Deepak Rajendraprasad

Bio: Deepak Rajendraprasad is an academic researcher from Indian Institutes of Technology. The author has contributed to research in topics: Chordal graph & Dimension (graph theory). The author has an hindex of 13, co-authored 67 publications receiving 513 citations. Previous affiliations of Deepak Rajendraprasad include University of Haifa & Indian Institute of Science.

Inapproximability of Rainbow Colouring

Abstract: A rainbow colouring of a connected graph G is a colouring of the edges of G such that every pair of vertices in G is connected by at least one path in which no two edges are coloured the same. The minimum number of colours required to rainbow colour G is called its rainbow connection number. Chakraborty, Fischer, Matsliah and Yuster have shown that it is NP-hard to compute the rainbow connection number of graphs [J. Comb. Optim., 2011]. Basavaraju, Chandran, Rajendraprasad and Ramaswamy have reported an (r+3)-factor approximation algorithm to rainbow colour any graph of radius r [Graphs and Combinatorics, 2012]. In this article, we use a result of Guruswami, Hastad and Sudan on the NP-hardness of colouring a 2-colourable 4-uniform hypergraph using constantly many colours [SIAM J. Comput., 2002] to show that for every positive integer k, it is NP-hard to distinguish between graphs with rainbow connection number 2k+2 and 4k+2. This, in turn, implies that there cannot exist a polynomial time algorithm to rainbow colour graphs with less than twice the optimum number of colours, unless P=NP. The authors have earlier shown that the rainbow connection number problem remains NP-hard even when restricted to the class of chordal graphs, though in this case a 4-factor approximation algorithm is available [COCOON, 2012]. In this article, we improve upon the 4-factor approximation algorithm to design a linear-time algorithm that can rainbow colour a chordal graph G using at most 3/2 times the minimum number of colours if G is bridgeless and at most 5/2 times the minimum number of colours otherwise. Finally we show that the rainbow connection number of bridgeless chordal graphs cannot be polynomial-time approximated to a factor less than 5/4, unless P=NP.

Testing for forbidden order patterns in an array

Abstract: In this paper, we study testing of sequence properties that are defined by forbidden order patterns. A sequence f : {1, . . . , n} → ℝ of length n contains a pattern π ∈ 𝔖k (𝔖k is the group of permutations of k elements), iff there are indices i1 f(iy) whenever π(x) > π(y). If f does not contain π, we say f is π-free. For example, for π = (2, 1), the property of being π-free is equivalent to being non-decreasing, i.e. monotone. The property of being (k, k − 1, . . . , 1)-free is equivalent to the property of having a partition into at most k − 1 non-decreasing subsequences.Let π ∈ 𝔖k, k constant, be a (forbidden) pattern. Assuming f is stored in an array, we consider the property testing problem of distinguishing the case that f is π-free from the case that f differs in more than ϵn places from any π-free sequence. We show the following results: There is a clear dichotomy between the monotone patterns and the non-monotone ones:• For monotone patterns of length k, i.e., (k, k − 1, . . . , 1) and (1, 2, . . . , k), we design non-adaptive one-sided error ϵ-tests of (ϵ−1 log n)O(k2) query complexity.• For non-monotone patterns, we show that for any size-k non-monotone π, any non-adaptive one-sided error ϵ-test requires at least [EQUATION] queries. This general lower bound can be further strengthened for specific non-monotone k-length patterns to Ω(n1−2/(k+1)).On the other hand, there always exists a non-adaptive one-sided error ϵ-test for π ∈ 𝔖k with O(ϵ−1/kn1−1/k) query complexity. Again, this general upper bound can be further strengthened for specific non-monotone patterns. E.g., for π = (1, 3, 2), we describe an ϵ-test with (almost tight) query complexity of [EQUATION].Finally, we show that adaptivity can make a big difference in testing non-monotone patterns, and develop an adaptive algorithm that for any π ∈ 𝔖3, tests π-freeness by making (ϵ−1 log n)O(1) queries.For all algorithms presented here, the running times are linear in their query complexity.

Rainbow Connection Number and Connected Dominating Sets

Abstract: Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).

Rainbow colouring of split and threshold Graphs

Abstract: A rainbow colouring of a connected graph is a colouring of the edges of the graph, such that every pair of vertices is connected by at least one path in which no two edges are coloured the same. Such a colouring using minimum possible number of colours is called an optimal rainbow colouring, and the minimum number of colours required is called the rainbow connection number of the graph. A Chordal Graph is a graph in which every cycle of length more than 3 has a chord. A Split Graph is a chordal graph whose vertices can be partitioned into a clique and an independent set. A threshold graph is a split graph in which the neighbourhoods of the independent set vertices form a linear order under set inclusion. In this article, we show the following:

Separation Dimension of Graphs and Hypergraphs

Abstract: Separation dimension of a hypergraph H, denoted by $$\pi (H)$$?(H), is the smallest natural number k so that the vertices of H can be embedded in $$\mathbb {R}^k$$Rk 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. In this paper, we study the separation dimension of hypergraphs and graphs.

Property Testing and its connection to Learning and Approximation

Abstract: 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.

Rainbow connections of graphs -- A survey

Abstract: 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

Abstract: 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.

Improved Testing Algorithms for Monotonicity.

Abstract: 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 \(\tilde{O}((n^2/\epsilon) \cdot \vert\Sigma\vert^2 \cdot \vert\Xi\vert)\).

Property Testing Lower Bounds via Communication Complexity.

Abstract: 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.