61 research outputs found
Packing Plane Perfect Matchings into a Point Set
Given a set of points in the plane, where is even, we consider
the following question: How many plane perfect matchings can be packed into
? We prove that at least plane perfect matchings
can be packed into any point set . For some special configurations of point
sets, we give the exact answer. We also consider some extensions of this
problem
Hypergraph matchings and designs
We survey some aspects of the perfect matching problem in hypergraphs, with
particular emphasis on structural characterisation of the existence problem in
dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC
The hamburger theorem
We generalize the ham sandwich theorem to measures in as
follows. Let be absolutely continuous finite
Borel measures on . Let for , and assume that . Assume that for every . Then there
exists a hyperplane such that each open halfspace defined by
satisfies for every
and . As a
consequence we obtain that every -colored set of points in
such that no color is used for more than points can be
partitioned into disjoint rainbow -dimensional simplices.Comment: 11 pages, 2 figures; a new proof of Theorem 8, extended concluding
remark
Dependent k-Set Packing on Polynomoids
Specialized hereditary systems, e.g., matroids, are known to have many applications in algorithm design. We define a new notion called d-polynomoid as a hereditary system (E, ? ? 2^E) so that every two maximal sets in ? have less than d elements in common. We study the problem that, given a d-polynomoid (E, ?), asks if the ground set E contains ? disjoint k-subsets that are not in ?, and obtain a complexity trichotomy result for all pairs of k ? 1 and d ? 0. Our algorithmic result yields a sufficient and necessary condition that decides whether each hypergraph in some classes of r-uniform hypergraphs has a perfect matching, which has a number of algorithmic applications
A Geometric Theory for Hypergraph Matching
We develop a theory for the existence of perfect matchings in hypergraphs
under quite general conditions. Informally speaking, the obstructions to
perfect matchings are geometric, and are of two distinct types: 'space
barriers' from convex geometry, and 'divisibility barriers' from arithmetic
lattice-based constructions. To formulate precise results, we introduce the
setting of simplicial complexes with minimum degree sequences, which is a
generalisation of the usual minimum degree condition. We determine the
essentially best possible minimum degree sequence for finding an almost perfect
matching. Furthermore, our main result establishes the stability property:
under the same degree assumption, if there is no perfect matching then there
must be a space or divisibility barrier. This allows the use of the stability
method in proving exact results. Besides recovering previous results, we apply
our theory to the solution of two open problems on hypergraph packings: the
minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's
conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we
prove the exact result for tetrahedra and the asymptotic result for Fischer's
conjecture; since the exact result for the latter is technical we defer it to a
subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical
Society. 101 pages. v2: minor changes including some additional diagrams and
passages of expository tex
Planar Bichromatic Bottleneck Spanning Trees
Given a set P of n red and blue points in the plane, a planar bichromatic spanning tree of P is a geometric spanning tree of P, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree T, such that the length of the longest edge in T is minimized. In this paper, we show that this problem is NP-hard for points in general position. Our main contribution is a polynomial-time (8?2)-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck ? can be converted to a planar bichromatic spanning tree of bottleneck at most 8?2 ?
Multipartite hypergraphs achieving equality in Ryser's conjecture
A famous conjecture of Ryser is that in an -partite hypergraph the
covering number is at most times the matching number. If true, this is
known to be sharp for for which there exists a projective plane of order
. We show that the conjecture, if true, is also sharp for the smallest
previously open value, namely . For , we find the minimal
number of edges in an intersecting -partite hypergraph that has
covering number at least . We find that is achieved only by linear
hypergraphs for , but that this is not the case for . We
also improve the general lower bound on , showing that .
We show that a stronger form of Ryser's conjecture that was used to prove the
case fails for all . We also prove a fractional version of the
following stronger form of Ryser's conjecture: in an -partite hypergraph
there exists a set of size at most , contained either in one side of
the hypergraph or in an edge, whose removal reduces the matching number by 1.Comment: Minor revisions after referee feedbac
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