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Learning Bayesian Networks is NPHard
, 1994
"... Algorithms for learning Bayesian networks from data have two components: a scoring metric and a search procedure. The scoring metric computes a score reflecting the goodnessoffit of the structure to the data. The search procedure tries to identify network structures with high scores. Heckerman et ..."
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Cited by 191 (2 self)
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there is a Bayesian networkamong those where each node has at most k parentsthat has a relative posterior probability greater than a given constant is NPcomplete, when the BDe metric is used. 1 Introduction Recently, many researchers have begun to investigate methods for learning Bayesian networks
A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 778 (5 self)
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Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhard
Terrain Guarding is NPHard
, 2009
"... A set G of points on a 1.5dimensional terrain, also known as an xmonotone polygonal chain, is said to guard the terrain if every point on the terrain is seen by a point in G. Two points on the terrain see each other if and only if the line segment between them is never strictly below the terrain. ..."
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Cited by 8 (0 self)
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. The minimum terrain guarding problem asks for a minimum guarding set for the given input terrain. Using a reduction from PLANAR 3SAT we prove that the decision version of this problem is NPhard. This solves a significant open problem and complements recent positive approximability results
The approximability of NPhard problems
 In Proceedings of the Annual ACM Symposium on Theory of Computing
, 1998
"... Many problems in combinatorial optimization are NPhard (see [60]). This has forced researchers to explore techniques for dealing with NPcompleteness. Some have considered algorithms that solve “typical” ..."
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Cited by 17 (0 self)
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Many problems in combinatorial optimization are NPhard (see [60]). This has forced researchers to explore techniques for dealing with NPcompleteness. Some have considered algorithms that solve “typical”
Exact algorithms for NPhard problems: A survey
 Combinatorial Optimization  Eureka, You Shrink!, LNCS
"... Abstract. We discuss fast exponential time solutions for NPcomplete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NPcomplete problems includes the travelling salesman problem, schedu ..."
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Cited by 152 (3 self)
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, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more. 1
Inflating Balls is NPhard
, 2008
"... A collection C of balls in R d is δinflatable if it is isometric to the intersection U ∩ E of some ddimensional affine subspace E with a collection U of (d + δ)dimensional balls that are disjoint and have equal radius. We give a quadratictime algorithm to recognize 1inflatable collections of b ..."
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Cited by 1 (1 self)
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of balls in any fixed dimension, and show that recognizing δinflatable collections of ddimensional balls is NPhard for δ ≥ 2 and d ≥ 3 if the balls ’ centers and radii are given by numbers of the form a + b p c + d √ e, where a,..., e are integers.
GPSGRecognition is NPHard
, 1986
"... Proponents of generalized phrase structure grammar (GPSG) cite its weak contextfree generative power as proof of the computational tractability of GPSGRecognition. Since contextfree languages (CFLs) can be parsed in time proportional to the cube of the sentence length, and GPSGs only generate CFL ..."
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Cited by 5 (1 self)
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recognition problem for the GPSGs of Gazdar (1981) is NPhard. Crucially, the time to parse a sentence of a CFL can be the product of sentence length cubed and contextfree grammar size squared, and the GPSG grammar can result in an exponentially large set of derived contextfree rules. A central object
Instance Complexity of NPhard sets
, 1999
"... Instance complexity was introduced by Orponen, Ko, Schöning, and Watanabe [7, 14, 15] as a measure of the complexity of individual instances of a decision problem. Comparing instance complexity to Kolmogorov complexity (i.e. the inherent complexity of a string), they introduced the notion of ph ..."
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instances, and conjectured that every set not in P has phard instances. Whereas this conjecture is still unsettled, Fortnow and Kummer [4] proved that NPhard sets have phard instances, unless P=NP. The unbounded version of the conjecture was proven wrong by Kummer [9], and therefore it cannot
SELFCONCORDANCE IS NPHARD
"... Abstract. We give an elementary proof of a somewhat curious result, namely, that deciding whether a convex function is selfconcordant is in general an intractable problem. 1. ..."
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Abstract. We give an elementary proof of a somewhat curious result, namely, that deciding whether a convex function is selfconcordant is in general an intractable problem. 1.
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