Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Characterization of $1$-sided error Strongly-Testable Graph Properties for bounded-degree graphs, including an appendix
299
0
0.0
(
0
)
Added by
Ilan Newman
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We study property testing of (di)graph properties in bounded-degree graph models. The study of graph properties in bounded-degree models is one of the focal directions of research in property testing in the last 15 years. However, despite of the many results and the extensive research effort, there is no characterization of the properties that are strongly-testable (i.e., testable with constant query complexity) even for $1$-sided error tests. The bounded-degree model can naturally be generalized to directed graphs resulting in two models that were considered in the literature. The first contains the directed graphs in which the outdegree is bounded but the indegree is not restricted. In the other, both the outdegree and indegree are bounded. We give a characterization of the $1$-sided error strongly-testable {em monotone} graph properties, and the $1$-sided error strongly-testable {em hereditary} graph properties in all the bounded-degree directed and undirected graphs models.
rate research
Read More
We show that for any odd $k$ and any instance of the Max-kXOR constraint satisfaction problem, there is an efficient algorithm that finds an assignment satisfying at least a $frac{1}{2} + Omega(1/sqrt{D})$ fraction of constraints, where $D$ is a bound on the number of constraints that each variable occurs in. This improves both qualitatively and quantitatively on the recent work of Farhi, Goldstone, and Gutmann (2014), which gave a emph{quantum} algorithm to find an assignment satisfying a $frac{1}{2} + Omega(D^{-3/4})$ fraction of the equations. For arbitrary constraint satisfaction problems, we give a similar result for triangle-free instances; i.e., an efficient algorithm that finds an assignment satisfying at least a $mu + Omega(1/sqrt{D})$ fraction of constraints, where $mu$ is the fraction that would be satisfied by a uniformly random assignment.
We study quantum algorithms for testing bipartiteness and expansion of bounded-degree graphs. We give quantum algorithms that solve these problems in time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing expansion, we also prove an Omega(N^(1/4)) quantum query lower bound, thus ruling out the possibility of an exponential quantum speedup. Our quantum algorithms follow from a combination of classical property testing techniques due to Goldreich and Ron, derandomization, and the quantum algorithm for element distinctness. The quantum lower bound is obtained by the polynomial method, using novel algebraic techniques and combinatorial analysis to accommodate the graph structure.
A (not necessarily proper) vertex colouring of a graph has clustering $c$ if every monochromatic component has at most $c$ vertices. We prove that planar graphs with maximum degree $Delta$ are 3-colourable with clustering $O(Delta^2)$. The previous best bound was $O(Delta^{37})$. This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $Delta$ that exclude a fixed minor are 3-colourable with clustering $O(Delta^5)$. The best previous bound for this result was exponential in $Delta$.
Answering some questions of Gutman, we show that, except for four specific trees, every connected graph G of order n, with no cycle of order 4 and with maximum degree at most 3, has energy greater that its order. Here, the energy of a graph is the sum of the moduli of its eigenvalues. We give more general theorems and state two conjectures.
For a graph $G=(V,E)$, $kin mathbb{N}$, and a complex number $w$ the partition function of the univariate Potts model is defined as [ {bf Z}(G;k,w):=sum_{phi:Vto [k]}prod_{substack{uvin E phi(u)=phi(v)}}w, ] where $[k]:={1,ldots,k}$. In this paper we give zero-free regions for the partition function of the anti-ferromagnetic Potts model on bounded degree graphs. In particular we show that for any $Deltain mathbb{N}$ and any $kgeq eDelta+1$, there exists an open set $U$ in the complex plane that contains the interval $[0,1)$ such that ${bf Z}(G;k,w) eq 0$ for any $win U$ and any graph $G$ of maximum degree at most $Delta$. (Here $e$ denotes the base of the natural logarithm.) For small values of $Delta$ we are able to give better results. As an application of our results we obtain improved bounds on $k$ for the existence of deterministic approximation algorithms for counting the number of proper $k$-colourings of graphs of small maximum degree.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
