Showing papers in "Combinatorica in 1990"

How to draw a planar graph on a grid

Abstract: Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fary embedding (i.e., straight-line embedding) on the 2n−4 byn−2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fary embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

Abstract: Letλi(L), λi(L*) denote the successive minima of a latticeL and its reciprocal latticeL*, and let [b1,..., b n ] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that Open image in new window and Open image in new window , where Open image in new window andγj denotes Hermite's constant. As a consequence the inequalities Open image in new window are obtained forn≥7. Given a basisB of a latticeL in ℝ m of rankn andx∃ℝ m , we define polynomial time computable quantitiesλ(B) andΜ(x,B) that are lower bounds for λ1(L) andΜ(x,L), whereΜ(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL*, then λ1(L)≤γ n * λ(B) and Open image in new window .

A deterministic view of random sampling and its use in geometry

Abstract: The combination of divide-and-conquer and random sampling has proven very effective in the design of fast geometric algorithms. A flurry of efficient probabilistic algorithms have been recently discovered, based on this happy marriage. We show that all those algorithms can be derandomized with only polynomial overhead. In the process we establish results of independent interest concerning the covering of hypergraphs and we improve on various probabilistic bounds in geometric complexity. For example, givenn hyperplanes ind-space and any integerr large enough, we show how to compute, in polynomial time, a simplicial packing of sizeO(r d ) which coversd-space, each of whose simplices intersectsO(n/r) hyperplanes.

Applications of matrix methods to the theory of lower bounds in computational complexity

Abstract: We present some criteria for obtaining lower bounds for the formula size of Boolean functions. In the monotone case we get the boundn Ω(logn) for the function “MINIMUM COVER” using methods considerably simpler than all previously known. In the general case we are only able to prove that the criteria yield an exponential lower bound when applied to almost all functions. Some connections with graph complexity and communication complexity are also given.

Entropy splitting for antiblocking corners and -perfect graphs

Abstract: We characterize pairs of convex setsA, B in thek-dimensional space with the property that every probability distribution (p1,...,p k ) has a repsesentationp i =a l .b i , a∃A, b∃B.

On the number of halving planes

Abstract: Let S subset-of R3 be an n-set in general position. A plane containing three of the points is called a halving plane if it dissects S into two parts of equal cardinality. It is proved that the number of halving planes is at most O(n2.998).As a main tool, for every set Y of n points in the plane a set N of size O(n4) is constructed such that the points of N are distributed almost evenly in the triangles determined by Y.

Approximate Inclusion-Exclusion

Abstract: The Inclusion-Exclusion formula expresses the size of a union of a family of sets in terms of the sizes of intersections of all subfamilies. This paper considers approximating the size of the union when intersection sizes are known for only some of the subfamilies, or when these quantities are given to within some error, or both. In particular, we consider the case when allk-wise intersections are given for everyk≤K. It turns out that the answer changes in a significant way aroundK=√n: ifK≤O(√n) then any approximation may err by a factor of θ(n/K 2), while ifK≥ Ω(√n) it is shown how to approximate the size of the union to within a multiplicative factor of $$1 \pm e^{ - \Omega (K/\sqrt n )} $$ . When the sizes of all intersections are only given approximately, good bounds are derived on how well the size of the union may be approximated. Several applications for Boolean function are mentioned in conclusion.

Triangles in space or building (and analyzing) castles in the air

Abstract: We show that the total combinatorial complexity of all non-convex cells in an arrangement ofn (possibly intersecting) triangles in 3-space isO(n 7/3 logn) and that this bound is almost tight in the worst case. Our bound significantly improves a previous nearly cubic bound of Pach and Sharir. We also present a (nearly) worst-case optimal randomized algorithm for calculating a single cell of the arrangement and an alternative less efficient, but still subcubic algorithm for calculating all non-convex cells, analyze some special cases of the problem where improved bounds (and faster algorithms) can be obtained, and describe applications of our results to translational motion planning for polyhedra in 3-space.

An acyclicity theorem for cell complexes in d dimension

Abstract: LetC be a cell complex ind-dimensional Euclidean space whose faces are obtained by orthogonal projection of the faces of a convex polytope ind+ 1 dimensions. For example, the Delaunay triangulation of a finite point set is such a cell complex. This paper shows that the in_front/behind relation defined for the faces ofC with respect to any fixed viewpointx is acyclic. This result has applications to hidden line/surface removal and other problems in computational geometry.

