Showing papers in "Discrete and Computational Geometry in 1988"

The algebraic degree of geometric optimization problems

Abstract: In this paper we apply Galois methods to certain fundamentalgeometric optimization problems whose exact computational complexity has been an open problem for a long time. In particular we show that the classic Weber problem, along with theline-restricted Weber problem and itsthree-dimensional version are in general not solvable by radicals over the field of rationals. One direct consequence of these results is that for these geometric optimization problems there existsno exact algorithm under models of computation where the root of an algebraic equation is obtained using arithmetic operations and the extraction ofkth roots. This leaves only numerical or symbolic approximations to the solutions, where the complexity of the approximations is shown to be primarily a function of the algebraic degree of the optimum solution point.

On the complexity of polyhedral separability

Abstract: It is NP-complete to recognize whether two sets of points in general space can be separated by two hyperplanes. It is NP-complete to recognize whether two sets of points in the plane can be separated withk lines. For every fixedk in any fixed dimension, it takes polynomial time to recognize whether two sets of points can be separated withk hyperplanes.

Planar realizations of nonlinear Davenport-Schinzel sequences by segments

Abstract: LetG={l1,...,ln} be a collection ofn segments in the plane, none of which is vertical. Viewing them as the graphs of partially defined linear functions ofx, letYG be their lower envelope (i.e., pointwise minimum).YG is a piecewise linear function, whose graph consists of subsegments of the segmentsli. Hart and Sharir [7] have shown thatYG consists of at mostO(n?(n)) segments (where?(n) is the extremely slowly growing inverse Ackermann's function). We present here a construction of a setG ofn segments for whichYG consists ofΩ(n?(n)) subsegments, proving that the Hart-Sharir bound is tight in the worst case. Another interpretation of our result is in terms of Davenport-Schinzel sequences: the sequenceEG of indices of segments inG in the order in which they appear alongYG is a Davenport-Schinzel sequence of order 3, i.e., no two adjacent elements ofEG are equal andEG contains no subsequence of the forma ...b ...a ...b ...a. Hart and Sharir have shown that the maximal length of such a sequence composed ofn symbols is ?(n?(n)). Our result shows that the lower bound construction of Hart and Sharir can be realized by the lower envelope ofn straight segments, thus settling one of the main open problems in this area.

Nonobtuse triangulation of polygons

Abstract: We show how to triangulate a polygon without using any obtuse triangles. Such triangulations can be used to discretize partial differential equations in a way that guarantees that the resulting matrix is Stieltjes, a desirable property both for computation and for theoretical analysis. A simple divide-and-conquer approach would fail because adjacent subproblems cannot be solved independently, but this can be overcome by careful subdivision. Overlay a square grid on the polygon, preferably with the polygon vertices at grid points. Choose boundary cells so they can be triangulated without propagating irregular points to adjacent cells. The remaining interior is rectangular and easily triangulated. Small angles can also be avoided in these constructions.

Many triangulated spheres

Abstract: Lets(d, n) be the number of triangulations withn labeled vertices ofSd?1, the (d?1)-dimensional sphere. We extend a construction of Billera and Lee to obtain a large family of triangulated spheres. Our construction shows that logs(d, n)?C1(d)n[(d?1)/2], while the known upper bound is logs(d, n)≤C2(d)n[d/2] logn. Letc(d, n) be the number of combinatorial types of simpliciald-polytopes withn labeled vertices. (Clearly,c(d, n)≤s(d, n).) Goodman and Pollack have recently proved the upper bound: logc(d, n)≤d(d+1)n logn. Combining this upper bound forc(d, n) with our lower bounds fors(d, n), we obtain, for everyd?5, that limn??(c(d, n)/s(d, n))=0. The cased=4 is left open. (Steinitz's fundamental theorem asserts thats(3,n)=c(3,n), for everyn.) We also prove that, for everyb?4, limd??(c(d, d+b)/s(d, d+b))=0. (Mani proved thats(d, d+3)=c(d, d+3), for everyd.) Lets(n) be the number of triangulated spheres withn labeled vertices. We prove that logs(n)=20.69424n(1+o(1)). The same asymptotic formula describes the number of triangulated manifolds withn labeled vertices.

On the classification of toric fano varieties

Abstract: Toric Fano varieties are algebraic varieties associated with a special class of convex polytopes inR?'. We extend results of V. E. Voskresenskij and A. A. Klyachko about the classification of such varieties using a purely combinatorial method of proof.

On optimal realizations of finite metric spaces by graphs

Abstract: Graph realizations of finite metric spaces have widespread applications, for example, in biology, economics, and information theory. The main results of this paper are:1.Finding optimal realizations of integral metrics (which means all distances are integral) is NP-complete.2.There exist metric spaces with a continuum of optimal realizations. Furthermore, two conditions necessary for a weighted graph to be an optimal realization are given and an extremal problem arising in connection with the realization problem is investigated.

Valuations and polarity

Abstract: Given a collection[Figure not available: see fulltext.] of convex polytopes, let?([Figure not available: see fulltext.]) denote the set of all convex transversals of[Figure not available: see fulltext.]. If[Figure not available: see fulltext.] and ? are two such collections, of finite cardinality, then there is a simple, arithmetical condition which holds precisely when ?([Figure not available: see fulltext.])=?(?). Another such condition, involving what we call the "Sallee-Shephard mapping," characterizes those pairs[Figure not available: see fulltext.] and ? for which ?(?([Figure not available: see fulltext.]))=?(?). As these results are established, several distributive lattices involving convex sets are introduced, and relationships between their valuation modules are determined. In particular, it is proven that the Sallee-Shephard mapping is an isomorphism of the additive, abelian group of simple functions generated by the characteristic functions of the open, convex sets and that generated by those of the closed, convex sets.

