135 research outputs found
Unique Minimal Liftings for Simplicial Polytopes
For a minimal inequality derived from a maximal lattice-free simplicial
polytope in , we investigate the region where minimal liftings are
uniquely defined, and we characterize when this region covers . We then
use this characterization to show that a minimal inequality derived from a
maximal lattice-free simplex in with exactly one lattice point in the
relative interior of each facet has a unique minimal lifting if and only if all
the vertices of the simplex are lattice points.Comment: 15 page
Optimal General Matchings
Given a graph and for each vertex a subset of the
set , where denotes the degree of vertex
in the graph , a -factor of is any set such that
for each vertex , where denotes the number of
edges of incident to . The general factor problem asks the existence of
a -factor in a given graph. A set is said to have a {\em gap of
length} if there exists a natural number such that and . Without any restrictions the
general factor problem is NP-complete. However, if no set contains a gap
of length greater than , then the problem can be solved in polynomial time
and Cornuejols \cite{Cor} presented an algorithm for finding a -factor, if
it exists. In this paper we consider a weighted version of the general factor
problem, in which each edge has a nonnegative weight and we are interested in
finding a -factor of maximum (or minimum) weight. In particular, this
version comprises the minimum/maximum cardinality variant of the general factor
problem, where we want to find a -factor having a minimum/maximum number of
edges.
We present an algorithm for the maximum/minimum weight -factor for the
case when no set contains a gap of length greater than . This also
yields the first polynomial time algorithm for the maximum/minimum cardinality
-factor for this case
Identically self-blocking clutters
A clutter is identically self-blocking if it is equal to its blocker. We prove that every identically self-blocking clutter different from is nonideal. Our proofs borrow tools from Gauge Duality and Quadratic Programming. Along the way we provide a new lower bound for the packing number of an arbitrary clutter
Idealness of k-wise intersecting families
A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that, for some integer k â„ 4, every k-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it for k = 4 for the class of binary clutters. Two key ingredients for our proof are Jaegerâs 8-flow theorem for graphs, and Seymourâs characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975. We also discuss connections to the chromatic number of a clutter, projective geometries over the two-element field, uniform cycle covers in graphs, and quarter-integral packings of value two in ideal clutters
Branching on multi-aggregated variables
open5siopenGamrath, Gerald; Melchiori, Anna; Berthold, Timo; Gleixner, Ambros M.; Salvagnin, DomenicoGamrath, Gerald; Melchiori, Anna; Berthold, Timo; Gleixner, Ambros M.; Salvagnin, Domenic
Maximum gradient embeddings and monotone clustering
Let (X,d_X) be an n-point metric space. We show that there exists a
distribution D over non-contractive embeddings into trees f:X-->T such that for
every x in X, the expectation with respect to D of the maximum over y in X of
the ratio d_T(f(x),f(y)) / d_X(x,y) is at most C (log n)^2, where C is a
universal constant. Conversely we show that the above quadratic dependence on
log n cannot be improved in general. Such embeddings, which we call maximum
gradient embeddings, yield a framework for the design of approximation
algorithms for a wide range of clustering problems with monotone costs,
including fault-tolerant versions of k-median and facility location.Comment: 25 pages, 2 figures. Final version, minor revision of the previous
one. To appear in "Combinatorica
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