Igor Shinkar

On the conditional hardness of coloring a 4-colorable graph with super-constant number of colors

For 3 ≤ q < Q we consider the ApproxColoring(q,Q) problem of deciding whether χ(G) ≤ q or &chi(G) ≥ Q for a given graph G. Hardness of this problem was shown in [7] for q = 3,4 and arbitrary large constant Q under variants of the Unique Games Conjecture [10]. We extend this result to values of Q that depend on the size of a given graph. The extension depends on the parameters of the conjectures we consider. Following the approach of [7], we find that a careful calculation of the parameters gives hardness of coloring a 4-colorable graph with lgc(lg(n)) colors for some constant c > 0. By improving the analysis of the reduction we show that under related conjectures it is hard to color a 4-colorable graph with lgc(n) colors for some constant c > 0. The main technical contribution of the paper is a variant of the Majority is Stablest Theorem, which says that among all balanced functions whose each coordinate has o(1) influence, the Majority function has the largest noise stability. We adapt the theorem for our applications to get a better dependency between the parameters required for the reduction.

Acquaintance Time of a Graph

We define the following parameter of connected graphs. For a given graph

An ~O(n) Queries Adaptive Tester for Unateness

G

A Tight Upper Bound on Acquaintance Time of Graphs

we place one agent in each vertex of

Two-sided error proximity oblivious testing

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Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball

. Every pair of agents sharing a common edge is declared to be acquainted. In each round we choose some matching of

A counterexample to monotonicity of relative mass in random walks

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Zero-Fixing Extractors for Sub-Logarithmic Entropy

(not necessarily a maximal matching), and for each edge in the matching the agents on this edge swap places. After the swap, again, every pair of agents sharing a common edge become acquainted, and the process continues. We define the \emph{acquaintance time} of a graph

On percolation and NP-hardness

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On Percolation and

, denoted by