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119
Free Bits, PCPs and NonApproximability  Towards Tight Results
, 1996
"... This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems. ..."
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Cited by 206 (41 self)
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This paper continues the investigation of the connection between proof systems and approximation. The emphasis is on proving tight nonapproximability results via consideration of measures like the "free bit complexity" and the "amortized free bit complexity" of proof systems.
Vertex Cover: Further Observations and Further Improvements
 Journal of Algorithms
, 1999
"... Recently, there have been increasing interests and progresses in lowering the worst case time complexity for wellknown NPhard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several simple and new techniques are int ..."
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Cited by 151 (15 self)
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Recently, there have been increasing interests and progresses in lowering the worst case time complexity for wellknown NPhard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several simple and new techniques are introduced, which lead to an improved algorithm of time O(kn + 1:271 k k 2 ) for the problem. Our algorithm also induces improvement on previous algorithms for the Independent Set problem on graphs of small degree. 1 Introduction Many optimization problems from industrial applications are NPhard. According to the NPcompleteness theory [10], these problems cannot be solved in polynomial time unless P = NP. However, this fact does not obviate the need for solving these problems for their practical importance. There has been a number of approaches to attacking the NPhardness of optimization problems, including approximation algorithms, heuristic algorithms, and average time analysis. Recent...
Simple heuristics for unit disk graphs
 NETWORKS
, 1995
"... Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NPhard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring ..."
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Cited by 125 (6 self)
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Unit disk graphs are intersection graphs of circles of unit radius in the plane. We present simple and provably good heuristics for a number of classical NPhard optimization problems on unit disk graphs. The problems considered include maximum independent set, minimum vertex cover, minimum coloring and minimum dominating set. We also present an online coloring heuristic which achieves a competitive ratio of 6 for unit disk graphs. Our heuristics do not need a geometric representation of unit disk graphs. Geometric representations are used only in establishing the performance guarantees of the heuristics. Several of our approximation algorithms can be extended to intersection graphs of circles of arbitrary radii in the plane, intersection graphs of regular polygons, and to intersection graphs of higher dimensional regular objects.
THE PRIMALDUAL METHOD FOR APPROXIMATION ALGORITHMS AND ITS APPLICATION TO NETWORK DESIGN PROBLEMS
"... The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent researc ..."
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Cited by 120 (7 self)
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The primaldual method is a standard tool in the design of algorithms for combinatorial optimization problems. This chapter shows how the primaldual method can be modified to provide good approximation algorithms for a wide variety of NPhard problems. We concentrate on results from recent research applying the primaldual method to problems in network design.
A unified approach to approximating resource allocation and scheduling
 Journal of the ACM
, 2000
"... We present a general framework for solving resource allocation and scheduling problems. Given a resource of fixed size, we present algorithms that approximate the maximum throughput or the minimum loss by a constant factor. Our approximation factors apply to many problems, among which are: (i) real ..."
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Cited by 116 (16 self)
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We present a general framework for solving resource allocation and scheduling problems. Given a resource of fixed size, we present algorithms that approximate the maximum throughput or the minimum loss by a constant factor. Our approximation factors apply to many problems, among which are: (i) realtime scheduling of jobs on parallel machines, (ii) bandwidth allocation for sessions between two endpoints, (iii) general caching, (iv) dynamic storage allocation, and (v) bandwidth allocation on optical line and ring topologies. For some of these problems we provide the first constant factor approximation algorithm. Our algorithms are simple and efficient and are based on the localratio technique. We note that they can equivalently be interpreted within the primaldual schema.
Improved Approximation Algorithms for the Vertex Cover Problem in Graphs and Hypergraphs
, 1999
"... We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a performa ..."
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Cited by 92 (6 self)
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We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a performance ratio of 2  (1  o(1)) 2 ln ln \Delta ln \Delta for large \Delta, which improves the previously known ratio of 2 \Gamma log \Delta+O(1) \Delta obtained by HalldÃ³rsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For kuniform hypergraphs with n vertices, we achieve a ratio of k \Gamma (1 \Gamma o(1)) k ln ln n ln n for large n, and for kuniform hypergraphs with maximum degree at most \Delta, the algorithm achieves a ratio of k \Gamma (1 \Gamma o(1)) k(k\Gamma1) ln ln \Delta ln \Delta for large \Delta. These results considerably improve the previous best ratio of k(1\Gammac=\Delta 1 k\Gamma1 ) for bounded degree kuniform hypergraphs, and k(1 \Gamma c=n k\Gamma1 k ) for general kuniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of HalldÃ³rsson.
The Importance of Being Biased
, 2002
"... The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1 ..."
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Cited by 85 (8 self)
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The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1
PolynomialTime Approximation Schemes for Geometric Graphs
, 2001
"... A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weigh ..."
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Cited by 71 (4 self)
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A disk graph is the intersection graph of a set of disks with arbitrary diameters in the plane. For the case that the disk representation is given, we present polynomialtime approximation schemes (PTASs) for the maximum weight independent set problem (selecting disjoint disks of maximum total weight) and for the minimum weight vertex cover problem in disk graphs. These are the first known PTASs for NPhard optimization problems on disk graphs. They are based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible. The PTASs for disk graphs represent a common generalization of previous results for planar graphs and unit disk graphs. They can be extended to intersections graphs of other "disklike" geometric objects (such as squares or regular polygons), also in higher dimensions.
A 2approximation algorithm for the undirected feedback vertex set problem
 SIAM J. Discrete Math
, 1999
"... Abstract. A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. The problem considered is that of finding a minimum feedback vertex set given a weighted and undirected graph. We present a simple and efficient approximation algorithm ..."
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Cited by 67 (0 self)
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Abstract. A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. The problem considered is that of finding a minimum feedback vertex set given a weighted and undirected graph. We present a simple and efficient approximation algorithm with performance ratio of at most 2, improving previous best bounds for either weighted or unweighted cases of the problem. Any further improvement on this bound, matching the best constant factor known for the vertex cover problem, is deemed challenging. The approximation principle, underlying the algorithm, is based on a generalized form of the classical local ratio theorem, originally developed for approximation of the vertex cover problem, and a more flexible style of its application.
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 65 (9 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