Many commonly-used auction mechanisms are “maximal-in-range”. We show that any maximalin-range mechanism for n bidders and m items cannot both approximate the social welfare with a ratio better than min(n,m) for any constant η < 1/2 and run in polynomial time, unless NP ⊆ P/poly. This significantly improves upon a previous bound on the achievable social welfare of polynomial time maximal-in-ran...

Semi-unification is a common generalization of unification and matching. The semiunification problem is to decide solvability of finite sets of equations s = t and inequations s̃ ≤i t̃ between first-order terms, with different inequality relations ≤i, i ∈ I. A solution consists of a substitution T0 and residual substitutions Ti, i ∈ I, such that , respectively, T0(s) = T0(t) and Ti(T0(s̃)) = T0(t̃)...

{0, 1}-valued random variables X1, . . . , Xn are drawn independently each from Bernoulli distribution with parameter p = 0.1. Define Pn := P( 1 n ∑n i=1Xi ≤ 0.2). (a) For n = 1 to 30 calculate and plot the below in the same plot (see [1, section 6.1] for definition of Hoeffding and Bernstein inequalities): i. Exact value of Pn (binomial distribution). ii. Normal approximation for Pn. iii. Hoef...

An inequality g{x) 2i 0 is often said to be a reverse convex constraint if the function g is continuous and convex. The feasible regions for linear program with an additional reverse convex constraint are generally non-convex and disconnected. In this paper a convergent algorithm for solving such a linear problem is proposed. The method is based upon a combination of the branch and bound proced...

A Feynman-Kac type formula of relativistic Schrödinger operators with unbounded vector potential and spin 1/2 is given in terms of a three-component process consisting of Brownian motion, a Poisson process and a subordinator. This formula is obtained for unbounded magnetic fields and magnetic field with zeros. From this formula an energy comparison inequality is derived. Spatial decay of bound ...

We consider random graphs such that each edge is determined by an independent random variable, where the probability of each edge is not assumed to be equal. We use a Chernoff inequality for matrices to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent ...

Preliminaries Before we venture into Chernoff bound, let us recall two simple bounds on the probability that a random variable deviates from the mean by a certain amount: Markov's inequality and Chebyshev's inequality. Markov's inequality only applies to non-negative random variables and gives us a bound depending on the expectation of the random variable.

Vertex k-center problem introduces the notion to recognize k locations as centers in a given network of n connected nodes holding the condition of triangle inequality. This paper presents an efficient algorithm that provides a better solution for vertex k-center problem. Anticipatory Bound Selection Procedure (ABSP) is deployed to find the initial threshold distance (or radius) and eradicating ...

Model complexities of shallow (i.e., one-hidden-layer) networks representing highly varying multivariable f 1;1g-valued functions are studied in terms of variational norms tailored to dictionaries of network units. It is shown that bounds on these norms define classes of functions computable by networks with constrained numbers of hidden units and sizes of output weights. Estimates of probabili...

We use the following two well known bounds in our proofs. Lemma 1 (Chernoff-Hoeffding bound). Let X1, · · · , Xn be random variables with common support [0, 1] and E[Xi] = μ. Let Sn = X1 + · · ·+Xn. Then for all t ≥ 0, Pr[Sn ≥ nμ+ t] ≤ e−2t /n and Pr[Sn ≤ nμ− t] ≤ e−2t /n Lemma 2 (Bernstein inequality). Let X1, . . . , Xn be independent zero-mean random variables. If for all 1 ≤ i ≤ n, |Xi| ≤ k...