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

Summary, in English

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 nst/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. Shortly afterwards, Alon and Gutner (TALG 2010) showed that these problems have Ω(n^{⌊st/2⌋}) and Ω(n^{⌊k/2⌋}) lower bounds when counting by color coding. Here we show that one can do better, namely, we show that the “meet-in-the-middle” exponent st/2 can be beaten and give an algorithm that counts in time n^{0.4547st+O(1)} for t a multiple of three. This implies algorithms for counting occurrences of a fixed subgraph on k vertices and pathwidth p ≪ k in an n-vertex graph in n^{0.4547k+2p+O(1)} time, improving on the three mentioned algorithms for paths and matchings, and circumventing the color-coding lower bound.

• Author: Andreas Björklund; Petteri Kaski; Lukasz Kowalik
• Editor: Chandra Chekuri
• Department/s: Department of Computer Science; Parallel Systems
• Publishing year: 2014
• Language: English
• Pages: 594-603
• Publication/Series: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
• Links: http://dx.doi.org/10.1137/1.9781611973402.45
• Document type: Conference paper
• Publisher: Society for Industrial and Applied Mathematics
• Topic: Computer Science
• Conference name: 25th Annual ACM-SIAM Symposium on Discrete Algorithms
• Conference date: 2014-01-05
• Status: Published
• Project: Exact algorithms
• Research group: Algorithms
• ISBN/ISSN/Other: ISBN: 978-1-61197-338-9