Results 1  10
of
48
Dominating Sets in Planar Graphs: BranchWidth and Exponential Speedup
, 2002
"... Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance. ..."
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Cited by 69 (18 self)
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Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance.
Digraph measures: Kelly decompositions, games and orderings
"... We consider various wellknown, equivalent complexity measures for graphs such as elimination orderings, ktrees and cops and robber games and study their natural translations to digraphs. We show that on digraphs all these measures are also equivalent and induce a natural connectivity measure. We i ..."
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Cited by 36 (5 self)
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We consider various wellknown, equivalent complexity measures for graphs such as elimination orderings, ktrees and cops and robber games and study their natural translations to digraphs. We show that on digraphs all these measures are also equivalent and induce a natural connectivity measure. We introduce a decomposition for digraphs and an associated width, Kellywidth, which is equivalent to the aforementioned measure. We demonstrate its usefulness by exhibiting a number of potential applications including polynomialtime algorithms for NPcomplete problems on graphs of bounded Kellywidth, and complexity analysis of asymmetric matrix factorization. Finally, we compare the new width to other known decompositions of digraphs.
Subexponential parameterized algorithms
 Computer Science Review
"... We give a review of a series of techniques and results on the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch (or tree) decomposition of the input graph whose width is bounded by a sublinear ..."
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Cited by 34 (16 self)
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We give a review of a series of techniques and results on the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch (or tree) decomposition of the input graph whose width is bounded by a sublinear function of the parameter and, second, use this decomposition to solve the problem in time that is single exponential to this bound. The main tool for the first step is Bidimensionality Theory. Here we present the potential, but also the boundaries, of this theory. For the second step, we describe recent techniques, associating the analysis of subexponential algorithms to combinatorial bounds related to Catalan numbers. As a result, we have 2 O( √ k) · n O(1) time algorithms for a wide variety of parameterized problems on graphs, where n is the size of the graph and k is the parameter. 1
Solving connectivity problems parameterized by treewidth in single exponential time (Extended Abstract)
, 2011
"... For the vast majority of local problems on graphs of small treewidth (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c tw V  O(1) time algorithms, where tw is the treewidth of the input g ..."
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Cited by 34 (7 self)
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For the vast majority of local problems on graphs of small treewidth (where by local we mean that a solution can be verified by checking separately the neighbourhood of each vertex), standard dynamic programming techniques give c tw V  O(1) time algorithms, where tw is the treewidth of the input graph G = (V, E) and c is a constant. On the other hand, for problems with a global requirement (usually connectivity) the best–known algorithms were naive dynamic programming schemes running in at least tw tw time. We breach this gap by introducing a technique we named Cut&Count that allows to produce c tw V  O(1) time Monte Carlo algorithms for most connectivitytype problems, including HAMILTONIAN PATH, STEINER TREE, FEEDBACK VERTEX SET and CONNECTED DOMINATING SET. These results have numerous consequences in various fields, like parameterized complexity, exact and approximate algorithms on planar and Hminorfree graphs and exact algorithms on graphs of bounded degree. The constant c in our algorithms is in all cases small, and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. In contrast to the problems aiming to minimize the number of connected components that we solve using Cut&Count as mentioned above, we show that, assuming the Exponential Time Hypothesis, the aforementioned gap cannot be breached for some problems that aim to maximize the number of connected components like CYCLE PACKING.
Dynamic programming on tree decompositions using generalised fast subset convolution
 Proceedings of the 17th Annual European Symposium on Algorithms, ESA 2009
"... Abstract. In this paper, we show that algorithms on tree decompositions can be made faster with the use of generalisations of fast subset convolution. Amongst others, this gives algorithms that, for a graph, given with a tree decomposition of width k, solve the dominated set problem in O(nk23k) tim ..."
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Cited by 21 (1 self)
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Abstract. In this paper, we show that algorithms on tree decompositions can be made faster with the use of generalisations of fast subset convolution. Amongst others, this gives algorithms that, for a graph, given with a tree decomposition of width k, solve the dominated set problem in O(nk23k) time and the problem to count the number of perfect matchings in O∗(2k) time. Using a generalisation of fast subset convolution, we obtain faster algorithms for all [ρ, σ]domination problems with finite or cofinite ρ and σ on tree decompositions. These include many well known graph problems. We give additional results on many more graph covering and partitioning problems. 1
FixedParameter Tractability Results for FullDegree Spanning Tree and Its Dual
 In Proc. the 2nd International Workshop on Parameterized and Exact Computation (IWPEC), Springer LNCS
, 2006
"... We provide firsttime fixedparameter tractability results for the NPhard problems Maximum FullDegree Spanning Tree and MinimumVertex Feedback Edge Set. These problems are dual to each other: In Maximum FullDegree Spanning Tree, the task is to find a spanning tree for a given graph that maximizes ..."
