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33
Improved approximation algorithms for minimum weight vertex separators
 In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, FOCS’89
, 1989
"... vertex separators ..."
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
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
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
Bidimensionality and EPTAS
"... Bidimensionality theory appears to be a powerful framework for 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 problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] e ..."
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Cited by 12 (5 self)
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Bidimensionality theory appears to be a powerful framework for 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 problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al [SODA 2010] established a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. One of the important conditions required in the framework developed by Demaine and Hajiaghayi [SODA 2005] is that to obtain an EPTAS for a graph optimization problem Π, we have to know a constantfactor approximation algorithm for Π. Our approach eliminates this strong requirement, which makes it amenable to more problems. At the heart of our framework is a decomposition lemma which states that for “most ” bidimensional problems, there is a polynomial time algorithm which given an Hminorfree graph G as input and an ɛ> 0 outputs a vertex set X of size ɛ · OP T such that the treewidth of G \ X is O(1/ɛ). Here, OP T is the objective function value of the problem in question This allows us to obtain EPTASs on (apex)minorfree graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including Cycle Packing, VertexH
Contraction Bidimensionality: The Accurate Picture
 Proceedings of the 17th Annual European Symposium on Algorithms, Lecture Notes in Computer Science
, 2009
"... Abstract. We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth. As a consequence of our combinatorial results we unify and significantly simplify contraction bidimensionality theory—the meta algorithmic framework to design ..."
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Cited by 9 (5 self)
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Abstract. We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth. As a consequence of our combinatorial results we unify and significantly simplify contraction bidimensionality theory—the meta algorithmic framework to design efficient parameterized and approximation algorithms for contraction closed parameters. 1
Subexponential Algorithms for Partial Cover Problems
"... Partial Cover problems are optimization versions of fundamental and well studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number (k) of vertices, rather than covering all edges (or vertic ..."
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Cited by 7 (2 self)
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Partial Cover problems are optimization versions of fundamental and well studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number (k) of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by k. It was recently shown by Amini et. al. [FSTTCS 08] that Partial Vertex Cover and Partial Dominating Set are fixed parameter tractable on large classes of sparse graphs, namely Hminor free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time 2 O(k) n O(1). During the last decade there has been an extensive study on parameterized subexponential algorithms. In particular, it was shown that the classical Vertex Cover and Dominating Set problems can be solved in subexponential time on Hminor free graphs. The techniques developed to obtain subexponential algorithms for classical problems do not apply to partial cover problems. It was left as an open problem by Amini et al. [FSTTCS 08] whether there is a subexponential algorithm for Partial Vertex Cover and Partial Dominating Set. In this paper, we answer the question affirmatively by solving both problems in time 2 O( √ k) n O(1) not only on planar graphs but also on much larger classes of graphs, namely, apexminor free graphs. Compared to previously known algorithms for these problems our algorithms are significantly faster and simpler. 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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Cited by 7 (6 self)
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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)
The Complexity of PolynomialTime Approximation
 TO APPEAR IN JOURNAL THEORY OF COMPUTING SYSTEMS
, 2006
"... In 1996, Khanna and Motwani [KM96] proposed three logicbased optimization problems constrained by planar structure, and offered the hypothesis that these putatively fundamental problems might provide insight into characterizing the class of optimization problems that admit a polynomialtime approxi ..."
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Cited by 6 (0 self)
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In 1996, Khanna and Motwani [KM96] proposed three logicbased optimization problems constrained by planar structure, and offered the hypothesis that these putatively fundamental problems might provide insight into characterizing the class of optimization problems that admit a polynomialtime approximation scheme (PTAS). The main contribution of this paper is to explore this program from the point of view of parameterized complexity. Problems of optimization are naturally parameterized by the parameter k = 1/ɛ. An optimization problem admits a PTAS if and only if there is an algorithm Φ, that, given an input of size n, and a parameter value k = 1/ɛ, produces a solution that is within a multiplicative factor of (1 + ɛ) of optimal, in a running time that is polynomial for every fixed value of ɛ. This definition admits the possibility that the degree of the polynomial that bounds the running time of Φ may increase, even quite dramatically, as a function of k = 1/ɛ. In fact, amongst the PTASs that are currently known, for an error of 20%, polynomial time bounds no better than O(n 2000) are quite common [Dow03]. Viewing k = 1/ɛ as a problem parameter, in the sense of parameterized complexity, leads naturally to the question of whether an efficient PTAS (EPTAS) might be possible for a given optimization problem. An EPTAS is simply an FPT algorithm with respect to this parameter. We offer a number of results concerning the problems Planar TMIN, Planar TMAX and Planar MPSAT defined by Khanna and Motwani: (1) We show that each of these problems of approximation, naturally parameterized by k = 1/ɛ, is hard for W [1], and thus it is highly unlikely that they admit EPTASs. (One could also interpret this as indicating that PTASs for these problems are unlikely to be a useful way of coping with intractability in the sense of [GJ79].) (2) We show that there are EPTASs for some subproblems described by syntactic restrictions, and establish some limits on how far these positive results can be extended.