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54
Parameterized Complexity
, 1998
"... the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs ..."
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Cited by 1213 (77 self)
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the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs into the toolkit of every algorithm designer. The purpose of the seminar was to bring together leading experts from all over the world, and from the diverse areas of computer science that have been attracted to this new framework. The seminar was intended as the rst larger international meeting with a specic focus on parameterized complexity, and it hopefully serves as a driving force in the development of the eld. 1 We had 49 participants from Australia, Canada, India, Israel, New Zealand, USA, and various European countries. During the workshop 25 lectures were given. Moreover, one night session was devoted to open problems and Thursday was basically used for problem discussion
On problems without polynomial kernels
 LECT. NOTES COMPUT. SCI
, 2007
"... Kernelization is a strong and widelyapplied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomialtime transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parame ..."
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Cited by 143 (17 self)
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Kernelization is a strong and widelyapplied technique in parameterized complexity. In a nutshell, a kernelization algorithm, or simply a kernel, is a polynomialtime transformation that transforms any given parameterized instance to an equivalent instance of the same problem, with size and parameter bounded by a function of the parameter in the input. A kernel is polynomial if the size and parameter of the output are polynomiallybounded by the parameter of the input. In this paper we develop a framework which allows showing that a wide range of FPT problems do not have polynomial kernels. Our evidence relies on hypothesis made in the classical world (i.e. nonparametric complexity), and evolves around a new type of algorithm for classical decision problems, called a distillation algorithm, which might be of independent interest. Using the notion of distillation algorithms, we develop a generic lowerbound engine which allows us to show that a variety of FPT problems, fulfilling certain criteria, cannot have polynomial kernels unless the polynomial hierarchy collapses. These problems include kPath, kCycle, kExact Cycle, kShort Cheap Tour, kGraph Minor Order Test, kCutwidth, kSearch Number, kPathwidth, kTreewidth, kBranchwidth, and several optimization problems parameterized by treewidth or cliquewidth.
Fixed Parameter Algorithms for Dominating Set and Related Problems on Planar Graphs
, 2002
"... We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. ..."
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Cited by 112 (22 self)
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We present an algorithm that constructively produces a solution to the kdominating set problem for planar graphs in time O(c . To obtain this result, we show that the treewidth of a planar graph with domination number (G) is O( (G)), and that such a tree decomposition can be found in O( (G)n) time. The same technique can be used to show that the kface cover problem ( find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c n) time, where c 1 = 3 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of kdominating set, e.g., kindependent dominating set and kweighted dominating set.
Subexponential parameterized algorithms on graphs of boundedgenus and Hminorfree Graphs
"... ... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossing ..."
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Cited by 63 (22 self)
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... Building on these results, we develop subexponential fixedparameter algorithms for dominating set, vertex cover, and set cover in any class of graphs excluding a fixed graph H as a minor. Inparticular, this general category of graphs includes planar graphs, boundedgenus graphs, singlecrossingminorfree graphs, and anyclass of graphs that is closed under taking minors. Specifically, the running time is 2O(pk)nh, where h is a constant depending onlyon H, which is polynomial for k = O(log² n). We introducea general approach for developing algorithms on Hminorfreegraphs, based on structural results about Hminorfree graphs at the
Tight lower bounds for certain parameterized NPhard problems
 Proc. 19th Annual IEEE Conference on Computational Complexity (CCC’04
, 2004
"... Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of wellknown NPhard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on deptht circuits cannot be solve ..."
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Cited by 62 (10 self)
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Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of wellknown NPhard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on deptht circuits cannot be solved in time no(k) poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)st level W [t − 1] of the Whierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NPhard problems, including weighted sat, dominating set, hitting set, set cover, and feature set, cannot be solved in time no(k) poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the Whierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted qsat (for any fixed q ≥ 2), clique, and independent set, cannot be solved in time no(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k poly(m) or O(n k). 1
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.
On the parameterized complexity of multipleinterval graph problems
 Theor. Comput. Sci
"... Abstract. Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multipleinterval graphs, often allowing for more robustness in the modeling of the specifi ..."
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Cited by 50 (8 self)
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Abstract. Multipleinterval graphs are a natural generalization of interval graphs where each vertex may have more than one interval associated with it. Many applications of interval graphs also generalize to multipleinterval graphs, often allowing for more robustness in the modeling of the specific application. With this motivation in mind, a recent systematic study of optimization problems in multipleinterval graphs was initiated. In this sequel, we study multipleinterval graph problems from the perspective of parameterized complexity. The problems under consideration are kIndependent Set, kDominating Set, and kClique, which are all known to be W[1]hard for general graphs, and NPcomplete for multipleinterval graphs. We prove that kClique is in FPT, while kIndependent Set and kDominating Set are both W[1]hard. We also prove that kIndependent Dominating Set, a hybrid of the two above problems, is also W[1]hard. Our hardness results hold even when each vertex is associated with at most two intervals, and all intervals have unit length. Furthermore, as an interesting byproduct of our hardness results, we develop a useful technique for showing W[1]hardness via a reduction from the kMulticolored Clique problem, a variant of kClique. We believe this technique has interest in its own right, as it should help in simplifying W[1]hardness results which are notoriously hard to construct and technically tedious.
Parameterized Complexity for the Skeptic
 In Proc. 18th IEEE Annual Conference on Computational Complexity
, 2003
"... The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity. ..."
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Cited by 41 (1 self)
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The goal of this article is to provide a tourist guide, with an eye towards structural issues, to what I consider some of the major highlights of parameterized complexity.
Strong computational lower bounds via parameterized complexity
, 2006
"... We develop new techniques for deriving strong computational lower bounds for a class of wellknown NPhard problems. This class includes weighted satisfiability, dominating set, hitting set, set cover, clique, and independent set. For example, although a trivial enumeration can easily test in time O ..."
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Cited by 39 (5 self)
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We develop new techniques for deriving strong computational lower bounds for a class of wellknown NPhard problems. This class includes weighted satisfiability, dominating set, hitting set, set cover, clique, and independent set. For example, although a trivial enumeration can easily test in time O(n k) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f(k)n o(k) for any function f, even if we restrict the parameter values to be bounded by an arbitrarily small function of n. Under the same assumption, we prove that even if we restrict the parameter values k to be of the order Θ(µ(n)) for any reasonable function µ, no algorithm of running time n o(k) can test if a graph of n vertices has a clique of size k. Similar strong lower bounds on the computational complexity are also derived for other NPhard problems in the above class. Our techniques can be further extended to derive computational lower bounds on polynomial time approximation schemes for NPhard optimization problems. For example, we prove that the NPhard distinguishing substring selection problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f(1/ɛ)n o(1/ɛ) for any function f unless an unlikely collapse occurs in parameterized complexity theory.