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12
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 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 of Geometric Problems
, 2007
"... This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter in ..."
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Cited by 13 (5 self)
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This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixedparameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixedparameter intractability results are surveyed as well. Finally, we give some directions for future research.
Computing the Independence Number of Intersection Graphs
"... Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the in ..."
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Cited by 11 (0 self)
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Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the independence number, α(GC), of the intersection graph GC of C, obtained by connecting two elements of C with an edge if and only if their intersection is nonempty. This is known to be an NPhard task even for systems of segments in the plane with at most two different slopes. The best known polynomial time approximation algorithm for systems of arbitrary segments is due to Agarwal and Mustafa, and returns in the worst case an n 1/2+o(1)approximation for α. Using extensions of the LiptonTarjan separator theorem, we improve this result and present, for every ɛ> 0, a polynomial time algorithm for computing α(GC) with approximation ratio at most n ɛ. In contrast, for general graphs, for any ɛ> 0 it is NPhard to approximate the independence number within a factor of n 1−ɛ. We also give a subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane. 1
Parameterized Complexity of Stabbing Rectangles and Squares in the Plane
, 2009
"... The NPcomplete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the ..."
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Cited by 6 (1 self)
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The NPcomplete geometric covering problem Rectangle Stabbing is defined as follows: Given a set of horizontal and vertical lines in the plane, a set of rectangles in the plane, and a positive integer k, select at most k of the lines such that every rectangle is intersected by at least one of the selected lines. While it is known that the problem can be approximated in polynomial time with a factor of two, its parameterized complexity with respect to the parameter k was open so far—only its generalization to three or more dimensions was known to be W[1]hard. Giving two fixedparameter reductions, one from the W[1]complete problem Multicolored Clique and one to the W[1]complete problem Short Turing Machine Acceptance, we prove that Rectangle Stabbing is W[1]complete with respect to the parameter k, which in particular means that there is no hope for fixedparameter tractability with respect to the parameter k. Our reductions show also the W[1]completeness of the more general problem Set Cover on instances that “almost have the consecutiveones property”, that is, on instances whose matrix representation has at most two blocks of 1s per row. For the special case of Rectangle Stabbing where all rectangles are squares of the same size we can also show W[1]hardness, while the parameterized complexity of the special case where the input consists of rectangles that do not overlap is open. By giving an algorithm running in (4k + 1) k · n O(1) time, we show that Rectangle Stabbing is fixedparameter tractable in the still NPhard case where both these restrictions apply.
Domination in Geometric Intersection Graphs
 PROC. LATIN 2008, LNCS 4957
, 2008
"... For intersection graphs of disks and other fat objects, polynomialtime approximation schemes are known for the independent set and vertex cover problems, but the existing techniques were not able to deal with the dominating set problem except in the special case of unitsize objects. We present ap ..."
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Cited by 6 (1 self)
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For intersection graphs of disks and other fat objects, polynomialtime approximation schemes are known for the independent set and vertex cover problems, but the existing techniques were not able to deal with the dominating set problem except in the special case of unitsize objects. We present approximation algorithms and inapproximability results that shed new light on the approximability of the dominating set problem in geometric intersection graphs. On the one hand, we show that for intersection graphs of arbitrary fat objects, the dominating set problem is as hard to approximate as for general graphs. For intersection graphs of arbitrary rectangles, we prove APXhardness. On the other hand, we present a new general technique for deriving approximation algorithms for various geometric intersection graphs, yielding constantfactor approximation algorithms for rregular polygons, where r is an arbitrary constant, for pairwise homothetic triangles, and for rectangles with bounded aspect ratio. For arbitrary fat objects with bounded ply, we get a (3 + )approximation algorithm.
The computer journal special issue on parameterized complexity: Foreword by the guest editors
 The Computer Journal
, 2008
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Parameterized complexity in multipleinterval graphs: domination
 In Proceedings of the 6th International Symposium on Parameterized and Exact Computation
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WHierarchies Defined by Symmetric Gates
 THEORY COMPUT SYST
"... The classes of the Whierarchy are the most important classes of intractable problems in parameterized complexity. These classes were originally defined via the weighted satisfiability problem for Boolean circuits. Here, besides the Boolean connectives we consider connectives such as majority, not ..."
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Cited by 2 (0 self)
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The classes of the Whierarchy are the most important classes of intractable problems in parameterized complexity. These classes were originally defined via the weighted satisfiability problem for Boolean circuits. Here, besides the Boolean connectives we consider connectives such as majority, notallequal, and unique. For example, a gate labelled by the majority connective outputs TRUE if more than half of its inputs are TRUE. For any finite set C of connectives we construct the corresponding W(C)hierarchy. We derive some general conditions which guarantee that the Whierarchy and the W(C)hierarchy coincide levelwise. If C only contains the majority connective then the first levels of the hierarchies coincide. We use this to show that a variant of the parameterized vertex cover problem, the majority vertex cover problem, is W[1]complete.
Parameterized Domination in Circle Graphs
 THEORY COMPUT SYST
, 2013
"... A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 42(1):51–63, 1993] proved that DOMINATING SET, CONNECTED DOMINATING SET, and TOTAL DOMINATING SET are NPcomplete in circle graphs. To the best of our knowledge, nothing was known about the paramete ..."
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Cited by 1 (1 self)
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A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 42(1):51–63, 1993] proved that DOMINATING SET, CONNECTED DOMINATING SET, and TOTAL DOMINATING SET are NPcomplete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction: