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.

Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl, in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl for 2-vertex-connected outerplanar graphs will work for all outer planar graphs.

2-connecting Outerplanar Graphs without Blowing Up the Pathwidth

Given a connected outerplanar graph G with pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). As a consequence, we get a constant factor approximation algorithm to compute a straight line planar drawing of any outerplanar graph, with its vertices placed on a two dimensional grid of minimum height. This settles an open problem raised by Biedl [3].

Conditions for the existence of heterochromatic Hamiltonian paths and cycles in edge colored graphs are well investigated in literature. A related problem in this domain is to obtain good lower bounds for the length of a maximum heterochromatic path in an edge colored graph G . This problem is also well explored by now and the lower bounds are often specified as functions of the minimum color degree of G -the minimum number of distinct colors occurring at edges incident to any vertex of G -denoted by ? ( G ) .Initially, it was conjectured that the lower bound for the length of a maximum heterochromatic path for an edge colored graph G would be ? 2 ? ( G ) 3 ? . Chen and Li (2005) showed that the length of a maximum heterochromatic path in an edge colored graph G is at least ? ( G ) - 1 , if 1 ? ? ( G ) ? 7 , and at least ? 3 ? ( G ) 5 ? + 1 , if ? ( G ) ? 8 . They conjectured that the tight lower bound would be ? ( G ) - 1 and demonstrated some examples which achieve this bound. An unpublished manuscript from the same authors (Chen, Li) reported to show that if ? ( G ) ? 8 , then G contains a heterochromatic path of length at least ? 2 ? ( G ) 3 ? + 1 .In this paper, we give lower bounds for the length of a maximum heterochromatic path in edge colored graphs without small cycles. We show that if G has no four cycles, then it contains a heterochromatic path of length at least ? ( G ) - o ( ? ( G ) ) and if the girth of G is at least 4 log 2 ( ? ( G ) ) + 2 , then it contains a heterochromatic path of length at least ? ( G ) - 2 , which is only one less than the bound conjectured by Chen and Li (2005). Other special cases considered include lower bounds for the length of a maximum heterochromatic path in edge colored bipartite graphs and triangle-free graphs: for triangle-free graphs we obtain a lower bound of ? 5 ? ( G ) 6 ? and for bipartite graphs we obtain a lower bound of ? 6 ? ( G ) - 3 7 ? .In this paper, it is also shown that if the coloring is such that G has no heterochromatic triangles, then G contains a heterochromatic path of length at least ? 13 ? ( G ) 17 ? . This improves the previously known ? 3 ? ( G ) 4 ? bound obtained by Chen and Li (2011). We also give a relatively shorter and simpler proof showing that any edge colored graph G contains a heterochromatic path of length at least ? 2 ? ( G ) 3 ? .

Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs

A $d$-hypertree on $[n]$ is a maximal acyclic $d$-dimensional simplicial complex with full $(d-1)$-skeleton on the vertex set $[n]$. Alternatively, in the language of algebraic topology, it is a minimal $d$-dimensional simplicial complex $T$ (assuming full $(d-1)$-skeleton) such that $\tilde{H}_{d-1}(T;\mathbb{F})=0$. 
The $d$-hypertrees are a basic object in combinatorial theory of simplicial complexes. They have been studied; and yet, many of their structural aspects remain poorly understood. 
In this paper we study the boundaries $\partial_d T$ of $d$-hypertrees, and the fundamental $d$-cycles defined by them. Our findings include: 
1. A full characterization of $\partial_d T$ over $\mathbb{F}_2$ for $d \leq 2$, and some partial results for $d \geq 3$. 
2. Lower bounds on the maximum size of a largest simple $d$-cycle on $[n]$. In particular, for $d=2$, we construct a {\em Hamiltonian $d$-cycle} $H$ on $[n]$, i.e., a simple $d$-cycle of size ${{n-1} \choose d} + 1$. For $d\geq 3$, we construct a simple $d$-cycle of size ${{n-1} \choose d} - O(n^{d-2})$. 
3. Observing that the maximum of the expected distance between two vertices chosen uniformly at random in a tree ($1$-hypertree) on $[n]$ is at most $\thicksim n/3$, attained on Hamiltonian paths, we ask a similar question about $d$-hypertrees. "How large can be the {\em average} size of a fundamental cycle of a $d$-hypertree $T$ (i.e., the expected size of the dependency created by adding a $d$-simplex on $[n]$, chosen uniformly at random, to $T$)?" For every $d \in \mathbb{N}$, we construct an infinite family of $d$-hypertrees $\{T\}$ with the average size of a fundamental cycle at least $c_d\, |T| \,=\, c_d\,{n-1 \choose d}$, where $c_d$ is a constant depending on the dimension $d$ alone.

Deepak Rajendraprasad

Papers

Upper bound on cubicity in terms of boxicity for graphs of low chromatic number

2-connecting Outerplanar Graphs without Blowing Up the Pathwidth

2-connecting Outerplanar Graphs without Blowing Up the Pathwidth

Heterochromatic paths in edge colored graphs without small cycles and heterochromatic-triangle-free graphs

Boundaries of Hypertrees, and Hamiltonian Cycles in Simplicial Complexes