44 research outputs found
The Poset of Hypergraph Quasirandomness
Chung and Graham began the systematic study of k-uniform hypergraph
quasirandom properties soon after the foundational results of Thomason and
Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in
the early work on k-uniform hypergraph quasirandomness is that properties that
are equivalent for graphs are not equivalent for hypergraphs, and thus
hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past
two decades, there has been an intensive study of these disparate notions of
quasirandomness for hypergraphs, and an open problem that has emerged is to
determine the relationship between them.
Our main result is to determine the poset of implications between these
quasirandom properties. This answers a recent question of Chung and continues a
project begun by Chung and Graham in their first paper on hypergraph
quasirandomness in the early 1990's.Comment: 43 pages, 1 figur
Algorithms to Approximate Column-Sparse Packing Problems
Column-sparse packing problems arise in several contexts in both
deterministic and stochastic discrete optimization. We present two unifying
ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain
improved approximation algorithms for some well-known families of such
problems. As three main examples, we attain the integrality gap, up to
lower-order terms, for known LP relaxations for k-column sparse packing integer
programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set
packing (Bansal et al., Algorithmica, 2012), and go "half the remaining
distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and
Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM
Transactions of Algorithm
Exact Bounds for Some Hypergraph Saturation Problems
Let W_n(p,q) denote the minimum number of edges in an n x n bipartite graph G
on vertex sets X,Y that satisfies the following condition; one can add the
edges between X and Y that do not belong to G one after the other so that
whenever a new edge is added, a new copy of K_{p,q} is created. The problem of
bounding W_n(p,q), and its natural hypergraph generalization, was introduced by
Balogh, Bollob\'as, Morris and Riordan. Their main result, specialized to
graphs, used algebraic methods to determine W_n(1,q).
Our main results in this paper give exact bounds for W_n(p,q), its hypergraph
analogue, as well as for a new variant of Bollob\'as's Two Families theorem. In
particular, we completely determine W_n(p,q), showing that if 1 <= p <= q <= n
then
W_n(p,q) = n^2 - (n-p+1)^2 + (q-p)^2.
Our proof applies a reduction to a multi-partite version of the Two Families
theorem obtained by Alon. While the reduction is combinatorial, the main idea
behind it is algebraic
Tree-like Queries in OWL 2 QL: Succinctness and Complexity Results
This paper investigates the impact of query topology on the difficulty of
answering conjunctive queries in the presence of OWL 2 QL ontologies. Our first
contribution is to clarify the worst-case size of positive existential (PE),
non-recursive Datalog (NDL), and first-order (FO) rewritings for various
classes of tree-like conjunctive queries, ranging from linear queries to
bounded treewidth queries. Perhaps our most surprising result is a
superpolynomial lower bound on the size of PE-rewritings that holds already for
linear queries and ontologies of depth 2. More positively, we show that
polynomial-size NDL-rewritings always exist for tree-shaped queries with a
bounded number of leaves (and arbitrary ontologies), and for bounded treewidth
queries paired with bounded depth ontologies. For FO-rewritings, we equate the
existence of polysize rewritings with well-known problems in Boolean circuit
complexity. As our second contribution, we analyze the computational complexity
of query answering and establish tractability results (either NL- or
LOGCFL-completeness) for a range of query-ontology pairs. Combining our new
results with those from the literature yields a complete picture of the
succinctness and complexity landscapes for the considered classes of queries
and ontologies.Comment: This is an extended version of a paper accepted at LICS'15. It
contains both succinctness and complexity results and adopts FOL notation.
The appendix contains proofs that had to be omitted from the conference
version for lack of space. The previous arxiv version (a long version of our
DL'14 workshop paper) only contained the succinctness results and used
description logic notatio
Reasoning with Inconsistent Information
In this thesis we are concerned with developing formal and representational mechanisms for reasoning with inconsistent information. Strictly speaking there are two conceptually distinct senses in which we are interested in reasoning with inconsistent information. In one sense, we are interested in using logical deduction to draw inferences in a symbolic system. More specifically, we are interested in mechanisms that can continue to perform deduction in a reasonable manner despite the threat of inconsistencies as a direct result of errors or misrepresentations. So in this sense we are interested in inconsistency-tolerant or paraconsistent deduction.
…
¶ In this thesis we adopt a novel framework to unify both logic-as-deduction and logic-as-representation approaches to reasoning with inconsistent information. …