New algorithms for linear k-matroid intersection and matroid k-parity problems
We present algorithms for the k-Matroid Intersection Problem and for the Matroid k-Parity Problem when the matroids are represented over the field of rational numbers and k > 2. The computational complexity of the algorithms is linear in the cardinality n and singly exponential in the rank r of the matroids. Thus if n grows faster than a linear function in r (this is the case for most combinatorial applications) then the algorithms are asymptotically faster than exhaustive search and provide the best known worst-case complexity. If r = O(log n) then the algorithms have polynomial-time complexity. As an application, we prove that for any fixed k one can determine in polynomial time whether there exist O(log n) pairwise disjoint edges in a given uniform k-hypergraph on n vertices. Our approach extends known methods of linear algebra developed earlier for the case k = 2. Using the generalized Binet-Cauchy formula and its analogue for the Pfaffian we reduce in O(nr{sup 2k}) time the k-Matroid intersection Problem to computation of the hyperdeterminant of a 2k-dimensional r x ... x r tensor and the Matroid k-Parity Problem to computation of the hyperpfaffian of a 2k-dimensional 2r x ... 2r tensor. We use dynamic programming to compute these invariants of tensors using O(r{sup 2k}4{sup rk}) and O(r{sup 2k+1}4{sup r}) arithmetic operations correspondingly.
- OSTI ID:
- 35811
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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