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12
Limits and applications of group algebras for parameterized problems
 In Automata, Languages and Programming: ThirtySixth International Colloquium (ICALP
, 2009
"... The algebraic framework introduced in [Koutis, Proc. of the 35 th ICALP 2008] reduces several combinatorial problems in parameterized complexity to the problem of detecting multilinear degreek monomials in polynomials presented as circuits. The best known (randomized) algorithm for this problem req ..."
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The algebraic framework introduced in [Koutis, Proc. of the 35 th ICALP 2008] reduces several combinatorial problems in parameterized complexity to the problem of detecting multilinear degreek monomials in polynomials presented as circuits. The best known (randomized) algorithm for this problem requires only O ∗ (2 k) time and oracle access to an arithmetic circuit, i.e. the ability to evaluate the circuit on elements from a suitable group algebra. This algorithm has been used to obtain the best known algorithms for several parameterized problems. In this paper we use communication complexity to show that the O ∗ (2 k) algorithm is essentially optimal within this evaluation oracle framework. On the positive side, we give new applications of the method: finding a copy of a given tree on k nodes, a spanning tree with at least k leaves, a minimum set of nodes that dominate at least t nodes, and an mdimensional kmatching. In each case we achieve a faster algorithm than what was known. We also apply the algebraic method to problems in exact counting. Among other results, we show that a combination of dynamic programming and a variation of the algebraic method can break the trivial upper bounds for exact parameterized counting in fairly general settings. 1
KERNEL(S) FOR PROBLEMS WITH NO KERNEL: ON OUTTREES WITH MANY LEAVES (EXTENDED ABSTRACT)
 STACS 2009
, 2009
"... 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 9 (4 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 has cubic size and is obtained using extremal combinatorics. For the kLeafOutBranching problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a nontrivial fashion. However, our positive results for Rooted kLeafOutBranching immediately imply that the seemingly intractable kLeafOutBranching problem admits a data reduction to n independent O(k³) kernels. These two results, tractability and intractability side by side, are the first ones separating manytoone kernelization from Turing kernelization. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.
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
Spanning directed trees with many leaves
 SIAM J. Discrete Math
"... Abstract. The Directed Maximum Leaf OutBranching problem is to find an outbranching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in outbranchings. We show that – every strong ..."
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Abstract. The Directed Maximum Leaf OutBranching problem is to find an outbranching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in outbranchings. We show that – every strongly connected nvertex digraph D with minimum indegree at least 3 has an outbranching with at least (n/4) 1/3 − 1 leaves; – if a strongly connected digraph D does not contain an outbranching with k leaves, then the pathwidth of its underlying graph UG(D) is O(k log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k. The last result implies that it can be decided in time 2 O(k log2 k) · n O(1) whether a strongly connected digraph on n vertices has an outbranching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1). 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)
An exact algorithm for the maximum leaf spanning tree problem
 PROC. FOURTH INTERNATIONAL WORKSHOP ON PARAMETERIZED AND EXACT COMPUTATION. LECTURE NOTES IN COMPUTER SCIENCE
, 2009
"... Given an undirected graph with n nodes, the Maximum Leaf Spanning Tree problem is to find a spanning tree with as many leaves as possible. When parameterized in the number of leaves k, this problem can be solved in time O(4 k poly(n)) using a simple branching algorithm introduced by a subset of th ..."
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Cited by 3 (1 self)
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Given an undirected graph with n nodes, the Maximum Leaf Spanning Tree problem is to find a spanning tree with as many leaves as possible. When parameterized in the number of leaves k, this problem can be solved in time O(4 k poly(n)) using a simple branching algorithm introduced by a subset of the authors [12]. Daligault, Gutin, Kim, and Yeo [6] improved the branching and obtained a running time of O(3.72 k poly(n)). In this paper, we study the problem from an exponential time viewpoint, where it is equivalent to the Connected Dominating Set problem. Here, Fomin, Grandoni, and Kratsch showed how to break the Ω(2 n) barrier and proposed an O(1.9407 n)time algorithm [10]. In light of some useful properties of [12] and [6], we present a branching algorithm whose running time of O(1.8966 n) has been analyzed using the MeasureandConquer technique. Finally we provide a lower bound of Ω(1.4422 n) for the worst case running time of our algorithm.
Outbranchings with Extremal Number of Leaves
"... A subdigraph T of a digraph D is called an outtree if T is an oriented tree with just one vertex s of indegree zero. A spanning outtree is called an outbranching. A vertex x of an outbranching B is called a leaf if d + B (x) = 0. This is mainly a survey paper on outbranchings with minimum and ..."
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A subdigraph T of a digraph D is called an outtree if T is an oriented tree with just one vertex s of indegree zero. A spanning outtree is called an outbranching. A vertex x of an outbranching B is called a leaf if d + B (x) = 0. This is mainly a survey paper on outbranchings with minimum and maximum number of leaves. We give short proofs of some wellknown theorems. 1
Kernelization . . . with Positive Vertex Weights
, 2009
"... In this paper we consider a natural generalization of the wellknown Max Leaf Spanning Tree problem. In the generalized Weighted Max Leaf problem we get as input an undirected connected graph G, a rational number k not smaller than 1 and a weight function w: V ↦ → R≥1 on the vertices, and are aske ..."
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In this paper we consider a natural generalization of the wellknown Max Leaf Spanning Tree problem. In the generalized Weighted Max Leaf problem we get as input an undirected connected graph G, a rational number k not smaller than 1 and a weight function w: V ↦ → R≥1 on the vertices, and are asked whether a spanning tree T for G exists such that the combined weight of the leaves of T is at least k. We show that it is possible to transform an instance 〈G, w, k 〉 of Weighted Max Leaf in linear time into an equivalent instance 〈G ′ , w ′ , k ′ 〉 such that V (G ′)  ≤ 5.5k and k ′ ≤ k. In the context of fixed parameter complexity this means that Weighted Max Leaf admits a kernel with 5.5k vertices. The analysis of the kernel size is based on a new extremal result which shows that every graph G = (V, E) that excludes some simple substructures always contains a spanning tree with at least V /5.5 leaves.
An exact algorithm for the Maximum Leaf Spanning Tree problem ✩
"... Given an undirected graph with n vertices, the Maximum Leaf Spanning Tree problem is to find a spanning tree with as many leaves as possible. When parameterized in the number of leaves k, this problem can be solved in time O(4 k poly(n)) using a simple branching algorithm introduced by a subset of t ..."
Abstract
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Given an undirected graph with n vertices, the Maximum Leaf Spanning Tree problem is to find a spanning tree with as many leaves as possible. When parameterized in the number of leaves k, this problem can be solved in time O(4 k poly(n)) using a simple branching algorithm introduced by a subset of the authors [16]. Daligault, Gutin, Kim, and Yeo [6] improved the branching and obtained a running time of O(3.72 k poly(n)). In this paper, we study the problem from an exponential time viewpoint, where it is equivalent to the Connected Dominating Set problem. Here, Fomin, Grandoni, and Kratsch showed how to break the Ω(2 n) barrier and proposed an O(1.9407 n)time algorithm [11]. Based on some useful properties of [16] and [6], we present a branching algorithm whose running time of O(1.8966 n) has been analyzed using the MeasureandConquer technique. Finally we provide a lower bound of Ω(1.4422 n) for the worst case running time of our algorithm.
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