320 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
Exact Covers via Determinants
Given a -uniform hypergraph on vertices, partitioned in equal parts such that every hyperedge includes one vertex from each part, the -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. The notation hides factors
polynomial in and .
The general Exact Cover by -Sets problem asks the same when the partition constraint is dropped and arbitrary hyperedges of cardinality are permitted. We show it can be solved by a randomized polynomial space algorithm in time, where , and provide a general bound for larger .
Both results substantially improve on the previous best algorithms for these problems, especially for small . They follow from the new observation that Lov\u27asz\u27 perfect matching detection via determinants (Lov\u27asz, 1979) admits an embedding in the recently proposed inclusion--exclusion counting scheme for set covers, emph{despite} its inability to count the perfect matchings
Recommended from our members
Graph Theory
This was a workshop on graph theory, with a comprehensive approach. Highlights included the emerging theories of sparse graph limits and of infinite matroids, new techniques for colouring graphs on surfaces, and extensions of graph minor theory to directed graphs and to immersion
Recommended from our members
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
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