We consider profit-maximization problems for combinatorial auctions with non-single minded valuation functions and limited supply. We obtain fairly general results that relate the approximability of the profit-maximization problem to that of the corresponding social-welfare-maximization (SWM) problem, which is the problem of finding an allocation (S1, . . . , Sn) satisfying the capacity constra...

We study applications of discrete valuations to ideals in analytically irreducible domains, in particular applications to zero divisors modulo powers of ideals. We prove a uniform version of Izumi's theorem and calculate several examples illustrating it, such as for rational singularities. The paper contains a new criterion of analytic irreducibility, a new criterion of one-beredness, and a val...

We discuss valuations on convex sets of oriented hyperplanes in R. For d = 2, we prove an analogue of Hadwiger’s characterization theorem for continuous, rigid motion invariant valuations.

0 Introduction In this paper we study, for a certain type of topological rings A, the topological space Cont A of all equivalence classes of continuous valuations of A. The space ContA is defined as follows. Let v: A-+Fw{0} be a valuation of A, where F is an ordered multiplicative group generated by im(v)\{0}. On Fw [0} we introduce the topology such that U _~ F ~ {0} is open iff 0 q~ U or {xeF...

This expository paper contains history, definitions, constructions, and the basic properties of Rees valuations of ideals. A section is devoted to one-fibered ideals, that is, ideals with only one Rees valuation. Cutkosky [5] proved that there exists a two-dimensional complete Noetherian local integrally closed domain in which no zero-dimensional ideal is one-fibered. However, no concrete ring ...

A complete classification is established of Minkowski valuations on lattice polytopes that intertwine the special linear group over the integers and are translation invariant. In the contravariant case, the only such valuations are multiples of projection bodies. In the equivariant case, the only such valuations are generalized difference bodies combined with multiples of the newly defined disc...

The Alesker product turns the space of smooth translation-invariant valuations on convex bodies into a commutative associative unital algebra, satisfying Poincaré duality and hard Lefschetz theorem. In this article, version Hodge-Riemann relations for algebra is conjectured, conjecture proved in two particular situations: even valuations, 1-homogeneous valuations. latter result then used to ded...