68,249 research outputs found
Randomized Composable Core-sets for Distributed Submodular Maximization
An effective technique for solving optimization problems over massive data
sets is to partition the data into smaller pieces, solve the problem on each
piece and compute a representative solution from it, and finally obtain a
solution inside the union of the representative solutions for all pieces. This
technique can be captured via the concept of {\em composable core-sets}, and
has been recently applied to solve diversity maximization problems as well as
several clustering problems. However, for coverage and submodular maximization
problems, impossibility bounds are known for this technique \cite{IMMM14}. In
this paper, we focus on efficient construction of a randomized variant of
composable core-sets where the above idea is applied on a {\em random
clustering} of the data. We employ this technique for the coverage, monotone
and non-monotone submodular maximization problems. Our results significantly
improve upon the hardness results for non-randomized core-sets, and imply
improved results for submodular maximization in a distributed and streaming
settings.
In summary, we show that a simple greedy algorithm results in a
-approximate randomized composable core-set for submodular maximization
under a cardinality constraint. This is in contrast to a known impossibility result for (non-randomized) composable core-set. Our
result also extends to non-monotone submodular functions, and leads to the
first 2-round MapReduce-based constant-factor approximation algorithm with
total communication complexity for either monotone or non-monotone
functions. Finally, using an improved analysis technique and a new algorithm
, we present an improved -approximation algorithm
for monotone submodular maximization, which is in turn the first
MapReduce-based algorithm beating factor in a constant number of rounds
A Logical Characterization of Constant-Depth Circuits over the Reals
In this paper we give an Immerman's Theorem for real-valued computation. We
define circuits operating over real numbers and show that families of such
circuits of polynomial size and constant depth decide exactly those sets of
vectors of reals that can be defined in first-order logic on R-structures in
the sense of Cucker and Meer. Our characterization holds both non-uniformily as
well as for many natural uniformity conditions.Comment: 24 pages, submitted to WoLLIC 202
Implementability Among Predicates
Much work has been done to understand when given predicates (relations) on discrete variables can be conjoined to implement other predicates. Indeed, the lattice of "co-clones" (sets of predicates closed under conjunction, variable renaming, and existential quantification of variables) has been investigated steadily from the 1960's to the present. Here, we investigate a more general model, where duplicatability of values is not taken for granted. This model is motivated in part by large scale neural models, where duplicating a value is similar in cost to computing a function, and by quantum mechanics, where values cannot be duplicated. Implementations in this case are naturally given by a graph fragment in which vertices are predicates, internal edges are existentially quantified variables, and "dangling edges" (edges emanating from a vertex but not yet connected to another vertex) are the free variables of the implemented predicate. We examine questions of implementability among predicates in this scenario, and
we present the solution to all implementability problems for single predicates on up to three boolean values. However, we find that a variety of proof methods are required, and the question of implementability indeed becomes undecidable for larger predicates, although this is tricky to prove. We find that most predicates cannot implement the 3-way equality predicate, which reaffirms the view that duplicatability of values should not be assumed a priori
Computational Processes and Incompleteness
We introduce a formal definition of Wolfram's notion of computational process
based on cellular automata, a physics-like model of computation. There is a
natural classification of these processes into decidable, intermediate and
complete. It is shown that in the context of standard finite injury priority
arguments one cannot establish the existence of an intermediate computational
process
- …