We define multivariate polynomials over arbitrary (ordered) semirings in combination with (executable) operations like addition, multiplication, and substitution. We also define (weak) monotonicity of polynomials and comparison of polynomials where we provide standard estimations like absolute positiveness or the more recent approach of [3]. Moreover, it is proven that strongly normalizing (mon...

Given an in…nite hereditary property of graphs P; the principal extremal parameter of P is the value (P) = lim n!1 n 2 1 maxfe (G) : G 2 P and v (G) = ng: The Erd1⁄2os-Stone theorem gives (P) if P is monotone, but this result does not apply to hereditary P. Thus, one of the results of this note is to establish (P) for any hereditary property P: Similar questions are studied for the parameter (p...

There has been substantial interest in estimating the value of a graph parameter, i.e., of a real function defined on the set of finite graphs, by sampling a randomly chosen substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity qz = qz(ε) of an estimable par...

This short paper surveys recent results on the generation of implicitly given hypergraphs, as well as their applications in data mining, reliability theory, integer programming and combinatorics. More precisely, we consider a monotone property π over the subsets of a finite set V , the corresponding family Sπ of subsets satisfying property π, and the problem of generating (sequentially) the fam...

In the distribution-free property testing model, the distance between functions is measured with respect to an arbitrary and unknown probability distribution D over the input domain. We consider distribution-free testing of several basic Boolean function classes over {0, 1}, namely monotone conjunctions, general conjunctions, decision lists, and linear threshold functions. We prove that for eac...

We survey the results of Rubinfeld, Batu et al. ([2], [3]) on testing distributions for monotonicity, and testing distributions known to be monotone for uniformity. We extend some of their results to new partial orders, and provide evidence for some new conjectural lower bounds. Our results apply to various partial orders: bipartite graphs, lines, trees, grids, and hypercubes. Thesis Supervisor...

For a graph property P consider the following computational problem. Given an input graph G, what is the minimum number of edge modifications (additions and/or deletions) that one has to apply to G in order to turn it into a graph that satisfies P? Namely, what is the edit distance ∆(G,P) of a graph G from satisfying P. Clearly, the computational complexity of such a problem strongly depends on...