20 research outputs found
Minimum Weight Flat Antichains of Subsets
Building on classical theorems of Sperner and Kruskal-Katona, we investigate
antichains in the Boolean lattice of all subsets of
, where is flat, meaning that it contains
sets of at most two consecutive sizes, say , where contains only -subsets,
while contains only -subsets. Moreover, we assume
consists of the first -subsets in squashed
(colexicographic) order, while consists of all -subsets
not contained in the subsets in . Given reals , we
say the weight of is
. We characterize the minimum
weight antichains for any given , and we do the
same when in addition is a maximal antichain. We can then derive
asymptotic results on both the minimum size and the minimum Lubell function
Minimizing the regularity of maximal regular antichains of 2- and 3-sets
Let be a natural number. We study the problem to find the
smallest such that there is a family of 2-subsets and
3-subsets of with the following properties: (1)
is an antichain, i.e. no member of is a subset of
any other member of , (2) is maximal, i.e. for every
there is an with or , and (3) is -regular, i.e. every point
is contained in exactly members of . We prove lower
bounds on , and we describe constructions for regular maximal antichains
with small regularity.Comment: 7 pages, updated reference
Approximation in Databases
One source of partial information in databases is the need to combine information from several databases. Even if each database is complete for some world , the combined databases will not be, and answers to queries against such combined databases can only be approximated. In this paper we describe various situations in which a precise answer cannot be obtained for a query asked against multiple databases. Based on an analysis of these situations, we propose a classification of constructs that can be used to model approximations.
One of the main goals is to show that most of these models of approximations possess universality properties. The main motivation for doing this is applying the data-oriented approach, which turns universality properties into syntax, to obtain languages for approximations. We show that the languages arising from the universality properties have a number of limitations. In an attempt to overcome those limitations, we explain how all the languages can be embedded into a language for conjunctive and disjunctive sets from [21], and demonstrate its usefulness in querying independent databases
On the topology of the permutation pattern poset
The set of all permutations, ordered by pattern containment, forms a poset.
This paper presents the first explicit major results on the topology of
intervals in this poset. We show that almost all (open) intervals in this poset
have a disconnected subinterval and are thus not shellable. Nevertheless, there
seem to be large classes of intervals that are shellable and thus have the
homotopy type of a wedge of spheres. We prove this to be the case for all
intervals of layered permutations that have no disconnected subintervals of
rank 3 or more. We also characterize in a simple way those intervals of layered
permutations that are disconnected. These results carry over to the poset of
generalized subword order when the ordering on the underlying alphabet is a
rooted forest. We conjecture that the same applies to intervals of separable
permutations, that is, that such an interval is shellable if and only if it has
no disconnected subinterval of rank 3 or more. We also present a simplified
version of the recursive formula for the M\"obius function of decomposable
permutations given by Burstein et al.Comment: 33 pages, 4 figures. Incorporates changes suggested by the referees;
new open problems in Subsection 9.4. To appear in JCT(A
Experiment Selection for Causal Discovery
Randomized controlled experiments are often described as the most reliable tool available to scientists
for discovering causal relationships among quantities of interest. However, it is often unclear
how many and which different experiments are needed to identify the full (possibly cyclic) causal
structure among some given (possibly causally insufficient) set of variables. Recent results in the
causal discovery literature have explored various identifiability criteria that depend on the assumptions
one is able to make about the underlying causal process, but these criteria are not directly
constructive for selecting the optimal set of experiments. Fortunately, many of the needed constructions
already exist in the combinatorics literature, albeit under terminology which is unfamiliar to
most of the causal discovery community. In this paper we translate the theoretical results and apply
them to the concrete problem of experiment selection. For a variety of settings we give explicit
constructions of the optimal set of experiments and adapt some of the general combinatorics results
to answer questions relating to the problem of experiment selection
Combinatorics in Schubert varieties and Specht modules