Computational Complexity of Norm-Maximization

Abstract: This paper discusses the problem of maximizing a quasiconvex functionφ over a convex polytopeP inn-space that is presented as the intersection of a finite number of halfspaces. The problem is known to beNP-hard (for variablen) whenφ is thepth power of the classicalp-norm. The present reexamination of the problem establishesNP-hardness for a wider class of functions, and for thep-norm it proves theNP-hardness of maximization overn-dimensionalparallelotopes that are centered at the origin or have a vertex there. This in turn implies theNP-hardness of {−1, 1}-maximization and {0, 1}-maximization of a positive definite quadratic form. On the “good” side, there is an efficient algorithm for maximizing the Euclidean norm over an arbitraryrectangular parallelotope.

The asymptotic numbers of regular tournaments, Eulerian digraphs and Eulerian oriented graphs

Abstract: LetRT(n), ED(n) andEOG(n) be the number of labelled regular tournaments, labelled loop-free simple Eulerian digraphs, and labelled Eulerian oriented simple graphs, respectively, onn vertices. Then, asn→∞, , for anye>0. The last two families of graphs are also enumerated by their numbers of edges. The proofs use the saddle point method applied to appropriaten-dimensional integrals.

Geometric methods in the study of irregularities of distribution

Abstract: Letν be a signed measure on E d with νE d =0 and ¦ν¦Ed<∞. DefineD s(ν) as sup ¦νH¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Letν be supported by a finite pointsetp i. ThenD s(ν)>c d(δ1/δ 2)1/2{∑ i(νp i)2}1/2 whereδ 1 is the minimum distance between two distinctp i, andδ 2 is the maximum distance. The numberc d is an absolute dimensional constant. (The number .05 can be chosen forc 2 in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE 2, andp 1,p 2,...,p n be a set of points lying inD. If m if the usual area measure restricted toD, while γnP i=1/n defines an atomic measure γn, then independently of γn,nD s(m −γ n)≥ .0335n 1/4. Theorem B gives an improved solution to the Roth “disk segment problem” as described by Beck and Chen. Recent work by Beck shows thatnD s(m −γ n)≥cn 1/4(logn)−7/2.

The maximum number of Hamiltonian paths in tournaments

Abstract: Solving an old conjecture of Szele we show that the maximum number of directed Hamiltonian paths in a tournament onn vertices is at mostc · n 3/2 · n!/2 n−1, wherec is a positive constant independent ofn.

Colouring series-parallel graphs

Abstract: We establish a minimax formula for the chromatic index of series-parallel graphs; and also prove the correctness of a “greedy” algorithm for finding a vertex-colouring of a series-parallel graph.

Tilings of polygons with similar triangles

Abstract: We prove that if a polygon P is decomposed into finitely many similar triangles then the tangents of the angles of these triangles are algebraic over the field generated by the coordinates of the vertices of P. If P is a rectangle then, apart from four "sporadic" cases, the triangles of the decomposition must be right triangles. Three of these "sporadic" triangles tile the square. In any other decomposition of the square into similar triangles, the decomposition consists of right triangles with an acute angle alpha such that tan-alpha is a totally positive algebraic number. Most of the proofs are based on the following general theorem: if a convex polygon P is decomposed into finitely many triangles (not necessarily similar) then the coordinate system can be chosen in such a way that the coordinates of the vertices of P belong to the field generated by the cotangents of the angles of the triangles in the decomposition.

Variations on the theme of repeated distances

Abstract: We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ℝd, d≥4. We also give asymptotically tight upper bounds on the total number of occurrences of the “favourite” distances fromn points in ℝd, d≥4. Related results are proved for distances determined byn disjoint compact convex sets in ℝ2.

On low-dimensional faces that high-dimensional polytopes must have

Abstract: We prove that every five-dimensional polytope has a two-dimensional face which is a triangle or a quadrilateral. We state and discuss the following conjecture: For every integerk≥1 there is an integer f(k) such that everyd-polytope,d≥f(k), has ak-dimensional face which is either a simplex or combinatorially isomorphic to thek-dimensional cube.

Convexly independent sets

Abstract: A family of pairwise disjoint compact convex sets is called convexly independent, if none of its members is contained in the convex hull of the union of the other members of the family. The main result of the paper gives an upper bound for the maximum cardinalityh(k, n) of a family ℱ of mutually disjoint compact convex sets such that any subfamily of at mostk members of ℱ is convexly independent, but no subfamily of sizen is.

On recognizing integer polyhedra

Abstract: We give a simple proof that, determining whether a convex polytope has a fractional vertex, is NP-complete.

Necessary and sufficient conditions for hyperplane transversals

Abstract: We prove that a finite family ℬ={B 1,B 2, ...,B n } of connected compact sets in ℝ d has a hyperplane transversal if and only if for somek there exists a set of pointsP={p 1,p 2, ...,p n } (i.e., ak-dimensional labeling of the family) which spans ℝ k and everyk+2 sets of ℬ are met by ak-flat consistent with the order type ofP. This is a common generalization of theorems of Hadwiger, Katchalski, Goodman-Pollack and Wenger.

