Exact Covers via Determinants
Abstract
Given a kuniform hypergraph on n vertices, partitioned in k equal parts such that every hyperedge includes one vertex from each part, the kdimensional 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(k2)/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 ksets 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 inclusionexclusion counting scheme for set covers, despite its inability to count the perfect matchings.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 arXiv:
 arXiv:0910.0460
 Bibcode:
 2009arXiv0910.0460B
 Keywords:

 Computer Science  Data Structures and Algorithms