1,789 research outputs found
On set systems with restricted intersections modulo p and p-ary t-designs
We consider bounds on the size of families ℱ of subsets of a v-set subject to restrictions modulo a prime p on the cardinalities of the pairwise intersections. We improve the known bound when ℱ is allowed to contain sets of different sizes, but only in a special case. We show that if the bound for uniform families ℱ holds with equality, then ℱ is the set of blocks of what we call a p-ary t-design for certain values of t. This motivates us to make a few observations about p-ary t-designs for their own sake
Stable Intersections of Tropical Varieties
We give several characterizations of stable intersections of tropical cycles
and establish their fundamental properties. We prove that the stable
intersection of two tropical varieties is the tropicalization of the
intersection of the classical varieties after a generic rescaling. A proof of
Bernstein's theorem follows from this. We prove that the tropical intersection
ring of tropical cycle fans is isomorphic to McMullen's polytope algebra. It
follows that every tropical cycle fan is a linear combination of pure powers of
tropical hypersurfaces, which are always realizable. We prove that every stable
intersection of constant coefficient tropical varieties defined by prime ideals
is connected through codimension one. We also give an example of a realizable
tropical variety that is connected through codimension one but whose stable
intersection with a hyperplane is not.Comment: Revised version, to appear in Journal of Algebraic Combinatoric
Submodular Minimization Under Congruency Constraints
Submodular function minimization (SFM) is a fundamental and efficiently
solvable problem class in combinatorial optimization with a multitude of
applications in various fields. Surprisingly, there is only very little known
about constraint types under which SFM remains efficiently solvable. The
arguably most relevant non-trivial constraint class for which polynomial SFM
algorithms are known are parity constraints, i.e., optimizing only over sets of
odd (or even) cardinality. Parity constraints capture classical combinatorial
optimization problems like the odd-cut problem, and they are a key tool in a
recent technique to efficiently solve integer programs with a constraint matrix
whose subdeterminants are bounded by two in absolute value.
We show that efficient SFM is possible even for a significantly larger class
than parity constraints, by introducing a new approach that combines techniques
from Combinatorial Optimization, Combinatorics, and Number Theory. In
particular, we can show that efficient SFM is possible over all sets (of any
given lattice) of cardinality r mod m, as long as m is a constant prime power.
This covers generalizations of the odd-cut problem with open complexity status,
and with relevance in the context of integer programming with higher
subdeterminants. To obtain our results, we establish a connection between the
correctness of a natural algorithm, and the inexistence of set systems with
specific combinatorial properties. We introduce a general technique to disprove
the existence of such set systems, which allows for obtaining extensions of our
results beyond the above-mentioned setting. These extensions settle two open
questions raised by Geelen and Kapadia [Combinatorica, 2017] in the context of
computing the girth and cogirth of certain types of binary matroids
Characteristic and Ehrhart polynomials
Let A be a subspace arrangement and let chi(A,t) be the characteristic
polynomial of its intersection lattice L(A). We show that if the subspaces in A
are taken from L(B_n), where B_n is the type B Weyl arrangement, then chi(A,t)
counts a certain set of lattice points. One can use this result to study the
partial factorization of chi(A,t) over the integers and the coefficients of its
expansion in various bases for the polynomial ring R[t]. Next we prove that the
characteristic polynomial of any Weyl hyperplane arrangement can be expressed
in terms of an Ehrhart quasi-polynomial for its affine Weyl chamber. Note that
our first result deals with all subspace arrangements embedded in B_n while the
second deals with all finite Weyl groups but only their hyperplane
arrangements.Comment: 16 pages, 1 figure, Latex, to be published in J. Alg. Combin. see
related papers at http://www.math.msu.edu/~saga
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