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16
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 25 (13 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
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 color ..."
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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.
Local chromatic number, Ky Fan’s theorem, and circular colorings
 Combinatorica
"... The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex colorcritical sub ..."
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Cited by 18 (6 self)
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The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex colorcritical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [14] which generalizes the BorsukUlam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2 ⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [15].) This yields a lower bound of ⌈t/2 ⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier
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, Sarkari ..."
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Cited by 8 (1 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
On the chromatic number of some geometric type Kneser graphs
, 2004
"... We estimate the chromatic number of graphs whose vertex set is the set of edges of a complete geometric graph on n points, and adjacency is defined in terms of geometric disjointness or geometric intersection. ..."
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Cited by 7 (5 self)
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We estimate the chromatic number of graphs whose vertex set is the set of edges of a complete geometric graph on n points, and adjacency is defined in terms of geometric disjointness or geometric intersection.
Graph coloring manifolds
, 2005
"... We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovász. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds ..."
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Cited by 4 (0 self)
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We introduce a new and rich class of graph coloring manifolds via the Hom complex construction of Lovász. The class comprises examples of Stiefel manifolds, series of spheres and products of spheres, cubical surfaces, as well as examples of Seifert manifolds. Asymptotically, graph coloring manifolds provide examples of highly connected, highly symmetric manifolds. 1
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. ..."
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Cited by 4 (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.
THE BORSUKULAMPROPERTY, TUCKERPROPERTY AND CONSTRUCTIVE PROOFS IN COMBINATORICS
, 2005
"... This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tuckerproperty of a finite group G is introduced and its relation to the topological BorsukUlamproperty is ..."
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Cited by 3 (0 self)
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This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tuckerproperty of a finite group G is introduced and its relation to the topological BorsukUlamproperty is discussed. Applications of the Tuckerproperty in combinatorics are demonstrated.
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.