121 research outputs found

    On largest volume simplices and sub-determinants

    Full text link
    We show that the problem of finding the simplex of largest volume in the convex hull of nn points in Qd\mathbb{Q}^d can be approximated with a factor of O(log⁡d)d/2O(\log d)^{d/2} in polynomial time. This improves upon the previously best known approximation guarantee of d(d−1)/2d^{(d-1)/2} by Khachiyan. On the other hand, we show that there exists a constant c>1c>1 such that this problem cannot be approximated with a factor of cdc^d, unless P=NPP=NP. % This improves over the 1.091.09 inapproximability that was previously known. Our hardness result holds even if n=O(d)n = O(d), in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where cˉ\bar c is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a d×nd\times n matrix

    Approximating the multi-level bottleneck assignment problem.

    Get PDF
    We consider the multi-level bottleneck assignment problem (MBA). This problem is described in the recent book 'Assignment Problems' by Burkard et al. (2009) on pages 188-189. One of the applications described there concerns bus driver scheduling.We view the problem as a special case of a bottleneck m-dimensional multi-index assignment problem. We give approximation algorithms and inapproximability results, depending upon the completeness of the underlying graph. Keywords: bottleneck problem; multidimensional assignment; approximation; computational complexity; efficient algorithm.Bottleneck problem; Multidimensional assignment; Approximation; Computational complexity; Efficient algorithm;

    Improved Hardness of Approximation for Geometric Bin Packing

    Full text link
    The Geometric Bin Packing (GBP) problem is a generalization of Bin Packing where the input is a set of dd-dimensional rectangles, and the goal is to pack them into unit dd-dimensional cubes efficiently. It is NP-Hard to obtain a PTAS for the problem, even when d=2d=2. For general dd, the best known approximation algorithm has an approximation guarantee exponential in dd, while the best hardness of approximation is still a small constant inapproximability from the case when d=2d=2. In this paper, we show that the problem cannot be approximated within d1−ϔd^{1-\epsilon} factor unless NP=ZPP. Recently, dd-dimensional Vector Bin Packing, a closely related problem to the GBP, was shown to be hard to approximate within Ω(log⁥d)\Omega(\log d) when dd is a fixed constant, using a notion of Packing Dimension of set families. In this paper, we introduce a geometric analog of it, the Geometric Packing Dimension of set families. While we fall short of obtaining similar inapproximability results for the Geometric Bin Packing problem when dd is fixed, we prove a couple of key properties of the Geometric Packing Dimension that highlight the difference between Geometric Packing Dimension and Packing Dimension.Comment: 10 page

    Dagstuhl Reports : Volume 1, Issue 2, February 2011

    Get PDF
    Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-HĂŒbner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro PezzĂ©, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

    Approximating Smallest Containers for Packing Three-dimensional Convex Objects

    Get PDF
    We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are NP-hard so that we cannot expect to find exact polynomial time algorithms. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimal volume containers for the objects described
    • 

    corecore