In this paper, we consider the question of determining whether a function f has property P or is e-far from any function with property P. A property testing algorithm is given a sample of the value of f on instances drawn according to some distribution. In some cases, it is also allowed to query f on instances of its choice. We study this question for different properties and establish some connections to problems in learning theory and approximation.In particular, we focus our attention on testing graph properties. Given access to a graph G in the form of being able to query whether an edge exists or not between a pair of vertices, we devise algorithms to test whether the underlying graph has properties such as being bipartite, k-Colorable, or having a p-Clique (clique of density p with respect to the vertex set). Our graph property testing algorithms are probabilistic and make assertions that are correct with high probability, while making a number of queries that is independent of the size of the graph. Moreover, the property testing algorithms can be used to efficiently (i.e., in time linear in the number of vertices) construct partitions of the graph that correspond to the property being tested, if it holds for the input graph.

Property Testing and its connection to Learning and Approximation

The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems or questions.

Rainbow connections of graphs -- A survey

The concept of rainbow connection was introduced by Chartrand et al. [14] in 2008. It is interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems and questions.

Rainbow Connections of Graphs: A Survey

We present improved algorithms for testing monotonicity of functions. Namely, given the ability to query an unknown function f: Σ n ↦ Ξ, where Σ and Ξ are finite ordered sets, the test always accepts a monotone f, and rejects f with high probability if it is e-far from being monotone (i.e., every monotone function differs from f on more than an e fraction of the domain). For any e > 0, the query complexity of the test is O((n/e) · log ∣Σ ∣ · log ∣Ξ∣). The previous best known bound was $\tilde{O}((n^2/\epsilon) \cdot \vert\Sigma\vert^2 \cdot \vert\Xi\vert)$.

Improved Testing Algorithms for Monotonicity.

We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Omega(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

Property Testing Lower Bounds via Communication Complexity.

An {\em orientation} of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. Converting two-way traffic networks to one-way traffic networks and bidirectional communication networks to unidirectional communication networks are practical instances of graph orientations. In these contexts minimising the diameter of the resulting oriented graph is of prime interest. 
The $n$-star network topology was proposed as an alternative to the hypercube network topology for multiprocessor systems by Akers and Krishnamurthy [IEEE Trans. on Computers (1989)]. The $n$-star graph $S_n$ consists of $n!$ vertices, each labelled with a distinct permutation of $[n]$. Two vertices are adjacent if their labels differ exactly in the first and one other position. $S_n$ is an $(n-1)$-regular, vertex-transitive graph with diameter $\lfloor 3(n-1)/2 \rfloor$. Orientations of $S_n$, called unidirectional star graphs and distributed routing protocols over them were studied by Day and Tripathi [Information Processing Letters (1993)] and Fujita [The First International Symposium on Computing and Networking (CANDAR 2013)]. Fujita showed that the (directed) diameter of this unidirectional star graph $\overrightarrow{S_n}$ is at most $\lceil{5n/2}\rceil + 2$. 
In this paper, we propose a new distributed routing algorithm for the same $\overrightarrow{S_n}$ analysed by Fujita, which routes a packet from any node $s$ to any node $t$ at an undirected distance $d$ from $s$ using at most $\min\{4d+4, 2n+4\}$ hops. This shows that the (directed) diameter of $\overrightarrow{S_n}$ is at most $2n+4$. We also show that the diameter of $\overrightarrow{S_n}$ is at least $2n$ when $n \geq 7$, thereby showing that our upper bound is tight up to an additive factor.

Oriented Diameter of Star Graphs

Oriented diameter of star graphs

Characterization of B0-VPG Cocomparability Graphs and a 2D Visualization of their Posets

A collection of linear orders on X, say ${\mathscr{L}}$
 , is said to realize a partially ordered set (or poset) $\mathcal {P} = (X, \preceq )$
 if, for any two distinct x,y ∈ X, x ≼ y if and only if x ≺Ly, $\forall L \in {\mathscr{L}}$
 . We call ${\mathscr{L}}$
 a realizer of $\mathcal {P}$
 . The dimension of $\mathcal {P}$
 , denoted by $dim(\mathcal {P})$
 , is the minimum cardinality of a realizer of $\mathcal {P}$
 . A containment model $M_{\mathcal {P}}$
 of a poset $\mathcal {P}=(X,\preceq )$
 maps every x ∈ X to a set Mx such that, for every distinct x,y ∈ X, x ≼ y if and only if $M_{x} \varsubsetneq M_{y}$
 . We shall be using the collection (Mx)x∈X to identify the containment model $M_{\mathcal {P}}$
 . A poset $\mathcal {P}=(X,\preceq )$
 is a Containment order of Paths in a Tree (CPT poset), if it admits a containment model $M_{\mathcal {P}}=(P_{x})_{x \in X}$
 where every Px is a path of a tree T, which is called the host tree of the model. We show that if a poset $\mathcal {P}$
 admits a CPT model in a host tree T of maximum degree Δ and radius r, then $dim(\mathcal {P}) \leq \lg \lg {\Delta } + (\frac {1}{2} + o(1))\lg \lg \lg {\Delta } + \lg r + \frac {1}{2} \lg \lg r + \frac {1}{2}\lg \pi + 3$
 . This bound is asymptotically tight up to an additive factor of $\min \limits (\frac {1}{2}\lg \lg \lg {\Delta }, \frac {1}{2}\lg \lg r)$
 . Further, let $\mathcal {P}(1,2;n)$
 be the poset consisting of all the 1-element and 2-element subsets of [n] under ‘containment’ relation and let dim(1,2;n) denote its dimension. The proof of our main theorem gives a simple algorithm to construct a realizer for $\mathcal {P}(1,2;n)$
 whose cardinality is only an additive factor of at most $\frac {3}{2}$
 away from the optimum.

Dimension of CPT posets

The 'separation dimension' of a graph $G$ is the smallest natural number $k$ for which the vertices of $G$ can be embedded in $\mathbb{R}^k$ such that any pair of disjoint edges in $G$ can be separated by a hyperplane normal to one of the axes. Equivalently, it is the smallest possible cardinality of a family $\mathcal{F}$ of total orders of the vertices of $G$ such that for any two disjoint edges of $G$, there exists at least one total order in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. In general, the maximum separation dimension of a graph on $n$ vertices is $\Theta(\log n)$. In this article, we focus on bounded degree graphs and show that the separation dimension of a graph with maximum degree $d$ is at most $2^{9log^{\star} d} d$. We also demonstrate that the above bound is nearly tight by showing that, for every $d$, almost all $d$-regular graphs have separation dimension at least $\lceil d/2\rceil$.

Deepak Rajendraprasad

Papers

Oriented Diameter of Star Graphs

Oriented diameter of star graphs

Characterization of B0-VPG Cocomparability Graphs and a 2D Visualization of their Posets

Dimension of CPT posets

Separation dimension of bounded degree graphs