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26
On problems without polynomial kernels
 Lect. Notes Comput. Sci
, 2007
"... Abstract. 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 an ..."
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Cited by 69 (8 self)
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Abstract. 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. 1
Parameterized Complexity: Exponential SpeedUp for Planar Graph Problems
 in Electronic Colloquium on Computational Complexity (ECCC
, 2001
"... A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniqu ..."
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Cited by 61 (21 self)
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A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.
ConflictFree Colorings of Simple Geometric Regions with Applications to Frequency Assignment in Cellular Networks
, 2002
"... Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call Minimum ConflictFree Coloring (MinCFColoring). In its general form, the input of the MinCFcoloring problem is a set system (X, S), where each S 2 S is a subset of X . The ..."
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Cited by 45 (9 self)
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Motivated by a frequency assignment problem in cellular networks, we introduce and study a new coloring problem that we call Minimum ConflictFree Coloring (MinCFColoring). In its general form, the input of the MinCFcoloring problem is a set system (X, S), where each S 2 S is a subset of X . The output is a coloring of the sets in S that satisfies the following constraint: for every x 2 X there exists a color i and a unique set S 2 S, such that x 2 S and (S) = i. The goal is to minimize the number of colors used by the coloring .
Complexity of Consistent Query Answering in Databases under CardinalityBased and Incremental Repair Semantics
 In ICDT
, 2007
"... Abstract. Consistent Query Answering (CQA) is the problem of computing from a database the answers to a query that are consistent with respect to certain integrity constraints that the database, as a whole, may fail to satisfy. Consistent answers have been characterized as those that are invariant u ..."
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Cited by 30 (9 self)
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Abstract. Consistent Query Answering (CQA) is the problem of computing from a database the answers to a query that are consistent with respect to certain integrity constraints that the database, as a whole, may fail to satisfy. Consistent answers have been characterized as those that are invariant under certain minimal forms of restoration of the database consistency. In this paper we investigate algorithmic and complexity theoretic issues of CQA under database repairs that minimally departwrt the cardinality of the symmetric difference from the original database. Research on this kind of repairs has been suggested in the literature, but no systematic study had been done. Here we obtain first tight complexity bounds. We also address, considering for the first time a dynamic scenario for CQA, the problem of incremental complexity of CQA, that naturally occurs when an originally consistent database becomes inconsistent after the execution of a sequence of update operations. Tight bounds on incremental complexity are provided for various semantics under denial constraints, e.g. (a) minimum tuplebased repairs wrt cardinality, (b) minimal tuplebased repairs wrt set inclusion, and (c) minimum numerical aggregation of attributebased repairs. Fixed parameter tractability is also investigated in this dynamic context, where the size of the update sequence becomes the relevant parameter. 1
Graph separators: a parameterized view
 Journal of Computer and System Sciences
, 2001
"... Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. ..."
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Cited by 30 (14 self)
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Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p
Parameterized complexity and approximation algorithms
 Comput. J
, 2006
"... Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We ..."
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Cited by 28 (1 self)
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Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research. 1.
Lineartime reconstruction of Delaunay triangulations with applications
 In Proc. Annu. European Sympos. Algorithms, number 1284 in Lecture Notes Comput. Sci
, 1997
"... Many of the computational geometers' favorite data structures are planar graphs, canonically determined by a set of geometric data, that take \Theta(n log n) time to compute. Examples include 2d Delaunay triangulation, trapezoidations of segments, and constrained Voronoi diagrams, and 3d convex hu ..."
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Cited by 20 (3 self)
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Many of the computational geometers' favorite data structures are planar graphs, canonically determined by a set of geometric data, that take \Theta(n log n) time to compute. Examples include 2d Delaunay triangulation, trapezoidations of segments, and constrained Voronoi diagrams, and 3d convex hulls. Given such a structure, one can determine a permutation of the data in O(n) time such that the data structure can be reconstructed from the permuted data in O(n) time by a simple incremental algorithm. As a consequence, one can permute a data file to "hide" a geometric structure, such as a terrian model based on the Delaunay triangulation of a set of sampled points, without disrupting other applications. One can even include "importance" in the ordering so the incremental reconstruction produces approximate terrain models as the data is read or received. For the Delaunay triangulation, we can also handle input in degenerate position, even though the data structures may no longer be cano...
On problems without polynomial kernels (extended abstract
 ICALP (1), volume 5125 of LNCS
, 2008
"... Abstract. Kernelization is a central technique used in parameterized algorithms, and in other approaches for coping with NPhard problems. In this paper, we introduce a new method which allows us to show that many problems do not have polynomial size kernels under reasonable complexitytheoretic ass ..."
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Cited by 15 (1 self)
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Abstract. Kernelization is a central technique used in parameterized algorithms, and in other approaches for coping with NPhard problems. In this paper, we introduce a new method which allows us to show that many problems do not have polynomial size kernels under reasonable complexitytheoretic assumptions. These problems include k
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 9 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
Fast 3Coloring TriangleFree Planar Graphs
 12th Annual European Symposium (ESA’04), Lecture Notes In Computer Science, (c
, 2004
"... Abstract. We show the first o(n 2) algorithm for coloring vertices of trianglefree planar graphs using three colors. The time complexity of the algorithm is O(nlog n). Our approach can be also used to designO(npolylog n)time algorithms for two other similar coloring problems. A remarkable ingredien ..."
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Cited by 7 (0 self)
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Abstract. We show the first o(n 2) algorithm for coloring vertices of trianglefree planar graphs using three colors. The time complexity of the algorithm is O(nlog n). Our approach can be also used to designO(npolylog n)time algorithms for two other similar coloring problems. A remarkable ingredient of our algorithm is the data structure processing short path queries introduced recently in [9]. In this paper we show how to adapt it to the fully dynamic environment where edge insertions and deletions are allowed. 1