Limits and Applications of Group Algebras for Parameterized Problems

The algebraic framework introduced in [Koutis, Proc. of the 35th 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 (2k ) 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 (2k ) 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 al...

• Type: Conference
• Year: 2009
• Where: ICALP
• Authors: Ioannis Koutis, Ryan Williams
• Keywords: Algebraic Method, Exact Parameterized Counting, ICALP 2009, Parameterized Complexity, Theoretical Computer Science