirmatively an open problem