647 research outputs found
Exact Covers via Determinants
Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such
that every hyperedge includes one vertex from each part, the k-dimensional
matching problem asks whether there is a disjoint collection of the hyperedges
which covers all vertices. We show it can be solved by a randomized polynomial
space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors
polynomial in n and k.
When we drop the partition constraint and permit arbitrary hyperedges of
cardinality k, we obtain the exact cover by k-sets problem. We show it can be
solved by a randomized polynomial space algorithm in time O*(c_k^n), where
c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k.
Both results substantially improve on the previous best algorithms for these
problems, especially for small k, and follow from the new observation that
Lovasz' perfect matching detection via determinants (1979) admits an embedding
in the recently proposed inclusion-exclusion counting scheme for set covers,
despite its inability to count the perfect matchings
Outlaw distributions and locally decodable codes
Locally decodable codes (LDCs) are error correcting codes that allow for
decoding of a single message bit using a small number of queries to a corrupted
encoding. Despite decades of study, the optimal trade-off between query
complexity and codeword length is far from understood. In this work, we give a
new characterization of LDCs using distributions over Boolean functions whose
expectation is hard to approximate (in~~norm) with a small number of
samples. We coin the term `outlaw distributions' for such distributions since
they `defy' the Law of Large Numbers. We show that the existence of outlaw
distributions over sufficiently `smooth' functions implies the existence of
constant query LDCs and vice versa. We give several candidates for outlaw
distributions over smooth functions coming from finite field incidence
geometry, additive combinatorics and from hypergraph (non)expanders.
We also prove a useful lemma showing that (smooth) LDCs which are only
required to work on average over a random message and a random message index
can be turned into true LDCs at the cost of only constant factors in the
parameters.Comment: A preliminary version of this paper appeared in the proceedings of
ITCS 201
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
Embedding large subgraphs into dense graphs
What conditions ensure that a graph G contains some given spanning subgraph
H? The most famous examples of results of this kind are probably Dirac's
theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect
matchings are generalized by perfect F-packings, where instead of covering all
the vertices of G by disjoint edges, we want to cover G by disjoint copies of a
(small) graph F. It is unlikely that there is a characterization of all graphs
G which contain a perfect F-packing, so as in the case of Dirac's theorem it
makes sense to study conditions on the minimum degree of G which guarantee a
perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy
and Szemeredi have proved to be powerful tools in attacking such problems and
quite recently, several long-standing problems and conjectures in the area have
been solved using these. In this survey, we give an outline of recent progress
(with our main emphasis on F-packings, Hamiltonicity problems and tree
embeddings) and describe some of the methods involved
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