LIMITS and Applications of Group Algebras for Parameterized Problems

Abstract

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.

Deterministic single exponential time algorithms for connectivity problems parameterized by treewidth

“…The Cut & Count approach is one of the dynamic programming algorithms using a modulo 2 based transformation [3,4,16,15,18,23]. These algorithms give the smallest running times currently known, but have several disadvantages compared to traditional dynamic programming algorithms: (a) They are randomized.…”

Counting Thin Subgraphs via Packings Faster than Meet-in-the-Middle Time

“…Recalling the first family of linear equations from §2, from the proof of Lemma 6 and the structure of equations (19) and (21) we can observe that the right-hand side y i of the first family has trilinear rank at most (33) O n (3/2−γ)q+c whenever 0 ≤ i ≤ (3/2 − γ)q. Thus, using the first family we can show that the trilinear rank of x 0 is small by showing that the indeterminates x j for large values of j have low trilinear rank.…”

“…This result was accompanied, within the same year, of two publications presenting the same runtime restricted to counting paths and matchings. Koutis and Williams [19] and Björklund et al [6] describe different algorithms for the related problem of counting the number of t-tuples of disjoint sets that can be formed from a given family of s-subsets of an n-element universe in n st/2+O(1) time. Fomin et al [13] generalized the latter result into an algorithm that counts occurrences of a k-vertex pattern graph with pathwidth p in n k/2+2p+O(1) time.…”

“…Indeed, not only are the algorithms of Vassilevska and Williams [26], Koutis and Williams [19], and Björklund et al [6] fairly recent discoveries, but they all employ rather different techniques. Common to all three algorithms is however the need to consider the k/2-element subsets of the n-element vertex set, resulting in time n k/2+O (1) .…”

“…Vassilevska and Williams (STOC 2009) showed how to count simple paths on k vertices and matchings on k/2 edges in an n-vertex graph in time n k/2+O(1) . In the same year, two different algorithms with the same runtime were given by Koutis and Williams (ICALP 2009), and Björklund et al. (ESA 2009), via n st/2+O(1) -time algorithms for counting t-tuples of pairwise disjoint sets drawn from a given family of s-sized subsets of an n-element universe.…”