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47
Bidimensionality and Kernels
, 2010
"... Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi ..."
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Cited by 58 (23 self)
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Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this paper, we establish a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In parameterized complexity, each problem instance comes with a parameter k and the parameterized problem is said to admit a linear kernel if there is a polynomial time algorithm, called
Kernelization: New Upper and Lower Bound Techniques
 In Proc. of the 4th International Workshop on Parameterized and Exact Computation (IWPEC), volume 5917 of LNCS
, 2009
"... Abstract. In this survey, we look at kernelization: algorithms that transform in polynomial time an input to a problem to an equivalent input, whose size is bounded by a function of a parameter. Several results of recent research on kernelization are mentioned. This survey looks at some recent resu ..."
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Cited by 54 (0 self)
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Abstract. In this survey, we look at kernelization: algorithms that transform in polynomial time an input to a problem to an equivalent input, whose size is bounded by a function of a parameter. Several results of recent research on kernelization are mentioned. This survey looks at some recent results where a general technique shows the existence of kernelization algorithms for large classes of problems, in particular for planar graphs and generalizations of planar graphs, and recent lower bound techniques that give evidence that certain types of kernelization algorithms do not exist.
Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions
 IN PROCEEDINGS OF THE 13TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA 2005
, 2005
"... A divideandconquer strategy based on variations of the LiptonTarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on pla ..."
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Cited by 48 (18 self)
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A divideandconquer strategy based on variations of the LiptonTarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of noncrossing partitions. Compared to divideandconquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an O(26.903pn) time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time O(29.8594pn). Our approach can be used to design parameterized algorithms as well. For example we introduce the first 2O(pk)nO(1) time algorithm for parameterized Planar kcycle by showing that for a given k we can decide if a planar graph on n vertices has a cycle of length at least k in time O(213.6pkn + n3).
Locally excluding a minor
"... We introduce the concept of locally excluded minors. Graph classes locally excluding a minor are a common generalisation of the concept of excluded minor classes and of graph classes with bounded local treewidth. We show that firstorder modelchecking is fixedparameter tractable on any class of gr ..."
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Cited by 47 (13 self)
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We introduce the concept of locally excluded minors. Graph classes locally excluding a minor are a common generalisation of the concept of excluded minor classes and of graph classes with bounded local treewidth. We show that firstorder modelchecking is fixedparameter tractable on any class of graphs locally excluding a minor. This strictly generalises analogous results by Flum and Grohe on excluded minor classes and Frick and Grohe on classes with bounded local treewidth. As an important consequence of the proof we obtain fixedparameter algorithms for problems such as dominating or independent set on graph classes excluding a minor, where now the parameter is the size of the dominating set and the excluded minor. We also study graph classes with excluded minors, where the minor may grow slowly with the size of the graphs and show that again, firstorder modelchecking is fixedparameter tractable on any such class of graphs.
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.
Algorithmic MetaTheorems
 In M. Grohe and R. Neidermeier eds, International Workshop on Parameterized and Exact Computation (IWPEC), volume 5018 of LNCS
, 2008
"... Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of ..."
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Cited by 22 (6 self)
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Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of problems can be solved efficiently on every graph satisfying a certain property P”. A particularly well known example of a metatheorem is Courcelle’s theorem that every decision problem definable in monadic secondorder logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth [1]. The class C of problems can be defined in a number of different ways. One option is to state combinatorial or algorithmic criteria of problems in C. For instance, Demaine, Hajiaghayi and Kawarabayashi [5] showed that every minimisation problem that can be solved efficiently on graph classes of bounded treewidth and for which approximate solutions can be computed efficiently from solutions of certain subinstances, have a PTAS on any class of graphs excluding a fixed minor. While this gives a strong unifying explanation for PTAS of many
Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal
, 2010
"... We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2−ǫ) n m O(1) time, we show that fo ..."
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Cited by 19 (5 self)
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We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2−ǫ) n m O(1) time, we show that for any ǫ> 0; • INDEPENDENT SET cannot be solved in (2 − ǫ) tw(G) V (G)  O(1) time, • DOMINATING SET cannot be solved in (3 − ǫ) tw(G) V (G)  O(1) time, • MAX CUT cannot be solved in (2 − ǫ) tw(G) V (G)  O(1) time, • ODD CYCLE TRANSVERSAL cannot be solved in (3 − ǫ) tw(G) V (G)  O(1) time, • For any q ≥ 3, qCOLORING cannot be solved in (q − ǫ) tw(G) V (G)  O(1) time, • PARTITION INTO TRIANGLES cannot be solved in (2 − ǫ) tw(G) V (G)  O(1) time. Our lower bounds match the running times for the best known algorithms for the problems, up to the ǫ in the base.
Fast FAST
"... We present a randomized subexponential time, polynomial space parameterized algorithm for the kWeighted Feedback Arc Set in Tournaments (kFAST) problem. We also show that our algorithm can be derandomized by slightly increasing the running time. To derandomize our algorithm we construct a new kin ..."
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Cited by 16 (7 self)
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We present a randomized subexponential time, polynomial space parameterized algorithm for the kWeighted Feedback Arc Set in Tournaments (kFAST) problem. We also show that our algorithm can be derandomized by slightly increasing the running time. To derandomize our algorithm we construct a new kind of universal hash functions, that we coin universal coloring families. For integers m, k and r, a family F of functions from [m] to [r] is called a universal (m, k, r)coloring family if for any graph G on the set of vertices [m] with at most k edges, there exists an f ∈ F which is a proper vertex coloring of G. Our algorithm is the first nontrivial subexponential time parameterized algorithm outside the framework of bidimensionality.
The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number
"... In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A welldeveloped example is the investigation of how the parameter treewidth influences the complexity of (other) graph ..."
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Cited by 14 (7 self)
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In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A welldeveloped example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that realworld input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or wellunderstood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the wellstudied case of treewidth, we can ask: how hard is it to solve 3Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W [1]hard under this parameterization? We do two things: (1) We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G) ≤ k, based on the polynomialtime extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixedparameter tractable (FPT) runtimes O ∗ (f(k)). (2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.