The algebraic framework introduced in [Koutis, Proc. of the 35 th ICALP 2008] reduces several combinatorial problems in parameterized complexity to the problem of detecting multilinear degree-k monomials in polynomials presented as circuits. The best known (randomized) algorithm for this problem requires only O *(2 k ) time and oracle access to an arithmetic circuit, i.e. the ability to evaluate the circuit on elements from a suitable group algebra. This algorithm has been used to obtain the best known algorithms for several parameterized problems. In this paper we use communication complexity to show that the O *(2 k ) algorithm is essentially optimal within this evaluation oracle framework. On the positive side, we give new applications of the method: finding a copy of a given tree on k nodes, a spanning tree with at least k leaves, a minimum set of nodes that dominate at least t nodes, and an m-dimensional k-matching. In each case we achieve a faster algorithm than what was known. We also apply the algebraic method to problems in exact counting. Among other results, we show that a combination of dynamic programming and a variation of the algebraic method can break the trivial upper bounds for exact parameterized counting in fairly general settings. © 2009 Springer Berlin Heidelberg.
CITATION STYLE
Koutis, I., & Williams, R. (2009). Limits and applications of group algebras for parameterized problems. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 5555 LNCS, pp. 653–664). https://doi.org/10.1007/978-3-642-02927-1_54
Mendeley helps you to discover research relevant for your work.