An Intersection Inequality for Discrete Distributions and Related Generation Problems

Abstract

Abstract. Given two finite sets of points X , Y in R n which can be separated by a nonnegative linear function, and such that the componentwise minimum of any two distinct points in X is dominated by some point in Y, we show that |X | ≤ n|Y|. As a consequence of this result, we obtain quasi-polynomial time algorithms for generating all maximal integer feasible solutions for a given monotone system of separable inequalities, for generating all p-inefficient points of a given discrete probability distribution, and for generating all maximal empty hyper-rectangles for a given set of points in R n . This provides a substantial improvement over previously known exponential algorithms for these generation problems related to Integer and Stochastic Programming, and Data Mining. Furthermore, we give an incremental polynomial time generation algorithm for monotone systems with fixed number of separable inequalities, which, for the very special case of one inequality, implies that for discrete probability distributions with independent coordinates, both p-efficient and p-inefficient points can be separately generated in incremental polynomial time.

Properties and calculation of multivariate risk measures: MVaR and MCVaR

“…In Prékopa (2012) the term Multivariate Value-at-Risk (MVaR) was introduced as an alternative name for the collection of p-efficient points. Methods to generate elements of MVaR in the case of a continuously distributed ξ and the entire MVaR, in the discrete case, has already been existed in the literature (see, e.g., Prékopa (1995) and the references therein; Prékopa, Vizvári, Badics (1998); Boros et al (1998) ;Dentcheva, Prékopa, Ruszczcyński (2000), etc. ).…”

An Efficient Implementation of a Joint Generation Algorithm

“…Problems GEN(C, F π , X ) and GEN(C, G π , Y) arise in many practical applications and in a variety of fields, including artificial intelligence [14], game theory [18,19], reliability theory [8,12], database theory [9,14,17], integer programming [4,6,20], learning theory [1], and data mining [2,6,9]. Even though these two problems may be NP-hard in general (see e.g.…”

“…Then the sets F π3 and G π3 can be identified respectively with the set of maximal t-boxes (plus a polynomial number of non-boxes), and the set of minimal boxes of x ∈ B ⊆ C which contain at least k + 1 points of S in their interior. It is known [6] that the family of maximal t-boxes is (uniformly) dual-bounded:…”

An Efficient Implementation of a Quasi-polynomial Algorithm for Generating Hypergraph Transversals

“…The collection H d of minimal transversals is also called the dual or transversal hypergraph for H. The hypergraph transversal problem is the problem of generating all transversals of a given hypergraph. This problem has important applications in combinatorics [14], artificial intelligence [8], game theory [11,12], reliability theory [7], database theory [6,8,10], integer programming [3], learning theory [1], and data mining [2,5,6].…”

“…Problem GEN(C, A) has several interesting applications in integer programming and data mining, see [3,4,5] and the references therein. Extensions of the two hypergraph transversal algorithms mentioned above to solve problem DUAL(C, A, B) were given in [3].…”