An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every ǫ > 0, a connected graph with minimum degree at least ǫn has bounded rainbow connectivity, where the bound depends only on ǫ, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.
In this paper we define and examine the power of the conditional-sampling oracle in the context of distribution-property testing. The conditional-sampling oracle for a discrete distribution µ takes as input a subset S ⊂ [n] of the domain, and outputs a random sample i ∈ S drawn according to µ, conditioned on S (and independently of all prior samples). The conditional-sampling oracle is a natural generalization of the ordinary sampling oracle, in which S always equals [n].We show that with the conditional-sampling oracle, testing uniformity, testing identity to a known distribution, and testing any label-invariant property of distributions is easier than with the ordinary sampling oracle. On the other hand, we also show that for some distribution properties the sample-complexity remains near-maximal even with conditional sampling.tested with a constant number of samples (compared to Θ( √ n) unconditional samples even for uniformity [10,3]). The most general result of this paper (Section 6) is that any label-invariant property of distributions (a symmetric property in the terminology of [15]) can be tested using poly(log n) conditional samples. 1 On the other hand, there are properties for which testing remains almost as hard as possible even with conditional samples: We show a property of distributions that requires at least Ω(n) conditional samples to test (Section 8).Another feature that makes conditional-samples interesting is that in contrast to the testers using ordinary samples, which are non-adaptive by definition, adaptivity (and the algorithmic aspect of testing) in conditional-sampling model plays an important role. For instance, the aforementioned task of testing uniformity, while still possible with a much better sampling complexity than in the traditional model, cannot be done non-adaptively with a constant number of samples (see Section 7.2).Before we move to some motivating examples, let us address the concern of whether arbitrary conditioning is realistic: While the examples below do relate to arbitrary conditioning, sometimes one would like the conditioning to be more restricted, in some sense describable by fewer than the n bits required to describe the conditioning set S. In fact, many of our algorithms require less than that. For example, the adaptive uniformity test takes only unconditional samples and samples conditioned on a constant size set, so the description size per sample is in fact O(log n), as there are n O(1) possibilities. The adaptive general label invariant property tester takes only samples conditioned to dyadic intervals of [n], so here the description size is O(log n) as well. The non-adaptive tests do require general conditioning, as they pick uniformly random sets of prescribed sizes.
We develop a query-efficient sample extractor for juntas, that is,
We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any disjoint source-sink pairs on the directed hypercube can be connected with edge-disjoint paths, then monotonicity of functions f : {0, 1}n → R can be tested with O(n) queries, for any totally ordered range R. More generally, if at least an α(n) fraction of the pairs can always be connected with edge-disjoint paths then the query-complexity is O(n/α(n)).We construct a family of instances of = Ω(2 n ) pairs in n-dimensional hypercubes such that no more than roughly a 1 √ n fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [LR01], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≈ n 3/2 . Additionally, our construction can also be used to obtain a strong counterexample to Szymanski's conjecture on routing in the hypercube. In particular, we show that for any δ > 0, the n-dimensional hypercube is not n 1 2 −δ -realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths.We also prove a lower bound of Ω(n/ ) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.
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