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13
Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to ..."
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Cited by 22 (11 self)
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We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
Generalized Kneser coloring theorems with combinatorial proofs
 INVENTIONES MATH
, 2001
"... The Kneser conjecture (1955) was proved by Lovasz (1978) using the BorsukUlam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the BorsukUlam theorem and its extensions. Only in 2000, Matousek provided the rst combinatorial proof of t ..."
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Cited by 21 (4 self)
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The Kneser conjecture (1955) was proved by Lovasz (1978) using the BorsukUlam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the BorsukUlam theorem and its extensions. Only in 2000, Matousek provided the rst combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, AlonFranklLovasz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem.
The Hardness of 3Uniform Hypergraph Coloring
 In Proc. of the 43rd Annual IEEE Symposium on Foundations of Computer Science
, 2002
"... We prove that coloring a 3uniform 2colorable hypergraph with any constant number of colors is NPhard. The best known algorithm [20] colors such a graph using O(n ) colors. Our result immediately implies that for any constants k > 2 and c 2 > c 1 > 1, coloring a kuniform c 1 colorable hype ..."
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Cited by 20 (4 self)
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We prove that coloring a 3uniform 2colorable hypergraph with any constant number of colors is NPhard. The best known algorithm [20] colors such a graph using O(n ) colors. Our result immediately implies that for any constants k > 2 and c 2 > c 1 > 1, coloring a kuniform c 1 colorable hypergraph with c 2 colors is NPhard; leaving completely open only the k = 2 graph case.
A combinatorial proof of Kneser's conjecture
"... Kneser's conjecture, first proved by Lov'asz in 1978, states that the graph with all kelement subsets of f1; 2; : : : ; ng as vertices and with edges connecting disjoint sets has chromatic number n \Gamma 2k + 2. We derive this result from Tucker's combinatorial lemma on labeling the vertices of ..."
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Cited by 9 (0 self)
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Kneser's conjecture, first proved by Lov'asz in 1978, states that the graph with all kelement subsets of f1; 2; : : : ; ng as vertices and with edges connecting disjoint sets has chromatic number n \Gamma 2k + 2. We derive this result from Tucker's combinatorial lemma on labeling the vertices of special triangulations of the octahedral ball. By specializing a proof of Tucker's lemma, we obtain selfcontained purely combinatorial proof of Kneser's conjecture. 1 Introduction Let \Gamma [n] k \Delta denote the system of all kelement subsets of the set [n] = f1; 2; : : : ; ng. The Kneser graph KG(n; k) has vertex set \Gamma [n] k \Delta and edge set ffS; S 0 g : S; S 0 2 \Gamma [n] k \Delta ; S " S 0 = ;g. Kneser [8] conjectured in 1955 that (KG(n; k)) n \Gamma 2k + 2, n 2k 2, where denotes the chromatic number. This was proved in 1978 by Lov'asz [12], as one of the earliest and most spectacular applications of topological methods in combinatorics. Several...
On the chromatic number of Kneser hypergraphs
, 2000
"... We give a simple and elementary proof of Kr'iz's lower bound on the chromatic number of the Kneser rhypergraph of a set system S. 1 Introduction Let S be a system of subsets of a finite set X. The Kneser rhypergraph KG r (S) has S as the vertex set, and an rtuple (S 1 ; S 2 ; : : : ; S r ) of ..."
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Cited by 3 (1 self)
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We give a simple and elementary proof of Kr'iz's lower bound on the chromatic number of the Kneser rhypergraph of a set system S. 1 Introduction Let S be a system of subsets of a finite set X. The Kneser rhypergraph KG r (S) has S as the vertex set, and an rtuple (S 1 ; S 2 ; : : : ; S r ) of sets in S forms an edge if S i "S j = ; for all i 6= j. In particular, KG(S) = KG 2 (S) is the Kneser graph of S. Kneser [8] conjectured in 1955 that (KG( \Gamma [n] k \Delta )) n \Gamma 2k+2, n 2k, where \Gamma [n] k \Delta denotes the system of all kelement subsets of the set [n] = f1; 2; : : : ; ng, and denotes the chromatic number. This was proved in 1978 by Lov'asz [12], as one of the earliest and most spectacular applications of topological methods in combinatorics. Several other proofs have been published since then, all of them topological; among them, at least those of B'ar'any [2], Dol'nikov [5] (also see [6], [7]), and Sarkaria [13] can be regarded as substantially di...
On Valiant’s holographic algorithms
"... Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primit ..."
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Cited by 2 (2 self)
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Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.
On generalized Kneser hypergraph colorings
, 2006
"... In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs KGr sS, “generalized runiform Kneser hypergraphs with intersection multiplicities s. ” It generalized previous lower bounds by Kˇríˇz (1992/2000) for the case s = (1,...,1) without intersection mul ..."
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Cited by 2 (1 self)
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In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs KGr sS, “generalized runiform Kneser hypergraphs with intersection multiplicities s. ” It generalized previous lower bounds by Kˇríˇz (1992/2000) for the case s = (1,...,1) without intersection multiplicities, and by Sarkaria (1990) for S = ([n]) k. Here we discuss subtleties and difficulties that arise for intersection multiplicities si> 1: 1. In the presence of intersection multiplicities, there are two different versions of a “Kneser hypergraph,”depending on whether one admits hypergraph edges that are multisets rather than sets. We show that the chromatic numbers are substantially different for the two concepts of hypergraphs. The lower bounds of Sarkaria (1990) and Ziegler (2002) apply only to the multiset version. 2. The reductions to the case of prime r in the proofs by Sarkaria and by Ziegler work only if the intersection multiplicities are strictly smaller than the largest prime factor of r. Currently we have no valid proof for the lower bound result in the other cases. We also show that all uniform hypergraphs without multiset edges can be represented as generalized Kneser hypergraphs.
Topological Lower Bounds for the Chromatic Number: A Hierarchy
 JAHRESBER. DEUTSCH. MATH.VEREIN
, 2004
"... Kneser's conjecture, first proved by Lovasz in 1978, states that the graph with all k element subsets of 2, . . . , n} as vertices and all pairs of disjoint sets as edges has chromatic number n2k+ 2. Several other proofs have been published (by Barany, Schrijver, Dol'nikov, Sarkaria, Krz, Gr ..."
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Cited by 1 (0 self)
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Kneser's conjecture, first proved by Lovasz in 1978, states that the graph with all k element subsets of 2, . . . , n} as vertices and all pairs of disjoint sets as edges has chromatic number n2k+ 2. Several other proofs have been published (by Barany, Schrijver, Dol'nikov, Sarkaria, Krz, Greene, and others), all of them based on the BorsukUlam theorem from algebraic topology, but otherwise quite di#erent. Each can be extended to yield some lower bound on the chromatic number of an arbitrary graph. We show