Counting perfect matchings as fast as Ryser
Author
Summary, in English
We show that there is a polynomial space algorithm that counts the number of perfect matchings in an n-vertex graph in O*(2n/2) ⊂ O(1.415n) time. (O*(f(n)) suppresses functions polylogarithmic in f(n)). The previously fastest algorithms for the problem was the exponential space O*(((1 + √5)/2)n) ⊂ O(1.619n) time algorithm by Koivisto, and for polynomial space, the O(1.942n) time algorithm by Nederlof. Our new algorithm's runtime matches up to polynomial factors that of Ryser's 1963 algorithm for bipartite graphs. We present our algorithm in the more general setting of computing the hafnian over an arbitrary ring, analogously to Ryser's algorithm for permanent computation.
We also give a simple argument why the general exact set cover counting problem over a slightly superpolynomial sized family of subsets of an n element ground set cannot be solved in O*(2(1−ε1)n) time for any ε1 > 0 unless there are O*(2(1−ε2)n) time algorithms for computing an n x n 0/1 matrix permanent, for some ε2 > 0 depending only on ε1.
We also give a simple argument why the general exact set cover counting problem over a slightly superpolynomial sized family of subsets of an n element ground set cannot be solved in O*(2(1−ε1)n) time for any ε1 > 0 unless there are O*(2(1−ε2)n) time algorithms for computing an n x n 0/1 matrix permanent, for some ε2 > 0 depending only on ε1.
Department/s
Publishing year
2012
Language
English
Pages
914-921
Publication/Series
[Host publication title missing]
Links
Document type
Conference paper
Publisher
Society for Industrial and Applied Mathematics
Topic
- Computer Science
Conference name
ACM-SIAM Symposium on Discrete Algorithms
Conference date
2012-01-17 - 2012-01-19
Conference place
Kyoto, Japan
Status
Published
Project
- Exact algorithms
Research group
- Algorithms