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...s on degenerate graphs [13]. 2 Preliminaries Parameterized complexity and kernels. A parameterized problem Q is a subset of Σ∗×N. The second component of a tuple (x, k) ∈ Σ∗×N is called the parameter =-=[16,18]-=-. The set {1, 2, . . . , n} is abbreviated as [n]. For a finite set X and integer i we use ( X i ) to denote the collection of size-i subsets of X. Definition 1 (Generalized kernelization). Let Q,Q′ ⊆...

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...stances of an NP-complete graph problem into a Treewidth instance with O(n√t) vertices. The construction is a combination of three ingredients. We carefully inspect the properties of Arnborg et al.’s =-=[1]-=- NPcompleteness proof for Treewidth to obtain an NP-complete source problem called Cobipartite Graph Elimination that is amenable to composition. Its instances have a restricted form that ensures that...

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...adding edges between u and NG(v) while removing v. A graph H is a minor of a graph G, if H can be obtained from a subgraph of G by edge contractions. Treewidth and Elimination Orders. While treewidth =-=[2]-=- is commonly defined in terms of tree decompositions, for our purposes it is more convenient to 4 work with an alternative characterization in terms of elimination orders. Eliminating a vertex v in a ...

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...erally, preprocessing, is a vital step in speeding up resource-demanding computations in practical settings. In the context of theoretical analysis, the sparsification lemma due to Impagliazzo et al. =-=[21]-=- has proven to be an important asset for studying subexponentialtime algorithms. The work of Dell and van Melkebeek [15] on sparsification for Satisfiability has led to important advances in the area ...

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...Grant 306992 “Parameterized Approximation”. ar X iv :1 30 8. 36 65 v1s[ cs .C C]s1 6 A ugs20 13 Preprocessing procedures for Treewidth have been studied in applied [10,11,26] and theoretical settings =-=[3,7]-=-. A team including the current author obtained [7] a polynomial-time algorithm that takes an instance (G, k) of Treewidth, and produces in polynomial time a graph G′ such that tw(G) ≤ k if and only if...

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...algorithm that reduces Treewidth instances on n vertices to equivalent instances of an arbitrary problem, with O(n2−) bits, for some > 0, then NP ⊆ coNP/poly and the polynomial hierarchy collapses =-=[27]-=-. We prove this result by giving a particularly efficient form of or-cross-composition [9]. We embed the or of t n-vertex instances of an NP-complete graph problem into a Treewidth instance with O(n√t...

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...per and lower bounds on kernel sizes for Treewidth [vc]. Our new reduction rule for Treewidth [vc] relates to the old rules like the crown-rule for k-Vertex Cover relates to the high-degree Buss-rule =-=[12]-=-: by exploiting local optimality considerations, our reduction rule does not need to know the value of k. Related work. While there is an abundance of superpolynomial kernel lower bounds, few superlin...

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...ext of theoretical analysis, the sparsification lemma due to Impagliazzo et al. [21] has proven to be an important asset for studying subexponentialtime algorithms. The work of Dell and van Melkebeek =-=[15]-=- on sparsification for Satisfiability has led to important advances in the area of kernelization lower bounds. They proved that for all > 0 and q ≥ 3, assuming NP 6⊆ coNP/poly, there is no polynomia...

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...f obtaining kernels of bitsize O(k2−) for such parameterized problems, assuming NP 6⊆ coNP/poly. The kernel for Treewidth parameterized by vertex cover (Treewidth [vc]) obtained by Bodlaender et al. =-=[6]-=- contains O(vc3) vertices, and therefore has bitsize Ω(vc4). Motivated by the impossibility of obtaining kernels with O(vc2−) bits, and with the aim of developing new reduction rules that are useful ...

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...s whose elimination from the graph has a predictable effect on its treewidth. While finding such sets seems to be hard in general, we show that the q-expansion lemma, previously employed by Thomassé =-=[25]-=- and Fomin et al. [19], can be used to find them when the graph is large with respect to its vertex cover number. The resulting kernel shrinks Treewidth instances to O(vc2) vertices, allowing them to ...

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... work was supported by ERC Starting Grant 306992 “Parameterized Approximation”. ar X iv :1 30 8. 36 65 v1s[ cs .C C]s1 6 A ugs20 13 Preprocessing procedures for Treewidth have been studied in applied =-=[10,11,26]-=- and theoretical settings [3,7]. A team including the current author obtained [7] a polynomial-time algorithm that takes an instance (G, k) of Treewidth, and produces in polynomial time a graph G′ suc...

