Evasiveness and the Distribution of Prime Numbers

  title={Evasiveness and the Distribution of Prime Numbers},
  author={L{\'a}szl{\'o} Babai and Anandam Banerjee and Raghav Kulkarni and Vipul Naik},
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 $H$, "forbidden subgraph $H$" is eventually evasive and (b) all nontrivial monotone properties of graphs with $\le n^{3/2-\epsilon}$ edges are eventually evasive. ($n$ is the number of vertices.) While Chowla's conjecture is not known to follow from the Extended… 

A Lower Bound for the Complexity of Monotone Graph Properties

It is shown that all nontrivial monotone graph properties are evasive, i.e., have decision tree complexity $\binom{n}{2}$ and a lower bound of $\frac{1}{3}n^2-o( n^2)$ for general $n$ is proved.

Evasiveness through a circuit lens

This paper studies a weakening of the Evasiveness Conjecture called weak-EC, which asserts that every non-trivial monotone transitive Boolean function must have D(f) ≥ n1- ε, for every ε > 0.

Monotone Properties of k-Uniform Hypergraphs are Weakly Evasive

Inspired by the outline of the KQS approach, the general framework of "orbit augmentation sequences" of sets with group actions is formalized, showing that a parameter of such sequences is a lower bound on the decision-tree complexity for any nontrivial monotone property that is Γ-invariant for all groups involved in the orbit augmentation sequence, assuming all those groups are p-groups.

Any monotone property of 3-uniform hypergraphs is weakly evasive

Graph Properties in Node-Query Setting: Effect of Breaking Symmetry

The answer is no for any hereditary property with {finitely many} forbidden subgraphs by exhibiting a bound of $\Omega(n^{1/k})$ for some constant $k$ depending only on the property.

On the Sensitivity Complexity of k-Uniform Hypergraph Properties

This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Ω (nk/2), and shows that for many classes of transitive Boolean functions the minimum achievable sensitivity complexity can be O(N1/3).

On the Power of Parity Queries in Boolean Decision Trees

It is shown that the parity queries can be replaced by ordinary ones at the cost of the total influence aka average sensitivity per query, which is tight as demonstrated by the parity function.

Evasive properties of sparse graphs and some linear equations in primes

List of my favorite publications

[114] László Babai. Vertex-transitive graphs and vertex-transitive maps. games: A randomized proof system and a hierarchy of complexity classes. On the orders of primitive groups with restricted

Finite Groups and Complexity Theory: From Leningrad to Saint Petersburg via Las Vegas

This paper is a personal account of the author's journey through the evolution of some of these interconnections, culminating in recent definitive results on the matrix group membership problem.



A topological approach to evasiveness

A graph property G is a collection of graphs closed under isomorphism. G is said to be evasive if, for every possible local search strategy, there is at least one graph for which membership in G

Evasiveness of Subgraph Containment and Related Properties

It is proved that properties that are preserved under taking graph minors are evasive for all sufficiently large n, which greatly generalizes the evasiveness result for planarity.

On the time required to recognize properties of graphs: a problem

In a recent paper [i], Holt and Reingold have proved the following results: any algorithm which, given an n-node graph, detects whether or not the graph enjoys property P must in the worst case probe 0(n 2) entries of the incidence matrix.

Superpolynomial Lower Bounds for Monotone Span Programs

The results give the first superpolynomial lower bounds for linear secret sharing schemes and show that the perfect matching function can be computed by polynomial size (non-monotone) span programs over arbitrary fields.

The Least Prime in Certain Arithmetic Progressions

1 . S. Hedman, A First Course in Logic, Oxford University Press, New York, 2004. 2. D. Hobby and D. M. Silberger, Quotients of Primes, this Monthly 100 (1993) 50-52. 3. S. Marivani, On some

On the recognition complexity of some graph properties

By applying a topological approach due to Kahn, Saks and Sturtevant, we prove that all decreasing graph properties consisting of bipartite graphs only are elusive. This is an analogue to a well-known

Examples of Z-Acyclic and Contractible Vertex-Homogeneous Simplicial Complexes

  • F. Lutz
  • Mathematics
    Discret. Comput. Geom.
  • 2002
A five-dimensional example and further examples in higher dimensions are constructed, one of which is Oliver’s example of dimension 11, the only previously known example of a non-contractible Z -acyclic vertex-homogeneous simplicial complex.

Some Theorems in the Analytic Theory of Numbers

Multiplicative Number Theory

From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The