Results 1  10
of
1,650
Trading Group Theory for Randomness
, 1985
"... In a previous paper [BS] we proved, using the elements of the Clwory of nilyotenf yroupu, that some of the /undamcnla1 computational problems in mat & proup, belong to NP. These problems were also ahown to belong to CONP, assuming an unproven hypofhedi.9 concerning finilc simple Q ’ oup,. The a ..."
Abstract

Cited by 369 (9 self)
 Add to MetaCart
prove th:rt. in spite of their analogy with the polynomial time hierarchy, the finite levrls of this hierarchy collapse t,o Afsf=Ah42). Using a combinatorial lemma on finite groups [IIE], we construct a game by whirh t.he nondeterministic player (Merlin) is able to coavlnre the random player (Arthur
On sets of integers containing no k elements in arithmetic progression
, 1975
"... integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequen ..."
Abstract

Cited by 358 (1 self)
 Add to MetaCart
integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with bn> an9 which contains no arithmetic progression of three terms, but which intersects every infinite arithmetic progression. The finite form of van der Waerden's theorem goes as follows: For each positive integer n9 there exists a least integer f{n) with the property that if the integers from 1 to /(/?) are arbitrarily partitioned into two classes, then at least one class contains an arithmetic progression of « terms. (For a short proof, see the note of Graham and Rothschild [5].) However, the best upper bound on f{n) known at present is extremely poor. The best lower bound known, due to Berlekamp [3], asserts that/(«) < nln9 for n prime, which improves previous results of Erdös, Rado and W. Schmidt. More than 40 years ago, Erdös and Turân [4] considered the quantity rk{n)9 defined to be the greatest integer / for which there is a sequence of integers 0 < a \ < a2 < •• • < a; ^ n which does not contain an arithmetic progression of k terms. They were led to the investigation of rk{n) by several things. First of all the problem of estimating rk{n) is clearly interesting in itself. Secondly, rk{n) < n/2 would imply f{k) < 77, i.e., they hoped to improve the poor upper bound on f{k) by investigating rk{n). Finally, an old question in number theory asks if there are arbitrarily long arithmetic progressions of prime numbers. From rk{n) < %{rì) this would follow immediately. The hope was that this problem on primes could be attacked not by
The algorithmic aspects of the Regularity Lemma
 J. Algorithms
, 1994
"... The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show that ..."
Abstract

Cited by 113 (30 self)
 Add to MetaCart
The Regularity Lemma of Szemerédi is a result that asserts that every graph can be partitioned in a certain regular way. This result has numerous applications, but its known proof is not algorithmic. Here we first demonstrate the computational difficulty of finding a regular partition; we show
A parallel algorithmic version of the local lemma
, 1991
"... The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method for ..."
Abstract

Cited by 75 (10 self)
 Add to MetaCart
The Lovász Local Lemma is a tool that enables one to show that certain events hold with positive, though very small probability. It often yields existence proofs of results without supplying any efficient way of solving the corresponding algorithmic problems. J. Beck has recently found a method
Exponential Lower Bounds for the Pigeonhole Principle
, 1992
"... In this paper we prove an exponential lower bound on the size of boundeddepth Frege proofs for the pigeonhole principle (PHP). We also obtain an ~(log log rz)depth lower bound for any polynomialsized Frege proof of the pigeonhole principle. Our theorem nearly completes the search for the exact ..."
Abstract

Cited by 121 (27 self)
 Add to MetaCart
In this paper we prove an exponential lower bound on the size of boundeddepth Frege proofs for the pigeonhole principle (PHP). We also obtain an ~(log log rz)depth lower bound for any polynomialsized Frege proof of the pigeonhole principle. Our theorem nearly completes the search for the exact
Unifying duality theorems for width parameters in graphs and matroids II. General duality
, 2014
"... We prove a general duality theorem for tanglelike dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6]. 1 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We prove a general duality theorem for tanglelike dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6]. 1
Clustering with qualitative information
 In Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
, 2003
"... We consider the problem of clustering a collection of elements based on pairwise judgments of similarity and dissimilarity. Bansal, Blum and Chawla [1] cast the problem thus: given a graph G whose edges are labeled “+ ” (similar) or “− ” (dissimilar), partition the vertices into clusters so that ..."
Abstract

Cited by 123 (9 self)
 Add to MetaCart
We consider the problem of clustering a collection of elements based on pairwise judgments of similarity and dissimilarity. Bansal, Blum and Chawla [1] cast the problem thus: given a graph G whose edges are labeled “+ ” (similar) or “− ” (dissimilar), partition the vertices into clusters so that the number of pairs correctly (resp. incorrectly) classified with respect to the input labeling is maximized (resp. minimized). Complete graphs, where the classifier labels every edge, and general graphs, where some edges are not labeled, are both worth studying. We answer several questions left open in [1] and provide a sound overview of clustering with qualitative information. We give a factor 4 approximation for minimization on complete graphs, and a factor O(log n) approximation for general graphs. For the maximization version, a PTAS for complete graphs is shown in [1]; we give a factor 0.7664 approximation for general graphs, noting that a PTAS is unlikely by proving APXhardness. We also prove the APXhardness of minimization on complete graphs. 1.
EXPLORATIONS OF SPERNER’S LEMMA AND ITS CONNECTIONS TO BROUWER’S FIXED POINT THEOREM
"... Abstract. We discuss Sperner’s Lemma in the form of two different proofs. Connections can be made to graph theory and cochains in simplicial complexes. This result is then used to prove Brouwer’s Fixed Point Theorem in a nontraditional manner. Our method provides a more constructive approach to the ..."
Abstract
 Add to MetaCart
Abstract. We discuss Sperner’s Lemma in the form of two different proofs. Connections can be made to graph theory and cochains in simplicial complexes. This result is then used to prove Brouwer’s Fixed Point Theorem in a nontraditional manner. Our method provides a more constructive approach
A Blossoming Algorithm for Tree Volumes of Composite Digraphs
"... Abstract. A weighted composite digraph is obtained from some weighted digraph by replacing each vertex with a weighted digraph. In this paper, we give a beautiful combinatorial proof of the formula for forest volumes of composite digraphs obtained by KelmansPakPostnikov [7]. Moreover, a generaliz ..."
Abstract
 Add to MetaCart
Abstract. A weighted composite digraph is obtained from some weighted digraph by replacing each vertex with a weighted digraph. In this paper, we give a beautiful combinatorial proof of the formula for forest volumes of composite digraphs obtained by KelmansPakPostnikov [7]. Moreover, a
Results 1  10
of
1,650