7 research outputs found
LIPIcs
The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2
Improved Approximation Algorithms for Tverberg Partitions
Tverberg's theorem states that a set of points in can be
partitioned into \floor{n/(d+1)} sets with a common intersection. A point in
this intersection (aka Tverberg point) is a centerpoint of the input point set,
and the Tverberg partition provides a compact proof of this, which is
algorithmically useful.
Unfortunately, computing a Tverberg point exactly requires time.
We provide several new approximation algorithms for this problem, which improve
either the running time or quality of approximation, or both. In particular, we
provide the first strongly polynomial (in both and ) approximation
algorithm for finding a Tverberg point
Unique End of Potential Line
This paper studies the complexity of problems in PPAD PLS that have
unique solutions. Three well-known examples of such problems are the problem of
finding a fixpoint of a contraction map, finding the unique sink of a Unique
Sink Orientation (USO), and solving the P-matrix Linear Complementarity Problem
(P-LCP). Each of these are promise-problems, and when the promise holds, they
always possess unique solutions.
We define the complexity class UEOPL to capture problems of this type. We
first define a class that we call EOPL, which consists of all problems that can
be reduced to End-of-Potential-Line. This problem merges the canonical
PPAD-complete problem End-of-Line, with the canonical PLS-complete problem
Sink-of-Dag, and so EOPL captures problems that can be solved by a
line-following algorithm that also simultaneously decreases a potential
function.
Promise-UEOPL is a promise-subclass of EOPL in which the line in the
End-of-Potential-Line instance is guaranteed to be unique via a promise. We
turn this into a non-promise class UEOPL, by adding an extra solution type to
EOPL that captures any pair of points that are provably on two different lines.
We show that UEOPL EOPL CLS, and that all of our
motivating problems are contained in UEOPL: specifically USO, P-LCP, and
finding a fixpoint of a Piecewise-Linear Contraction under an -norm all
lie in UEOPL. Our results also imply that parity games, mean-payoff games,
discounted games, and simple-stochastic games lie in UEOPL.
All of our containment results are proved via a reduction to a problem that
we call One-Permutation Discrete Contraction (OPDC). This problem is motivated
by a discretized version of contraction, but it is also closely related to the
USO problem. We show that OPDC lies in UEOPL, and we are also able to show that
OPDC is UEOPL-complete.Comment: This paper substantially revises and extends the work described in
our previous preprint "End of Potential Line'' (arXiv:1804.03450). The
abstract has been shortened to meet the arXiv character limi
Unique End of Potential Line
This paper studies the complexity of problems in PPAD PLS that have unique solutions. Three well-known examples of such problems are the problem of finding a fixpoint of a contraction map, finding the unique sink of a Unique Sink Orientation (USO), and solving the P-matrix Linear Complementarity Problem (P-LCP). Each of these are promise-problems, and when the promise holds, they always possess unique solutions. We define the complexity class UEOPL to capture problems of this type. We first define a class that we call EOPL, which consists of all problems that can be reduced to End-of-Potential-Line. This problem merges the canonical PPAD-complete problem End-of-Line, with the canonical PLS-complete problem Sink-of-Dag, and so EOPL captures problems that can be solved by a line-following algorithm that also simultaneously decreases a potential function. Promise-UEOPL is a promise-subclass of EOPL in which the line in the End-of-Potential-Line instance is guaranteed to be unique via a promise. We turn this into a non-promise class UEOPL, by adding an extra solution type to EOPL that captures any pair of points that are provably on two different lines. We show that UEOPL EOPL CLS, and that all of our motivating problems are contained in UEOPL: specifically USO, P-LCP, and finding a fixpoint of a Piecewise-Linear Contraction under an -norm all lie in UEOPL. Our results also imply that parity games, mean-payoff games, discounted games, and simple-stochastic games lie in UEOPL. All of our containment results are proved via a reduction to a problem that we call One-Permutation Discrete Contraction (OPDC). This problem is motivated by a discretized version of contraction, but it is also closely related to the USO problem. We show that OPDC lies in UEOPL, and we are also able to show that OPDC is UEOPL-complete