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Cited by 13 (2 self)
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We provide firsttime fixedparameter tractability results for the NPhard problems Maximum FullDegree Spanning Tree and MinimumVertex Feedback Edge Set. These problems are dual to each other: In Maximum FullDegree Spanning Tree, the task is to find a spanning tree for a given graph that maximizes the number of vertices that preserve their degree. For MinimumVertex Feedback Edge Set, the task is to minimize the number of vertices that end up with a reduced degree. Parameterized by the solution size, we exhibit that MinimumVertex Feedback Edge Set is fixedparameter tractable and has a problem kernel with the number of vertices linearly depending on the parameter k. Our main contribution for Maximum FullDegree Spanning Tree, which is W[1]hard, is a linearsize problem kernel when restricted to planar graphs. Moreover, we present a dynamic programming algorithm for graphs of bounded treewidth. Keywords: Fixedparameter tractability, Problem kernel, Data reduction,
Fast subexponential algorithm for nonlocal problems on graphs of bounded genus, in
 Proc. of the 10th Scandinavian Workshop on Algorithm Theory, SWAT, in: LNCS
"... Abstract We give a general technique for designing fast subexponential algorithms for several graph problems whose instances are restricted to graphs of bounded genus. We use it to obtain time 2 O( √ n) algorithms for a wide family of problems such as Hamiltonian Cycle, Σembedded Graph Travelling ..."
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Cited by 12 (6 self)
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Abstract We give a general technique for designing fast subexponential algorithms for several graph problems whose instances are restricted to graphs of bounded genus. We use it to obtain time 2 O( √ n) algorithms for a wide family of problems such as Hamiltonian Cycle, Σembedded Graph Travelling Salesman Problem, Longest Cycle, and Max Leaf Tree. For our results, we combine planarizing techniques with dynamic programming on special type branch decompositions. Our techniques can also be used to solve parameterized problems. Thus, for example, we show how to find a cycle of length p (or to conclude that there is no such a cycle) on graphs of bounded genus in time 2 O( √ p) · n O(1) .
Dynamic programming and fast matrix multiplication
 OF LNCS
, 2006
"... We give a novel general approach for solving NPhard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We exemplify our approach on problems like Vertex Cover, Domina ..."
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Cited by 11 (3 self)
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We give a novel general approach for solving NPhard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We exemplify our approach on problems like Vertex Cover, Dominating Set and Longest Path. Our approach works faster than the usual dynamic programming solution for any vertex subset problem on graphs of bounded branchwidth. In particular, we obtain the currently fastest algorithms for Planar Vertex Cover of runtime O(2 2.52 √ n), for Planar Dominating Set of runtime exact O(2 3.99 √ n) and parameterized O(2 11.98 √ k) · n O(1) , and for Planar Longest Path of runtime O(2 5.58 √ n). The exponent of the running time is depending heavily on the running time of the fastest matrix multiplication algorithm that is currently o(n 2.376).
PLANAR SUBGRAPH ISOMORPHISM REVISITED
, 2009
"... Ten years after Eppstein’s results on planar subgraph isomorphism for ksized patterns, we improve the exponential term of the running time 2 O(k log k) · n of Eppstein’s algorithm to 2 O(k) (keeping the term in n linear!) Next to deciding subgraph isomorphism, we can construct a solution and enume ..."
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Cited by 11 (2 self)
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Ten years after Eppstein’s results on planar subgraph isomorphism for ksized patterns, we improve the exponential term of the running time 2 O(k log k) · n of Eppstein’s algorithm to 2 O(k) (keeping the term in n linear!) Next to deciding subgraph isomorphism, we can construct a solution and enumerate all solutions in the same asymptotic running time. We may list ω subgraphs with an additive term O(ωn) in the running time of our algorithm. For exact algorithms, this means we obtain a truly subexponential algorithm for patterns of size O ( √ n) of running time 2 O( √ n) imrpoving the former bound of 2 O( √ n log n)