126 research outputs found
Tight Euler tours in uniform hypergraphs - computational aspects
By a tight tour in a -uniform hypergraph we mean any sequence of its
vertices such that for all the set
is an edge of (where operations on
indices are computed modulo ) and the sets for are
pairwise different. A tight tour in is a tight Euler tour if it contains
all edges of . We prove that the problem of deciding if a given -uniform
hypergraph has a tight Euler tour is NP-complete, and that it cannot be solved
in time (where is the number of edges in the input hypergraph),
unless the ETH fails. We also present an exact exponential algorithm for the
problem, whose time complexity matches this lower bound, and the space
complexity is polynomial. In fact, this algorithm solves a more general problem
of computing the number of tight Euler tours in a given uniform hypergraph
Hamilton cycles in 5-connected line graphs
A conjecture of Carsten Thomassen states that every 4-connected line graph is
hamiltonian. It is known that the conjecture is true for 7-connected line
graphs. We improve this by showing that any 5-connected line graph of minimum
degree at least 6 is hamiltonian. The result extends to claw-free graphs and to
Hamilton-connectedness
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
A Harary-Sachs Theorem for Hypergraphs
We generalize the Harary-Sachs theorem to -uniform hypergraphs: the
codegree- coefficient of the characteristic polynomial of a uniform
hypergraph can be expressed as a weighted sum of subgraph counts
over certain multi-hypergraphs with edges. This includes a detailed
description of said multi-hypergraphs and a formula for their corresponding
weights, which we use to describe the low codegree coefficients of the
characteristic polynomial of a 3-uniform hypergraph. We further provide
explicit and asymptotic formulas for the contribution from -uniform
simplices and conclude by showing that the Harary-Sachs theorem for graphs is
indeed a special case of our main theorem.Comment: 30 page
Spectra of Uniform Hypergraphs
We present a spectral theory of hypergraphs that closely parallels Spectral
Graph Theory. A number of recent developments building upon classical work has
led to a rich understanding of "hyperdeterminants" of hypermatrices, a.k.a.
multidimensional arrays. Hyperdeterminants share many properties with
determinants, but the context of multilinear algebra is substantially more
complicated than the linear algebra required to address Spectral Graph Theory
(i.e., ordinary matrices). Nonetheless, it is possible to define eigenvalues of
a hypermatrix via its characteristic polynomial as well as variationally. We
apply this notion to the "adjacency hypermatrix" of a uniform hypergraph, and
prove a number of natural analogues of basic results in Spectral Graph Theory.
Open problems abound, and we present a number of directions for further study.Comment: 32 pages, no figure
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