1,214 research outputs found
Optimal Recombination in Genetic Algorithms
This paper surveys results on complexity of the optimal recombination problem
(ORP), which consists in finding the best possible offspring as a result of a
recombination operator in a genetic algorithm, given two parent solutions. We
consider efficient reductions of the ORPs, allowing to establish polynomial
solvability or NP-hardness of the ORPs, as well as direct proofs of hardness
results
Measuring sets in infinite groups
We are now witnessing a rapid growth of a new part of group theory which has
become known as "statistical group theory". A typical result in this area would
say something like ``a random element (or a tuple of elements) of a group G has
a property P with probability p". The validity of a statement like that does,
of course, heavily depend on how one defines probability on groups, or,
equivalently, how one measures sets in a group (in particular, in a free
group). We hope that new approaches to defining probabilities on groups
outlined in this paper create, among other things, an appropriate framework for
the study of the "average case" complexity of algorithms on groups.Comment: 22 page
Shannon Information and Kolmogorov Complexity
We compare the elementary theories of Shannon information and Kolmogorov
complexity, the extent to which they have a common purpose, and where they are
fundamentally different. We discuss and relate the basic notions of both
theories: Shannon entropy versus Kolmogorov complexity, the relation of both to
universal coding, Shannon mutual information versus Kolmogorov (`algorithmic')
mutual information, probabilistic sufficient statistic versus algorithmic
sufficient statistic (related to lossy compression in the Shannon theory versus
meaningful information in the Kolmogorov theory), and rate distortion theory
versus Kolmogorov's structure function. Part of the material has appeared in
print before, scattered through various publications, but this is the first
comprehensive systematic comparison. The last mentioned relations are new.Comment: Survey, LaTeX 54 pages, 3 figures, Submitted to IEEE Trans
Information Theor
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Streaming Algorithms Via Reductions
In the streaming algorithms model of computation we must process data in order and without enough memory to remember the entire input. We study reductions between problems in the streaming model with an eye to using reductions as an algorithm design technique. Our contributions include:
* Linear Transformation reductions, which compose with existing linear sketch techniques. We use these for small-space algorithms for numeric measurements of distance-from-periodicity, finding the period of a numeric stream, and detecting cyclic shifts.
* The first streaming graph algorithms in the sliding window\u27 model, where we must consider only the most recent L elements for some fixed threshold L. We develop basic algorithms for connectivity and unweighted maximum matching, then develop a variety of other algorithms via reductions to these problems.
* A new reduction from maximum weighted matching to maximum unweighted matching. This reduction immediately yields improved approximation guarantees for maximum weighted matching in the semistreaming, sliding window, and MapReduce models, and extends to the more general problem of finding maximum independent sets in p-systems.
* Algorithms in a stream-of-samples model which exhibit clear sample vs. space tradeoffs. These algorithms are also inspired by examining reductions. We provide algorithms for calculating F_k frequency moments and graph connectivity
Adversarially Robust Submodular Maximization under Knapsack Constraints
We propose the first adversarially robust algorithm for monotone submodular
maximization under single and multiple knapsack constraints with scalable
implementations in distributed and streaming settings. For a single knapsack
constraint, our algorithm outputs a robust summary of almost optimal (up to
polylogarithmic factors) size, from which a constant-factor approximation to
the optimal solution can be constructed. For multiple knapsack constraints, our
approximation is within a constant-factor of the best known non-robust
solution.
We evaluate the performance of our algorithms by comparison to natural
robustifications of existing non-robust algorithms under two objectives: 1)
dominating set for large social network graphs from Facebook and Twitter
collected by the Stanford Network Analysis Project (SNAP), 2) movie
recommendations on a dataset from MovieLens. Experimental results show that our
algorithms give the best objective for a majority of the inputs and show strong
performance even compared to offline algorithms that are given the set of
removals in advance.Comment: To appear in KDD 201
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