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, June 2011."June 2011." Cataloged from PDF version of thesis.Includes bibliographical references (p. 57-59).This thesis consists of two parts. Both parts are devoted to finding links between geometric/algebraic objects and combinatorial objects. In the first part of the thesis, we link Schubert varieties in the full flag variety with hyperplane arrangements. Schubert varieties are parameterized by elements of the Weyl group. For each element of the Weyl group, we construct certain hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial if and only if the Schubert variety is rationally smooth. For classical types the arrangements are (signed) graphical arrangements coning from (signed) graphs. Using this description, we also find an explicit combinatorial formula for the Poincaré polynomial in type A. The second part is about Specht modules of general diagram. For each diagram, we define a new class of polytopes and conjecture that the normalized volume of the polytope coincides with the dimension of the corresponding Specht module in many cases. We give evidences to this conjecture including the proofs for skew partition shapes and forests, as well as the normalized volume of the polytope for the toric staircase diagrams. We also define new class of toric tableaux of certain shapes, and conjecture the generating function of the tableaux is the Frobenius character of the corresponding Specht module. For a toric ribbon diagram, this is consistent with the previous conjecture. We also show that our conjecture is intimately related to Postnikov's conjecture on toric Specht modules and McNamara's conjecture of cylindric Schur positivity.by Hwanchul Yoo.Ph.D
A Theory of Sampling for Continuous-time Metric Temporal Logic
This paper revisits the classical notion of sampling in the setting of
real-time temporal logics for the modeling and analysis of systems. The
relationship between the satisfiability of Metric Temporal Logic (MTL) formulas
over continuous-time models and over discrete-time models is studied. It is
shown to what extent discrete-time sequences obtained by sampling
continuous-time signals capture the semantics of MTL formulas over the two time
domains. The main results apply to "flat" formulas that do not nest temporal
operators and can be applied to the problem of reducing the verification
problem for MTL over continuous-time models to the same problem over
discrete-time, resulting in an automated partial practically-efficient
discretization technique.Comment: Revised version, 43 pages
Space-Efficient Data Structures in the Word-RAM and Bitprobe Models
This thesis studies data structures in the word-RAM and bitprobe models, with an emphasis on space efficiency. In the word-RAM model of computation the space cost of a data structure is measured in terms of the number of w-bit words stored in memory, and the cost of answering a query is measured in terms of the number of read, write, and arithmetic operations that must be performed. In the bitprobe model, like the word-RAM model, the space cost is measured in terms of the number of bits stored in memory, but the query cost is measured solely in terms of the number of bit accesses, or probes, that are performed.
First, we examine the problem of succinctly representing a partially ordered set, or poset, in the word-RAM model with word size
Theta(lg n) bits. A succinct representation of a combinatorial object is one that occupies space matching the information theoretic lower bound to within lower order terms. We show how to represent a poset on n vertices using a data structure that occupies n^2/4 + o(n^2) bits, and can answer precedence (i.e., less-than) queries in
constant time. Since the transitive closure of a directed acyclic graph is a poset, this implies that we can support reachability
queries on an arbitrary directed graph in the same space bound. As far as we are aware, this is the first representation of an arbitrary directed graph that supports reachability queries in constant time,
and stores less than n choose 2 bits. We also consider several additional query operations.
Second, we examine the problem of supporting range queries on strings
of n characters (or, equivalently, arrays of
n elements) in the word-RAM model with word size Theta(lg n) bits. We focus on the specific problem of answering range majority queries: i.e., given a range, report the
character that is the majority among those in the range, if one exists. We show that these queries can be supported in constant time
using a linear space (in words) data structure. We generalize this
result in several directions, considering various frequency thresholds, geometric variants of the problem, and dynamism. These
results are in stark contrast to recent work on the similar range mode problem, in which the query operation asks for the mode (i.e., most frequent) character in a given range. The current best data structures for the range mode problem take soft-Oh(n^(1/2)) time per query for linear space data structures.
Third, we examine the deterministic membership (or dictionary) problem in the bitprobe model. This problem asks us to store a set of n elements drawn from a universe [1,u] such that membership queries
can be always answered in t bit probes. We present several new fully explicit results for this problem, in particular for the case
when n = 2, answering an open problem posed by Radhakrishnan, Shah, and Shannigrahi [ESA 2010]. We also present a general strategy for the membership problem that can be used to solve many related fundamental problems, such as rank, counting, and emptiness queries.
Finally, we conclude with a list of open problems and avenues for future work