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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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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.
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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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.
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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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. 1
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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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)
On the Directed DegreePreserving Spanning Tree Problem
"... Abstract. In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree with at most k vertices of reduced outdegree. This problem is a directed analog of the wellstudied Mi ..."
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Abstract. In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree with at most k vertices of reduced outdegree. This problem is a directed analog of the wellstudied MinimumVertex Feedback Edge Set problem. We show that this problem is fixedparameter tractable and admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k 2) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with runtime O ∗ (5.942 k). This adds the Reduced Degree Spanning Tree problem to the small list of directed graph problems for which fixedparameter tractable algorithms are known. Finally, we consider the dual of Reduced Degree Spanning Tree, that is, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree of D with at least k vertices of full outdegree. We show that this problem is W[1]hard on two important digraph classes: directed acyclic graphs and strongly connected digraphs. 1
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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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
Note on Maximal Bisection above Tight Lower Bound
"... In a graph G = (V, E), a bisection (X, Y) is a partition of V into sets X and Y such that X  ≤ Y  ≤ X+1. The size of (X, Y) is the number of edges between X and Y. In the Max Bisection problem we are given a graph G = (V, E) and are required to find a bisection of maximum size. It is not har ..."
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In a graph G = (V, E), a bisection (X, Y) is a partition of V into sets X and Y such that X  ≤ Y  ≤ X+1. The size of (X, Y) is the number of edges between X and Y. In the Max Bisection problem we are given a graph G = (V, E) and are required to find a bisection of maximum size. It is not hard to see that ⌈E/2 ⌉ is a tight lower bound on the maximum size of a bisection of G. We study parameterized complexity of the following parameterized problem called Max Bisection above Tight Lower Bound (MaxBisecATLB): decide whether a graph G = (V, E) has a bisection of size at least ⌈E/2 ⌉ + k, where k is the parameter. We show that this parameterized problem has a kernel with O(k 2) vertices and O(k 3) edges, i.e., every instance of MaxBisecATLB is equivalent to an instance of MaxBisecATLB on a graph with at most O(k 2) vertices and O(k 3) edges. 1
BEYOND BIDIMENSIONALITY: PARAMETERIZED SUBEXPONENTIAL ALGORITHMS ON DIRECTED GRAPHS
, 2010
"... Abstract. In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approa ..."
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Abstract. In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, kLeaf OutBranching, which is to find an oriented spanning tree with at least k leaves, we obtain an algorithm solving the problem in time 2 O( √ k log k) n + n O(1) on directed graphs whose underlying undirected graph excludes some fixed graph H as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2 O( √ k) n+n O(1). The second example is a generalization of the Directed Hamiltonian Path problem, namely kInternal OutBranching, which is to find an oriented spanning tree with at least k internal vertices. We obtain an algorithm solving the problem in time 2 O( √ k log k) + n O(1) on directed graphs whose underlying undirected graph excludes some fixed apex graph H as a minor. Finally, we observe that for any ε> 0, the kDirected Path problem is solvable in time O((1+ε) k n f(ε)), where f is some function of ε. Our methods are based on nontrivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs. 1.
BEYOND BIDIMENSIONALITY: PARAMETERIZED SUBEXPONENTIAL ALGORITHMS ON DIRECTED GRAPHS
"... Abstract. In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approa ..."
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Abstract. In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, kLeaf OutBranching, which is to find an oriented spanning tree with at least k leaves, we obtain an algorithm solving the problem in time 2 O( √ k log k) n + n O(1) on directed graphs whose underlying undirected graph excludes some fixed graph H as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2 O( √ k) n+n O(1). The second example is a generalization of the Directed Hamiltonian Path problem, namely kInternal OutBranching, which is to find an oriented spanning tree with at least k internal vertices. We obtain an algorithm solving the problem in time 2 O( √ k log k) + n O(1) on directed graphs whose underlying undirected graph excludes some fixed apex graph H as a minor. Finally, we observe that for any ε> 0, the kDirected Path problem is solvable in time O((1+ε) k n f(ε)), where f is some function of ε. Our methods are based on nontrivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs. 1.