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Register a new userCutting a Convex Polyhedron Out of a Sphere
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Added by
Masud Hasan
Publication date
2009
fields
Informatics Engineering
and research's language is
English
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Given a convex polyhedron $P$ of $n$ vertices inside a sphere $Q$, we give an $O(n^3)$-time algorithm that cuts $P$ out of $Q$ by using guillotine cuts and has cutting cost $O((log n)^2)$ times the optimal.
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Motivated by indoor localization by tripwire lasers, we study the problem of cutting a polygon into small-size pieces, using the chords of the polygon. Sever
Let $mathcal{P}$ be an $mathcal{H}$-polytope in $mathbb{R}^d$ with vertex set $V$. The vertex centroid is defined as the average of the vertices in $V$. We prove that computing the vertex centroid of an $mathcal{H}$-polytope is #P-hard. Moreover, we show that even just checking whether the vertex centroid lies in a given halfspace is already #P-hard for $mathcal{H}$-polytopes. We also consider the problem of approximating the vertex centroid by finding a point within an $epsilon$ distance from it and prove this problem to be #P-easy by showing that given an oracle for counting the number of vertices of an $mathcal{H}$-polytope, one can approximate the vertex centroid in polynomial time. We also show that any algorithm approximating the vertex centroid to emph{any} ``sufficiently non-trivial (for example constant) distance, can be used to construct a fully polynomial approximation scheme for approximating the centroid and also an output-sensitive polynomial algorithm for the Vertex Enumeration problem. Finally, we show that for unbounded polyhedra the vertex centroid can not be approximated to a distance of $d^{{1/2}-delta}$ for any fixed constant $delta>0$.
Inspired by the decomposition in the hybrid quantum-classical optimization algorithm we introduced in arXiv:1902.04215, we propose here a new (fully classical) approach to solving certain non-convex integer programs using Graver bases. This method is well suited when (a) the constraint matrix $A$ has a special structure so that its Graver basis can be computed systematically, (b) several feasible solutions can also be constructed easily and (c) the objective function can be viewed as many convex functions quilted together. Classes of problems that satisfy these conditions include Cardinality Boolean Quadratic Problems (CBQP), Quadratic Semi-Assignment Problems (QSAP) and Quadratic Assignment Problems (QAP). Our Graver Augmented Multi-seed Algorithm (GAMA) utilizes augmentation along Graver basis elements (the improvement direction is obtained by comparing objective function values) from these multiple initial feasible solutions. We compare our approach with a best-in-class commercially available solver (Gurobi). Sensitivity analysis indicates that the rate at which GAMA slows down as the problem size increases is much lower than that of Gurobi. We find that for several instances of practical relevance, GAMA not only vastly outperforms in terms of time to find the optimal solution (by two or three orders of magnitude), but also finds optimal solutions within minutes when the commercial solver is not able to do so in 4 or 10 hours (depending on the problem class) in several cases.
An unzipping of a polyhedron P is a cut-path through its vertices that unfolds P to a non-overlapping shape in the plane. It is an open problem to decide if every convex P has an unzipping. Here we show that there are nearly flat convex caps that have no unzipping. A convex cap is a top portion of a convex polyhedron; it has a boundary, i.e., it is not closed by a base.
We study several problems concerning convex polygons whose vertices lie in a Cartesian product (for short, grid) of two sets of n real numbers. First, we prove that every such grid contains a convex polygon with $Omega$(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d $in$ N), and obtain a tight lower bound of $Omega$(log d--1 n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the largest convex chain in a grid that contains no two points of the same x-or y-coordinate. We show how to efficiently approximate the maximum size of a supported convex polygon up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.
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