352 research outputs found
Lexicographic shellability, matroids and pure order ideals
In 1977 Stanley conjectured that the -vector of a matroid independence
complex is a pure -sequence. In this paper we use lexicographic shellability
for matroids to motivate a combinatorial strengthening of Stanley's conjecture.
This suggests that a pure -sequence can be constructed from combinatorial
data arising from the shelling. We then prove that our conjecture holds for
matroids of rank at most four, settling the rank four case of Stanley's
conjecture. In general, we prove that if our conjecture holds for all rank
matroids on at most elements, then it holds for all matroids
Universality theorems for inscribed polytopes and Delaunay triangulations
We prove that every primary basic semialgebraic set is homotopy equivalent to
the set of inscribed realizations (up to M\"obius transformation) of a
polytope. If the semialgebraic set is moreover open, then, in addition, we
prove that (up to homotopy) it is a retract of the realization space of some
inscribed neighborly (and simplicial) polytope. We also show that all algebraic
extensions of are needed to coordinatize inscribed polytopes.
These statements show that inscribed polytopes exhibit the Mn\"ev universality
phenomenon.
Via stereographic projections, these theorems have a direct translation to
universality theorems for Delaunay subdivisions. In particular, our results
imply that the realizability problem for Delaunay triangulations is
polynomially equivalent to the existential theory of the reals.Comment: 15 pages, 2 figure
Robust randomized matchings
The following game is played on a weighted graph: Alice selects a matching
and Bob selects a number . Alice's payoff is the ratio of the weight of
the heaviest edges of to the maximum weight of a matching of size at
most . If guarantees a payoff of at least then it is called
-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns
a -robust matching, which is best possible.
We show that Alice can improve her payoff to by playing a
randomized strategy. This result extends to a very general class of
independence systems that includes matroid intersection, b-matchings, and
strong 2-exchange systems. It also implies an improved approximation factor for
a stochastic optimization variant known as the maximum priority matching
problem and translates to an asymptotic robustness guarantee for deterministic
matchings, in which Bob can only select numbers larger than a given constant.
Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound
The Hilbert Zonotope and a Polynomial Time Algorithm for Universal Grobner Bases
We provide a polynomial time algorithm for computing the universal Gr\"obner
basis of any polynomial ideal having a finite set of common zeros in fixed
number of variables. One ingredient of our algorithm is an effective
construction of the state polyhedron of any member of the Hilbert scheme
Hilb^d_n of n-long d-variate ideals, enabled by introducing the Hilbert
zonotope H^d_n and showing that it simultaneously refines all state polyhedra
of ideals on Hilb^d_n
Solving the Maximum Popular Matching Problem with Matroid Constraints
We consider the problem of finding a maximum popular matching in a
many-to-many matching setting with two-sided preferences and matroid
constraints. This problem was proposed by Kamiyama (2020) and solved in the
special case where matroids are base orderable. Utilizing a newly shown matroid
exchange property, we show that the problem is tractable for arbitrary
matroids. We further investigate a different notion of popularity, where the
agents vote with respect to lexicographic preferences, and show that both
existence and verification problems become NP-hard, even in the -matching
case.Comment: 16 pages, 2 figure
Enumeration of PLCP-orientations of the 4-cube
The linear complementarity problem (LCP) provides a unified approach to many
problems such as linear programs, convex quadratic programs, and bimatrix
games. The general LCP is known to be NP-hard, but there are some promising
results that suggest the possibility that the LCP with a P-matrix (PLCP) may be
polynomial-time solvable. However, no polynomial-time algorithm for the PLCP
has been found yet and the computational complexity of the PLCP remains open.
Simple principal pivoting (SPP) algorithms, also known as Bard-type algorithms,
are candidates for polynomial-time algorithms for the PLCP. In 1978, Stickney
and Watson interpreted SPP algorithms as a family of algorithms that seek the
sink of unique-sink orientations of -cubes. They performed the enumeration
of the arising orientations of the -cube, hereafter called
PLCP-orientations. In this paper, we present the enumeration of
PLCP-orientations of the -cube.The enumeration is done via construction of
oriented matroids generalizing P-matrices and realizability classification of
oriented matroids.Some insights obtained in the computational experiments are
presented as well
Primitive Zonotopes
We introduce and study a family of polytopes which can be seen as a
generalization of the permutahedron of type . We highlight connections
with the largest possible diameter of the convex hull of a set of points in
dimension whose coordinates are integers between and , and with the
computational complexity of multicriteria matroid optimization.Comment: The title was slightly modified, and the determination of the
computational complexity of multicriteria matroid optimization was adde
- âŠ