A stable (or independent) set in a graph is a set of pairwise non-adjacent vertices. The stability number α(G) is the size of a maximum stable set in the graph G. There are three different kinds of structures that one can see observing behavior of stable sets of a graph: the enumerative structure, the intersection structure, and the exchange structure. The independence polynomial of G

We study the computational power of polynomial threshold functions, that is, threshold functions of real polynomials over the boolean cube. We provide two new results bounding the computational power of this model. Our first result shows that low-degree polynomial threshold functions cannot approximate any function with many influential variables. We provide a couple of examples where this tech...

A method is given for constructing elements in Fqn whose orders are larger than any polynomial in n when n becomes large. As a by-product a theorem on multiplicative independence of compositions of polynomials is proved.

Functions that are integrals of products generalized hypergeometric functions some kind considered. The conditions for the representability these as a polynomial in ones found. necessary and sufficient algebraic independence such established.

The stability number α(G) of the graph G is the size of a maximum stable set of G. If sk denotes the number of stable sets of cardinality k in graph G, then I(G;x) = s0 + s1x + ... + sαx α is the independence polynomial of G [12], where α = α(G) is the size of a maximum stable set. In this paper we prove that I(G;−1) satisfies |I(G;−1)| ≤ 2, where ν(G) equals the cyclomatic number of G, and the...

We describe NTRU, a new public key cryptosystem. NTRU features reasonably short, easily created keys, high speed, and low memory requirements. NTRU encryption and decryption use a mixing system suggested by polynomial algebra combined with a clustering principle based on elementary probability theory. The security of the NTRU cryptosystem comes from the interaction of the polynomial mixing syst...

We develop the first polynomial-time algorithm for co-training of homogeneous linear separators under weak dependence, a relaxation of the condition of independence given the label. Our algorithm learns from purely unlabeled data, except for a single labeled example to break symmetry of the two classes, and works for any data distribution having an inverse-polynomial margin and with center of m...

Let P(G;x,y) be the number of vertex colorings φ : V →{1, ...,x} of an undirected graph G = (V,E) such that for all edges {u,v} ∈ E the relations φ(u)≤ y and φ(v)≤ y imply φ(u) 6= φ(v). We show that P(G;x,y) is a polynomial in x and y which is closely related to Stanley’s chromatic symmetric function, and which simultaneously generalizes the chromatic polynomial, the independence polynomial, an...

4 We prove that the weak k-linkage problem is polynomial for every fixed k for totally Φ5 decomposable digraphs, under appropriate hypothesis on Φ. We then apply this and recent results 6 by Fradkin and Seymour (on the weak k-linkage problem for digraphs of bounded independence 7 number or bounded cut-width) to get polynomial algorithms for some class of digraphs like quasi8 transitive digraphs...

In this lecture we present a randomized parallel algorithm for generating a maximal independent set. We then show how to derandomize the algorithm using pairwise independence. For an input graph with n vertices, our goal is to devise an algorithm that works in time polynomial in log n and using polynomial in n processors. See Chapter 12.1 of Motwani and Raghavan [5] for background on parallel m...