Combinatoriald-tori with a large symmetry group

Abstract: We construct a sequence of combinatorial triangulations of thed-dimensional torus with 2d+1?1 vertices and with a vertex transitive group action. This generalizes well-known constructions in the casesd=2 (7-vertex torus) andd=3.

The orthogonal convex skull problem

Abstract: We give a combinatorial definition of the notion of a simple orthogonal polygon beingk-concave, wherek is a nonnegative integer. (A polygon is orthogonal if its edges are only horizontal or vertical.) Under this definition an orthogonal polygon which is 0-concave is convex, that is, it is a rectangle, and one that is 1-concave is orthoconvex in the usual sense, and vice versa. Then we consider the problem of computing an orthoconvex orthogonal polygon of maximal area contained in a simple orthogonal polygon. This is the orthogonal version of the potato peeling problem. AnO(n2) algorithm is presented, which is a substantial improvement over theO(n7) time algorithm for the general problem.

A sylvester theorem for conic sections

Abstract: If S is a finite set of points in the plane and no conic contains all points of S, then S determines a conic which contains exactly five points ofS.

Polygonizations of point sets in the plane

Abstract: We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons. We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n4) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n5) to compute a complete list ofO(n4) nondegenerate star-shaped polygonizations of the set ofn points.

On the contact dimensions of graphs

Abstract: Every simple graphG=(V, E) can be represented by a family of equal nonoverlapping spheres {Sv:v � V} in a Euclidean spaceRn in such a way thatuv � E if and only ifSu andSv touch each other. The smallest dimensionn needed to representG in such a way is called the contact dimension ofG and it is denoted by cd(G). We prove that (1) cd(T)<(7.3) log |T| for every treeT, and (2) $$m - 1 + \frac{n}{2}\left( {1 - \frac{n}{{2\pi m}}\left( {\sqrt {\frac{{n + 4\pi m}}{n}} - 1} \right)} \right)< cd(K_m + E_n ) \leqslant m - 1 + \left\lceil {\frac{n}{2}} \right\rceil ,$$ whereKm+En is the join of the complete graph of orderm and the empty graph of ordern. For the complete bipartite graphKn,n this implies (1.286)n�1

Tiling the torus and other space forms

Abstract: We consider graphs on two-dimensional space forms which are quotient graphs Γ/F, where Γ is an infinite, 3-connected, face, vertex, or edge transitive planar graph andF is a subgroup of Aut(Γ), all of whose elements act freely on Γ The enumeration of quotient graphs with transitivity properties reduces to computing the normalizers in Aut(Γ) of the subgroupsF Results include: all isogonal toriodal polyhedra belong to the two families found by Grunbaum and Shephard; there are no transitive graphs on the Mobius band; there is a graph on the Klein bottle whose automorphism group acts transitively on its faces, edges, and vertices

A decomposition theorem on Euclidean Steiner minimal trees

Abstract: The Euclidean Steiner minimal tree problem is known to be an NP-complete problem and current alogorithms cannot solve problems with more than 30 points. Thus decomposition theorems can be very helpful in extending the boundary of workable problems. There have been only two known decomposition theorems in the literature. This paper provides a 50% increase in the reservoir of decomposition theorems.

Arrangements of lines with a minimum number of triangles are simple

Abstract: Levi has shown that for every arrangement ofn lines in the real projective plane, there exist at leastn triangular faces, and Grunbaum has conjectured that equality can occur only for simple arrangements. In this note we prove this conjecture. The result does not hold for arrangements of pseudolines.

Nonorientable genus of nearly complete bipartite graphs

Abstract: LetG(m, n, k), m, n?3,k≤min(m, n), be the graph obtained from the complete bipartite graphKm,n by deleting an arbitrary set ofk independent edges, and let $$f(m,n,k) = [(m - 2)(n - 2) - k]/2.$$ It is shown that the nonorientable genus $$\tilde \gamma $$ (G(m, n, k)) of the graphG(m, n, k) is equal to the upper integer part off(m, n, k), except in trivial cases wheref(m, n, k)≤0 and possibly in some extreme cases, the graphsG(k, k, k) andG(k + 1,k, k). These cases are also discussed, obtaining some positive and some negative results. In particular, it is shown thatG(5, 4, 4) andG(5, 5, 5) have no nonorientable quadrilateral embedding.

Equidecomposability of polyhedra with reference to crystallographic groups

Abstract: This paper deals with the introduction of theoretical statements which are the results of studying equidecomposability of polyhedra with reference to discrete transformation groups Lattice polygons and paving polyhedra play the most important role The equivalence of volume equality and equidecomposability of any two of this polyhedra is shown for special discrete groups

Repeated Angles inE4

Abstract: Let there be givenn points in four-dimensional euclidean spaceE4. We show that the number of occurrences of the angleÂ? iso(n3) ifÂ? is not a right angle and Â?(n3) otherwise.

Polygonizations of point sets in the plane

Abstract: We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of p...