Randomized on-line scheduling on three processors

We consider a randomized on-line scheduling problem where each job has to be scheduled on any of m identical processors. The objective is to minimize the expected makespan. We show that the competitive ratio of any randomized algorithm for m=3 processors must be strictly greater than .     

Selecting distances in arrangements of hyperplanes spanned by points

In this paper we consider a problem of distance selection in the arrangement of hyperplanes induced by n given points. Given a set of n points in d-dimensional space and a number k, , determine the hyperplane that is spanned by d points and at distance ranked by k from the origin. For the planar case we present an O(nlog²n) runtime algorithm using parametric search partly different from the usual approach [N. Megiddo, J. ACM 30 (1983) 852]. We establish a connection between this problem in 3-d and the well-known 3SUM problem using an auxiliary problem of counting the number of vertices in the arrangement of n planes that lie between two sheets of a hyperboloid. We show that the 3-d problem is almost 3SUM-hard and solve it by an O(n²log²n) runtime algorithm. We generalize these results to the d-dimensional (d4) space and consider also a problem of enumerating distances.     

The Two-Dimensional Finite Bin Packing Problem. Part II: New lower and upper bounds

This paper is the second of a two part series and describes new lower and upper bounds for a more general version of the Two-Dimensional Finite Bin Packing Problem (2BP) than the one considered in Part I (see Boschetti and Mingozzi 2002). With each item is associated an input parameter specifying if it has a fixed orientation or it can be rotated by . This problem contains as special cases the oriented and non-oriented 2BP. The new lower bound is based on the one described in Part I for the oriented 2BP. The computational results on the test problems derived from the literature show the effectiveness of the new proposed lower and upper bounds.     

Solving Nonconvex Planar Location Problems by Finite Dominating Sets

It is well-known that some of the classical location problems with polyhedral gauges can be solved in polynomial time by finding a finite dominating set, i.e. a finite set of candidates guaranteed to contain at least one optimal location.In this paper it is first established that this result holds for a much larger class of problems than currently considered in the literature. The model for which this result can be proven includes, for instance, location problems with attraction and repulsion, and location-allocation problems.Next, it is shown that the approximation of general gauges by polyhedral ones in the objective function of our general model can be analyzed with regard to the subsequent error in the optimal objective value. For the approximation problem two different approaches are described, the sandwich procedure and the greedy algorithm. Both of these approaches lead - for fixed - to polynomial approximation algorithms with accuracy for solving the general model considered in this paper.     

Full analogy of Sharkovsky's theorem for lower semicontinuous maps

A full analogy of the celebrated Sharkovsky cycle coexistence theorem is established for lower semicontinuous (multivalued) maps on metrizable linear continua. This result is further extended to triangular maps.     

FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. A conference version of this article, containing a part of the results presented here, appeared in Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, Miami, FL, January 22–24, 2006, pp. 743–748. The first author gratefully acknowledges support from NSF grant DMS-0608785, a 2003 UC-Davis Chancellor’s fellow award, the Alexander von Humboldt foundation, and IMO Magdeburg. The remaining authors were supported by the European TMR network ADONET 504438.     

A unifying approach to integration for bounded positive charges

This paper deals with n-monotone functionals, which constitute a generalisation of n-monotone set functions. Using the notion of exactness of a functional, we introduce a new notion of lower and upper integral which subsumes as particular cases most of the approaches to integration in the literature. As a consequence, we can characterise which types of integrals can be used to calculate the natural extension (the lower envelope of all linear extensions) of a positive bounded charge.     

Approximate range searching

The range searching problem is a fundamental problem in computational geometry, with numerous important applications. Most research has focused on solving this problem exactly, but lower bounds show that if linear space is assumed, the problem cannot be solved in polylogarithmic time, except for the case of orthogonal ranges. In this paper we show that if one is willing to allow approximate ranges, then it is possible to do much better. In particular, given a bounded range Q of diameter w and >0, an approximate range query treats the range as a fuzzy object, meaning that points lying within distance w of the boundary of Q either may or may not be counted. We show that in any fixed dimension d, a set of n points in can be preprocessed in O(n+logn) time and O(n) space, such that approximate queries can be answered in O(logn(1/)^d) time. The only assumption we make about ranges is that the intersection of a range and a d-dimensional cube can be answered in constant time (depending on dimension). For convex ranges, we tighten this to O(logn+(1/)^d−1) time. We also present a lower bound for approximate range searching based on partition trees of Ω(logn+(1/)^d−1), which implies optimality for convex ranges (assuming fixed dimensions). Finally, we give empirical evidence showing that allowing small relative errors can significantly improve query execution times.     

Highly monodisperse crosslinked polymethylmethacrylate microparticles by dispersion polymerization

Highly monodisperse polymethylmethacrylate (PMMA) microparticles crosslinked with carboxylic group-containing urethane acrylates (CUA) were produced by simple dispersion polymerization in methanol solution. In contrast to conventional crosslinkers, the CUA employed as a crosslinker was excellent for maintaining the monodispersity of PMMA microparticles even at moderate crosslinker concentrations (to about 5 wt%). It was believed that the CUA helped form the monomer-swellable surface of primary particles, because of the structurally long tetramethylene oxide groups in the molecule. Carboxylic groups in the molecular backbone resulted in larger primary particles by increasing the solubility of the monomer mixture in the medium. Owing to these larger primary particles, the crosslinked PMMA particles showed lower polymerization rates than the linear ones during particle growth. However, at high CUA concentrations (about 10 wt%), bimodal distributions were observed. This was attributed to the high crosslinking density of the primary particle surfaces. Therefore, monomer diffusion toward the polymer phase was restricted, resulting in more favorable secondary nucleation in the medium. Received: 12 May 1998 Accepted: 19 August 1998     

On Overlays and Minimization Diagrams

The overlay of 2≤m≤d minimization diagrams of n surfaces in ℝ ^d is isomorphic to a substructure of a suitably constructed minimization diagram of mn surfaces in ℝ ^d+m−1. This elementary observation leads to a new bound on the complexity of the overlay of minimization diagrams of collections of d-variate semi-algebraic surfaces, a tight bound on the complexity of the overlay of minimization diagrams of collections of hyperplanes, and faster algorithms for constructing such overlays. Further algorithmic implications are discussed. Work by V. Koltun was supported by NSF CAREER award CCF-0641402. Work by M. Sharir was supported by NSF Grants CCR-00-98246 and CCF-05-14079, by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University.