Infinite spectra in the first order theory of graphs

Abstract: The psectrum Spec(A) of a sentenceA is, roughly, the set of those a for whichA has a threshold function at or nearp=n −a . Examples are given ofA with infinite spectra and with spectra of order typeΩ i for arbitraryi.

On the power of interaction

Abstract: LetIP[f(n)] be the class of languages recognized by interactive proofs withf(¦x¦) interactions. Babai [2] showed that all languages recognized by interactive proofs with a bounded number of interactions can be recognized by interactive proofs with only two interactions; i.e., for every constantk, IP[k] collapses toIP[2]. In this paper, we give evidence that interactive proofs with an unbounded number of interactions may be more powerful than interactive proofs with a bounded number of interactions. We show that for any polynomially bounded polynomial time computable functionf(n) and anyg(n)=o(f(n)) there exists an oracleB such thatIP B [f(n)] = ⊄ IP B [g(n)]. The techniques employed are extensions of the techniques for proving lower bounds on small depth circuits used in [6], [14] and [10].

Solution of an extremal problem for sets using resultants of polynomials

Abstract: A new, short proof is given of the following theorem of Bollobas: LetA 1,..., Ah andB 1,..., Bh be collections of sets with ∀ i ∶¦A i¦=r,¦Bi¦=s and ¦A i∩Bj¦=O if and only ifi=j, thenh≤( s r+s ). The proof immediately extends to the generalizations of this theorem obtained by Frankl, Alon and others.

Weight distribution of translates of MDS codes

Abstract: The following is a particular case of a theorem of Delsarte: the weight distribution of a translate of an MDS code is uniquely determined by its firstn−k terms. Here an explicit formula is derived from a completely different approach.

An upper bound for the number of Eulerian orientations of a regular graph

Abstract: We establish a new upper bound for the number of Eulerian orientations of a regular graph with even degrees.

A lower bound for the recognition of digraph properties

Abstract: The complexity of a digraph property is the number of entries of the adjacency matrix which must be examined by a decision tree algorithm to recognize the property in the worst case, Aanderaa and Rosenberg conjectured that there is a constante such that every digraph property which is monotone (not destroyed by the deletion of edges) and nontrivial (holds for some but not all digraphs) has complexity at leastev 2 wherev is the number of nodes in the digraph. This conjecture was proved by Rivest and Vuillemin and a lower bound ofv 2/4−o(v 2) was subsequently found by Kahn, Saks, and Sturtevant. Here we show a lower bound ofv 2/2−o(v 2). We also prove that a certain class of monotone, nontrivial bipartite digraph properties is evasive (requires that every entry in the adjacency matrix be examined in the worst case).

ON THE fg-COLORING OF GRAPHS

Abstract: This paper introduces a new type of edge-coloring of multigraphs, called anfg-coloring, in which each color appears at each vertexv no more thanf(v) times and at each set of multiple edges joining verticesv andw no more thang(vw) times. The minimum number of colors needed tofg-color a multigraphG is called thefg-chromatic index ofG. Various upper bounds are given on thefg-chromatic index. One of them is a generalization of Vizing's bound for the ordinary chromatic index. Our proof is constructive, and immediately yields a polynomial-time algorithm tofg-color a given multigraph using colors no more than twice thefg-chromatic index.

The caratheodory number for the k -core

Abstract: The k-core of the set S subset-of R(n) is the intersection of the convex hull of all sets A subset-of-or-equal-to S with \S\A\ less-than-or-equal-to k. The Caratheodory number of the k-core is the smallest integer f(d, k) with the property that x member-of core(k)S, S subset-of R(n) implies the existence of a subset T subset-of-or-equal-to S such that x member-of core(k)T and \T\ less-than-or-equal-to f(d, k). In this paper various properties of f(d, k) are established.

Packing paths in planar graphs

Abstract: A generalization of P Seymour's theorem on planar integral 2-commodity flows is given when the underlying graphG together with the demand graphH (a graph having edges that connect the corresponding terminal pairs) form a planar graph and the demand edges are on two faces ofG

Some notes about affine diameters of convex figures

Abstract: Given a pointx in a convex figureM, letγ(x) denote the number of all affine diameters ofM passing throughx. It is shown that, for a convex figureM, the following conditions are equivalent. (i) γ(x)≥2 for every pointx ∈ intM. (ii) eitherγ(x)≡3 orγ(x)≡∞ on intM. Furthermore, the setB={x ∈ intM:γ(x) is either odd or infinite } is dense inM.