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 orientation of an undirected graph $G$ is an assignment of exactly one direction to each edge of $G$. The oriented diameter of a graph $G$ is the smallest diameter among all the orientations of $G$. The maximum oriented diameter of a family of graphs $\mathscr{F}$ is the maximum oriented diameter among all the graphs in $\mathscr{F}$. Chvatal and Thomassen [JCTB, 1978] gave a lower bound of $\frac{1}{2}d^2+d$ and an upper bound of $2d^2+2d$ for the maximum oriented diameter of the family of $2$-edge connected graphs of diameter $d$. We improve this upper bound to $ 1.373 d^2 + 6.971d-1 $, which outperforms the former upper bound for all values of $d$ greater than or equal to $8$. For the family of $2$-edge connected graphs of diameter $3$, Kwok, Liu and West [JCTB, 2010] obtained improved lower and upper bounds of $9$ and $11$ respectively. For the family of $2$-edge connected graphs of diameter $4$, the bounds provided by Chvatal and Thomassen are $12$ and $40$ and no better bounds were known. By extending the method we used for diameter $d$ graphs, along with an asymmetric extension of a technique used by Chvatal and Thomassen, we have improved this upper bound to $21$.

An Improvement to Chv\'atal and Thomassen's Upper Bound for Oriented Diameter

The eternal vertex cover problem is a variant of the classical vertex cover problem where a set of guards on the vertices have to be dynamically reconfigured from one vertex cover to another in every round of an attacker-defender game. The minimum number of guards required to protect a graph from an infinite sequence of attacks is the eternal vertex cover number (evc) of the graph. It is known that, given a graph G and an integer k, checking whether ${\text {evc}}(G) \le k$ is NP-Hard. However, for any graph G, ${\text {mvc}}(G) \le {\text {evc}}(G) \le 2 {\text {mvc}}(G)$, where ${\text {mvc}}(G)$ is the minimum vertex cover number of G. Precise value of eternal vertex cover number is known only for certain very basic graph classes like trees, cycles and grids. Though a characterization is known for graphs for which ${\text {evc}}(G) = 2{\text {mvc}}(G)$, a characterization of graphs for which ${\text {evc}}(G) = {\text {mvc}}(G)$ remained open. Here, we achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For some graph classes including biconnected chordal graphs, our characterization leads to a polynomial time algorithm to precisely determine ${\text {evc}}(G)$ and to determine a safe strategy of guard movement in each round of the game with ${\text {evc}}(G)$ guards.

On Graphs with Minimal Eternal Vertex Cover Number

The separation dimension π(G) of a hypergraph G is the smallest natural number k for which the vertices of G can be embedded in Rk so that any pair of disjoint edges in G can be separated by a hyperplane normal to one of the axes. Equivalently, it is the cardinality of a smallest family F of total orders of V (G), such that for any two disjoint edges of G, there exists at least one total order in F in which all the vertices in one edge precede those in the other. Separation dimension is a monotone parameter; adding more edges cannot reduce the separation dimension of a hypergraph. In this article we discuss the influence of separation dimension and edge-density of a graph on one another. On one hand, we show that the maximum separation dimension of a k-degenerate graph on n vertices is O (k lg lgn) and that there exists a family of 2-degenerate graphs with separation dimension Ω (lg lg n). On the other hand, we show that graphs with bounded separation dimension cannot be very dense. Quantitatively, we prove that n-vertex graphs with separation dimension s have at most 3(4 lg n)s−2 · n edges. We do not believe that this bound is optimal and give a question and a remark on the optimal bound.

Separation dimension and sparsity

A linear ordering of the vertices of a graph G separates two edges of G if both the endpoints of one precede both the endpoints of the other in the order. We call two edges $$\{a,b\}$$
 and $$\{c,d\}$$
 of G strongly independent if the set of endpoints $$\{a,b,c,d\}$$
 induces a $$2K_2$$
 in G. The induced separation dimension of a graph G is the smallest cardinality of a family $$\mathcal {L}$$
 of linear orders of V(G) such that every pair of strongly independent edges in G are separated in at least one of the linear orders in $$\mathcal {L}$$
 . For each $$k \in \mathbb {N}$$
 , the family of graphs with induced separation dimension at most k is denoted by $${\text {ISD}}(k)$$
 . In this article, we initiate a study of this new dimensional parameter. The class $${\text {ISD}}(1)$$
 or, equivalently, the family of graphs which can be embedded on a line so that every pair of strongly independent edges are disjoint line segments, is already an interesting case. On the positive side, we give characterizations for chordal graphs in $${\text {ISD}}(1)$$
 which immediately lead to a polynomial time algorithm which determines the induced separation dimension of chordal graphs. On the negative side, we show that the recognition problem for $${\text {ISD}}(1)$$
 is NP-complete for general graphs. Nevertheless, we show that the maximum induced matching problem can be solved efficiently in $${\text {ISD}}(1)$$
 . We then briefly study $${\text {ISD}}(2)$$
 and show that it contains many important graph classes like outerplanar graphs, chordal graphs, circular arc graphs and polygon-circle graphs. Finally, we describe two techniques to construct graphs with large induced separation dimension. The first one is used to show that the maximum induced separation dimension of a graph on n vertices is $$\Theta (\lg n)$$
 and the second one is used to construct AT-free graphs with arbitrarily large induced separation dimension. The second construction is also used to show that, for every $$k \ge 2$$
 , the recognition problem for $${\text {ISD}}(k)$$
 is NP-complete even on AT-free graphs.

The Induced Separation Dimension of a Graph

A family F of permutations of the vertices of a hypergraph H is called "pairwise suitable" for H if, for every pair of disjoint edges in H, there exists a permutation in F in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for H is called the "separation dimension" of H and is denoted by \pi(H). Equivalently, \pi(H) is the smallest natural number k so that the vertices of H can be embedded in R^k such that any two disjoint edges of H can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph H is equal to the "boxicity" of the line graph of H. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.

Deepak Rajendraprasad

Papers

An Improvement to Chv\'atal and Thomassen's Upper Bound for Oriented Diameter

On Graphs with Minimal Eternal Vertex Cover Number

Separation dimension and sparsity

The Induced Separation Dimension of a Graph

Pairwise Suitable Family of Permutations and Boxicity