18,984 research outputs found
Upper and Lower Bounds on Long Dual-Paths in Line Arrangements
Given a line arrangement with lines, we show that there exists a
path of length in the dual graph of formed by its
faces. This bound is tight up to lower order terms. For the bicolored version,
we describe an example of a line arrangement with blue and red lines
with no alternating path longer than . Further, we show that any line
arrangement with lines has a coloring such that it has an alternating path
of length . Our results also hold for pseudoline
arrangements.Comment: 19 page
On the dual graph of Cohen-Macaulay algebras
Given a projective algebraic set X, its dual graph G(X) is the graph whose
vertices are the irreducible components of X and whose edges connect components
that intersect in codimension one. Hartshorne's connectedness theorem says that
if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We
present two quantitative variants of Hartshorne's result:
1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where
r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for
coordinate arrangements, it yields an algebraic extension of Balinski's theorem
for simplicial polytopes.)
2) If X is a canonically embedded arrangement of lines no three of which meet
in the same point, then the diameter of the graph G(X) is not larger than the
codimension of X. (The bound is sharp; for coordinate arrangements, it yields
an algebraic expansion on the recent combinatorial result that the Hirsch
conjecture holds for flag normal simplicial complexes.)Comment: Minor changes throughout, Remark 4.1 expanded, to appear in IMR
Multitriangulations, pseudotriangulations and primitive sorting networks
We study the set of all pseudoline arrangements with contact points which
cover a given support. We define a natural notion of flip between these
arrangements and study the graph of these flips. In particular, we provide an
enumeration algorithm for arrangements with a given support, based on the
properties of certain greedy pseudoline arrangements and on their connection
with sorting networks. Both the running time per arrangement and the working
space of our algorithm are polynomial.
As the motivation for this work, we provide in this paper a new
interpretation of both pseudotriangulations and multitriangulations in terms of
pseudoline arrangements on specific supports. This interpretation explains
their common properties and leads to a natural definition of
multipseudotriangulations, which generalizes both. We study elementary
properties of multipseudotriangulations and compare them to iterations of
pseudotriangulations.Comment: 60 pages, 40 figures; minor corrections and improvements of
presentatio
Posets arising as 1-skeleta of simple polytopes, the nonrevisiting path conjecture, and poset topology
Given any polytope and any generic linear functional , one
obtains a directed graph by taking the 1-skeleton of and
orienting each edge from to for .
This paper raises the question of finding sufficient conditions on a polytope
and generic cost vector so that the graph will
not have any directed paths which revisit any face of after departing from
that face. This is in a sense equivalent to the question of finding conditions
on and under which the simplex method for linear programming
will be efficient under all choices of pivot rules. Conditions on and are given which provably yield a corollary of the desired face
nonrevisiting property and which are conjectured to give the desired property
itself. This conjecture is proven for 3-polytopes and for spindles having the
two distinguished vertices as source and sink; this shows that known
counterexamples to the Hirsch Conjecture will not provide counterexamples to
this conjecture.
A part of the proposed set of conditions is that be the
Hasse diagram of a partially ordered set, which is equivalent to requiring non
revisiting of 1-dimensional faces. This opens the door to the usage of
poset-theoretic techniques. This work also leads to a result for simple
polytopes in which is the Hasse diagram of a lattice L that the
order complex of each open interval in L is homotopy equivalent to a ball or a
sphere of some dimension. Applications are given to the weak Bruhat order, the
Tamari lattice, and more generally to the Cambrian lattices, using realizations
of the Hasse diagrams of these posets as 1-skeleta of permutahedra,
associahedra, and generalized associahedra.Comment: new results for 3-polytopes and spindles added; exposition
substantially improved throughou
Four Soviets Walk the Dog-Improved Bounds for Computing the Fr\'echet Distance
Given two polygonal curves in the plane, there are many ways to define a
notion of similarity between them. One popular measure is the Fr\'echet
distance. Since it was proposed by Alt and Godau in 1992, many variants and
extensions have been studied. Nonetheless, even more than 20 years later, the
original algorithm by Alt and Godau for computing the Fr\'echet
distance remains the state of the art (here, denotes the number of edges on
each curve). This has led Helmut Alt to conjecture that the associated decision
problem is 3SUM-hard.