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...nstances have a restricted form that ensures that good solutions to the composed Treewidth instance cannot be obtained by combining partial solutions to two different inputs. Then, like Dell and Marx =-=[14]-=-, we use the layout of a 2 × √t table to embed t instances into a graph on O(nO(1)√t) vertices. For each way of choosing a cell in the top and bottom row, we embed one instance into the edge set induc...

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...spectively feedback vertex) number. On the other hand, the natural parameterization of Treewidth is trivially and-compositional, and therefore does not admit a polynomial kernel unless NP ⊆ coNP/poly =-=[3,17]-=-. These results give an indication of how far the vertex count of a Treewidth instance can efficiently be reduced in terms of various measures of its complexity. However, they do not tell us anything ...

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...n abundance of superpolynomial kernel lower bounds, few superlinear lower bounds are known for problems admitting polynomial kernels. There are results for hitting set problems [15], packing problems =-=[14,20]-=-, and for domination problems on degenerate graphs [13]. 2 Preliminaries Parameterized complexity and kernels. A parameterized problem Q is a subset of Σ∗×N. The second component of a tuple (x, k) ∈ Σ...

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...Grant 306992 “Parameterized Approximation”. ar X iv :1 30 8. 36 65 v1s[ cs .C C]s1 6 A ugs20 13 Preprocessing procedures for Treewidth have been studied in applied [10,11,26] and theoretical settings =-=[3,7]-=-. A team including the current author obtained [7] a polynomial-time algorithm that takes an instance (G, k) of Treewidth, and produces in polynomial time a graph G′ such that tw(G) ≤ k if and only if...

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... work was supported by ERC Starting Grant 306992 “Parameterized Approximation”. ar X iv :1 30 8. 36 65 v1s[ cs .C C]s1 6 A ugs20 13 Preprocessing procedures for Treewidth have been studied in applied =-=[10,11,26]-=- and theoretical settings [3,7]. A team including the current author obtained [7] a polynomial-time algorithm that takes an instance (G, k) of Treewidth, and produces in polynomial time a graph G′ suc...

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...trary problem, with O(n2−) bits, for some > 0, then NP ⊆ coNP/poly and the polynomial hierarchy collapses [27]. We prove this result by giving a particularly efficient form of or-cross-composition =-=[9]-=-. We embed the or of t n-vertex instances of an NP-complete graph problem into a Treewidth instance with O(n√t) vertices. The construction is a combination of three ingredients. We carefully inspect t...

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...mbines these three ingredients with an intricate analysis of the behavior of elimination orders on the constructed instance. As the treewidth of the constructed cobipartite graph equals its pathwidth =-=[23]-=-, the obtained sparsification lower bound for Treewidth also applies to Pathwidth. 2 Our sparsification lower bound has immediate consequences for parameterizations of Treewidth by graph parameters th...

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...om the graph has a predictable effect on its treewidth. While finding such sets seems to be hard in general, we show that the q-expansion lemma, previously employed by Thomassé [25] and Fomin et al. =-=[19]-=-, can be used to find them when the graph is large with respect to its vertex cover number. The resulting kernel shrinks Treewidth instances to O(vc2) vertices, allowing them to be encoded in O(vc3) b...

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... work was supported by ERC Starting Grant 306992 “Parameterized Approximation”. ar X iv :1 30 8. 36 65 v1s[ cs .C C]s1 6 A ugs20 13 Preprocessing procedures for Treewidth have been studied in applied =-=[10,11,26]-=- and theoretical settings [3,7]. A team including the current author obtained [7] a polynomial-time algorithm that takes an instance (G, k) of Treewidth, and produces in polynomial time a graph G′ suc...

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...perlinear lower bounds are known for problems admitting polynomial kernels. There are results for hitting set problems [15], packing problems [14,20], and for domination problems on degenerate graphs =-=[13]-=-. 2 Preliminaries Parameterized complexity and kernels. A parameterized problem Q is a subset of Σ∗×N. The second component of a tuple (x, k) ∈ Σ∗×N is called the parameter [16,18]. The set {1, 2, . ....

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... each way of choosing a cell in the top and bottom row, we embed one instance into the edge set induced by the vertices representing the two cells. Finally, we use ideas employed by Bodlaender et al. =-=[8]-=- in the superpolynomial lower bound for Treewidth parameterized by the vertex-deletion distance to a clique: we compose the input instances of Cobipartite Graph Elimination into a cobipartite graph to...