Clique is hard to approximate within n/sup 1-ε/

Håstad, Johan

doi:10.1109/sfcs.1996.548522

Cited by 464 publications

(539 citation statements)

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“…We should note that a much stronger hardness result is known about approximating the clique number, due to Håstad [Hås99] (whose proof requires much more advanced techniques). He showed that efficiently approximating ω(G) to within n 1−δ , for any δ > 0, implies that NP = RP (via a probabilistic reduction).…”

Section: Convergence In Entropymentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,498

1,355

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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

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Section: Convergence In Entropymentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,498

1,355

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“…The MaxIS is approximable within O(|V |/(log |V |) 2 ) [2], but not approximable within |V | 1−ε , for any ε > 0, unless P=NP [5].…”

Section: Basic Definitionsmentioning

confidence: 99%

On the Approximability of Splitting-SAT in 2-CNF Horn Formulas

Böckenhauer

Keller

2013

Lecture Notes in Computer Science

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Abstract. Splitting a variable in a Boolean formula means to replace an arbitrary set of its occurrences by a new variable. In the minimum splitting SAT problem, we ask for a minimum-size set of variables to be split in order to make the formula satisfiable. This problem is known to be APX-hard, even for 2-CNF formulas. We consider the case of 2-CNF Horn formulas, i. e., 2-CNF formulas without positive 2-clauses, and prove that this problem is APX-hard as well. We also analyze subcases of 2-CNF Horn formulas, where additional clause types are forbidden. While excluding negative 2-clauses admits a polynomial-time algorithm based on network flows, the splitting problem stays APX-hard for formulas consisting of positive 1-clauses and negative 2-clauses only. Instead of splitting as many variables as possible to make a formula satisfiable, one can also look at the dual problem of finding the maximum number of variables that can be assigned without violating a clause. We also study the approximability of this maximum assignment problem on 2-CNF Horn formulas. While the polynomially solvable subproblems are the same as for the splitting problem, the maximum assignment problem in general Horn formulas is as hard to approximate as the maximum independent set problem.

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“…The approximability of Independent Set has been studied intensively, leading to the following result: Proposition 1 ( [19,20]). The Independent Set problem has no polynomial time O(n 1− )-approximation algorithm for any > 0 unless P = NP.…”

Section: Hardness Of Approximationmentioning

confidence: 99%

Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

Jansen

2010

Lecture Notes in Computer Science

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Abstract. In this paper we consider a natural generalization of the well-known Max Leaf Spanning Tree problem. In the generalized Weighted Max Leaf problem we get as input an undirected connected graph G, a rational number k not smaller than 1 and a weight function w : V → Q ≥1 on the vertices, and are asked whether a spanning tree T for G exists such that the combined weight of the leaves of T is at least k. We show that it is possible to transform an instance G, w, k of Weighted Max Leaf in linear time into an equivalent instance G , w , k such that |V (G )| ≤ 5.5k and k ≤ k. In the context of fixed parameter complexity this means that Weighted Max Leaf admits a kernel with 5.5k vertices. The analysis of the kernel size is based on a new extremal result which shows that every graph G = (V, E) that excludes some simple substructures always contains a spanning tree with at least |V |/5.5 leaves. 1

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Clique is hard to approximate within n/sup 1-ε/

Cited by 464 publications

References 10 publications

Expander graphs and their applications

Expander graphs and their applications

On the Approximability of Splitting-SAT in 2-CNF Horn Formulas

Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

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