In recent work, Agarwal et al. show how to break the quadratic barrier for
the discrete version of the Fr\'echet distance, where one considers sequences
of points instead of polygonal curves. Building on their work, we give a
randomized algorithm to compute the Fr\'echet distance between two polygonal
curves in time on a pointer machine
and in time on a word RAM. Furthermore, we show that
there exists an algebraic decision tree for the decision problem of depth
, for some . We believe that this
reveals an intriguing new aspect of this well-studied problem. Finally, we show
how to obtain the first subquadratic algorithm for computing the weak Fr\'echet
distance on a word RAM.Comment: 34 pages, 15 figures. A preliminary version appeared in SODA 201
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch conjecture, posed in 1957, stated that the graph of a
-dimensional polytope or polyhedron with facets cannot have diameter
greater than . The conjecture itself has been disproved, but what we
know about the underlying question is quite scarce. Most notably, no polynomial
upper bound is known for the diameters that were conjectured to be linear. In
contrast, no polyhedron violating the conjecture by more than 25% is known.
This paper reviews several recent attempts and progress on the question. Some
work in the world of polyhedra or (more often) bounded polytopes, but some try
to shed light on the question by generalizing it to simplicial complexes. In
particular, we include here our recent and previously unpublished proof that
the maximum diameter of arbitrary simplicial complexes is in and
we summarize the main ideas in the polymath 3 project, a web-based collective
effort trying to prove an upper bound of type nd for the diameters of polyhedra
and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter
of simplicial complexes and abstractions of them, in preparation
On Rearrangement of Items Stored in Stacks
There are stacks, each filled with items, and one empty stack.
Every stack has capacity . A robot arm, in one stack operation (step),
may pop one item from the top of a non-empty stack and subsequently push it
onto a stack not at capacity. In a {\em labeled} problem, all items are
distinguishable and are initially randomly scattered in the stacks. The
items must be rearranged using pop-and-pushs so that in the end, the stack holds items , in that order, from the top to
the bottom for all . In an {\em unlabeled} problem, the
items are of types of each. The goal is to rearrange items so that
items of type are located in the stack for all . In carrying out the rearrangement, a natural question is to find the least
number of required pop-and-pushes.
Our main contributions are: (1) an algorithm for restoring the order of
items stored in an table using only column and row
permutations, and its generalization, and (2) an algorithm with a guaranteed
upper bound of steps for solving both versions of the stack
rearrangement problem when for arbitrary fixed
positive number . In terms of the required number of steps, the labeled and
unlabeled version have lower bounds
and , respectively
On the maximum size of an anti-chain of linearly separable sets and convex pseudo-discs
We show that the maximum cardinality of an anti-chain composed of
intersections of a given set of n points in the plane with half-planes is close
to quadratic in n. We approach this problem by establishing the equivalence
with the problem of the maximum monotone path in an arrangement of n lines. For
a related problem on antichains in families of convex pseudo-discs we can
establish the precise asymptotic bound: it is quadratic in n. The sets in such
a family are characterized as intersections of a given set of n points with
convex sets, such that the difference between the convex hulls of any two sets
is nonempty and connected.Comment: 10 pages, 3 figures. revised version correctly attributes the idea of
Section 3 to Tverberg; and replaced k-sets by "linearly separable sets" in
the paper and the title. Accepted for publication in Israel Journal of
Mathematic
On densities of lattice arrangements intersecting every i-dimensional affine subspace
In 1978, Makai Jr. established a remarkable connection between the
volume-product of a convex body, its maximal lattice packing density and the
minimal density of a lattice arrangement of its polar body intersecting every
affine hyperplane. Consequently, he formulated a conjecture that can be seen as
a dual analog of Minkowski's fundamental theorem, and which is strongly linked
to the well-known Mahler-conjecture.
Based on the covering minima of Kannan & Lov\'asz and a problem posed by
Fejes T\'oth, we arrange Makai Jr.'s conjecture into a wider context and
investigate densities of lattice arrangements of convex bodies intersecting
every i-dimensional affine subspace. Then it becomes natural also to formulate
and study a dual analog to Minkowski's second fundamental theorem. As our main
results, we derive meaningful asymptotic lower bounds for the densities of such
arrangements, and furthermore, we solve the problems exactly for the special,
yet important, class of unconditional convex bodies.Comment: 19 page
Cardy's Formula for Certain Models of the Bond-Triangular Type
We introduce and study a family of 2D percolation systems which are based on
the bond percolation model of the triangular lattice. The system under study
has local correlations, however, bonds separated by a few lattice spacings act
independently of one another. By avoiding explicit use of microscopic paths, it
is first established that the model possesses the typical attributes which are
indicative of critical behavior in 2D percolation problems. Subsequently, the
so called Cardy-Carleson functions are demonstrated to satisfy, in the
continuum limit, Cardy's formula for crossing probabilities. This extends the
results of S. Smirnov to a non-trivial class of critical 2D percolation
systems.Comment: 49 pages, 7 figure
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