603 research outputs found
Evasiveness and the Distribution of Prime Numbers
We confirm the eventual evasiveness of several classes of monotone graph
properties under widely accepted number theoretic hypotheses. In particular we
show that Chowla's conjecture on Dirichlet primes implies that (a) for any
graph , "forbidden subgraph " is eventually evasive and (b) all
nontrivial monotone properties of graphs with edges are
eventually evasive. ( is the number of vertices.)
While Chowla's conjecture is not known to follow from the Extended Riemann
Hypothesis (ERH, the Riemann Hypothesis for Dirichlet's functions), we show
(b) with the bound under ERH.
We also prove unconditional results: (a) for any graph , the query
complexity of "forbidden subgraph " is ; (b) for
some constant , all nontrivial monotone properties of graphs with edges are eventually evasive.
Even these weaker, unconditional results rely on deep results from number
theory such as Vinogradov's theorem on the Goldbach conjecture.
Our technical contribution consists in connecting the topological framework
of Kahn, Saks, and Sturtevant (1984), as further developed by Chakrabarti,
Khot, and Shi (2002), with a deeper analysis of the orbital structure of
permutation groups and their connection to the distribution of prime numbers.
Our unconditional results include stronger versions and generalizations of some
result of Chakrabarti et al.Comment: 12 pages (conference version for STACS 2010
One-Point Suspensions and Wreath Products of Polytopes and Spheres
It is known that the suspension of a simplicial complex can be realized with
only one additional point. Suitable iterations of this construction generate
highly symmetric simplicial complexes with various interesting combinatorial
and topological properties. In particular, infinitely many non-PL spheres as
well as contractible simplicial complexes with a vertex-transitive group of
automorphisms can be obtained in this way.Comment: 17 pages, 8 figure
Knots in collapsible and non-collapsible balls
We construct the first explicit example of a simplicial 3-ball B_{15,66} that
is not collapsible. It has only 15 vertices. We exhibit a second 3-ball
B_{12,38} with 12 vertices that is collapsible and evasive, but not shellable.
Finally, we present the first explicit triangulation of a 3-sphere S_{18, 125}
(with only 18 vertices) that is not locally constructible. All these examples
are based on knotted subcomplexes with only three edges; the knots are the
trefoil, the double trefoil, and the triple trefoil, respectively. The more
complicated the knot is, the more distant the triangulation is from being
polytopal, collapsible, etc. Further consequences of our work are:
(1) Unshellable 3-spheres may have vertex-decomposable barycentric
subdivisions.
(This shows the strictness of an implication proven by Billera and Provan.)
(2) For d-balls, vertex-decomposable implies non-evasive implies collapsible,
and for d=3 all implications are strict.
(This answers a question by Barmak.)
(3) Locally constructible 3-balls may contain a double trefoil knot as a
3-edge subcomplex.
(This improves a result of Benedetti and Ziegler.)
(4) Rudin's ball is non-evasive.Comment: 25 pages, 5 figures, 11 tables, references update
Any monotone property of 3-uniform hypergraphs is weakly evasive
© 2014 Elsevier B.V. For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [11] show that any non-constant monotone property P:{0,1}(n2)→{0,1} of n-vertex graphs has D(P)=Ω(n2).We extend their result to 3-uniform hypergraphs. In particular, we show that any non-constant monotone property P:{0,1}(n3)→{0,1} of n-vertex 3-uniform hypergraphs has D(P)=Ω(n3).Our proof combines the combinatorial approach of Rivest and Vuillemin with the topological approach of Kahn, Saks, and Sturtevant [6]. Interestingly, our proof makes use of Vinogradov's Theorem (weak Goldbach Conjecture), inspired by its recent use by Babai et al. [1] in the context of the topological approach. Our work leaves the generalization to k-uniform hypergraphs as an intriguing open question
Graph Properties in Node-Query Setting: Effect of Breaking Symmetry
The query complexity of graph properties is well-studied when queries are on the edges. We investigate the same when queries are on the nodes. In this setting a graph G = (V,E) on n vertices and a property P are given. A black-box access to an unknown subset S of V is provided via queries of the form "Does i belong to S?". We are interested in the minimum number of queries needed in the worst case in order to determine whether G[S] - the subgraph of G induced on S - satisfies P.
Our primary motivation to study this model comes from the fact that it allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on the hereditary graph properties - properties that are closed under deletion of vertices as well as edges. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on G, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., n.
We show that in the absence of any symmetry on G it can fall as low as O(n^{1/(d + 1)}) where d denotes the minimum possible degree of a minimal forbidden sub-graph for P. In particular, every hereditary property benefits at least quadratically. The main question left open is: Can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with finitely many forbidden subgraphs by exhibiting a bound of Omega(n^{1/k}) for a constant k depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing Omega(log(n)*log(log(n))) bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdos and Rado
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