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.

The rainbow connection number of a connected graph is the minimum number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. In this article we show that for every connected graph on n vertices with minimum degree δ, the rainbow connection number is upper bounded by 3n/(δ + 1) + 3. This solves an open problem from Schiermeyer (Combinatorial Algorithms, Springer, Berlin/Hiedelberg, 2009, pp. 432–437), improving the previously best known bound of 20n/δ (J Graph Theory 63 (2010), 185–191). This bound is tight up to additive factors by a construction mentioned in Caro et al. (Electr J Combin 15(R57) (2008), 1).

As an intermediate step we obtain an upper bound of 3n/(δ + 1) − 2 on the size of a connected two-step dominating set in a connected graph of order n and minimum degree δ. This bound is tight up to an additive constant of 2. This result may be of independent interest.

We also show that for every connected graph G with minimum degree at least 2, the rainbow connection number, rc(G), is upper bounded by Γc(G) + 2, where Γc(G) is the connected domination number of G. Bounds of the form diameter(G)⩽rc(G)⩽diameter(G) + c, 1⩽c⩽4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, asteroidal triple-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)⩽3·radius(G). In most of these cases, we also demonstrate the tightness of the bounds. © 2012 Wiley Periodicals, Inc.

Rainbow connection number and connected dominating sets

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) <= r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (Star graph for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) <= rk. 
It is known that computing rc(G) is NP-Hard [Chakraborty et al., 2009]. Here, we present a (r+3)-factor approximation algorithm which runs in O(nm) time and a (d+3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.

Rainbow Connection Number and Radius

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) ≤ r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (K 1,n for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) ≤ rk. Hitherto, the only reported upper bound on the rainbow connection number of bridgeless graphs is 4n/5 ? 1, where n is order of the graph (Caro et al. in Electron J Comb 15(1):Research paper 57, 13, 2008). It is known that computing rc(G) is NP-Hard (Chakraborty and fischer in J Comb Optim 1---18, 2009). Here, we present a (r + 3)-factor approximation algorithm which runs in O(nm) time and a (d + 3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.

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 ∈ &#x1d516;k (&#x1d516;k 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 ∈ &#x1d516;k, 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 ∈ &#x1d516;k 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 ∈ &#x1d516;3, 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.

Testing for forbidden order patterns in an array

The rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its edges, so that every pair of vertices is connected by at least one path in which no two edges are colored the same. Our main result is that $rc(G)\leq \lceil\frac{n}{2}\rceil$ for any 2-connected graph with at least three vertices. We conjecture that $rc(G)\leq n/\kappa+C$ for a $\kappa$-connected graph $G$ of order $n$, where $C$ is a constant, and prove the conjecture for certain classes of graphs. We also prove that $rc(G)\leq(2+\varepsilon)n/\kappa+23/\varepsilon^2$ for any $\varepsilon>0$.

/pdf/rainbow-connection-number-and-connectivity-397r4r8gv0.pdf

Deepak Rajendraprasad

Papers

Rainbow connection number and connected dominating sets

Rainbow Connection Number and Radius

Rainbow Connection Number and Radius

Testing for forbidden order patterns in an array

Rainbow Connection Number and Connectivity