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

Vassilevska and Williams (STOC’09) showed how to count simple paths on k vertices and matchings on k/2 edges in an n-vertex graph in time nk/2+O(1). In the same year, two different algorithms with the same runtime were given by Koutis and Williams (ICALP’09), and Björklund et al. (ESA’09), 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’10) 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—we show that the “meet-in-the-middle” exponent st/2 can be beaten and give an algorithm that counts in time n0.45470382st+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 n0.45470382k+2p+O(1) time, improving on the three mentioned algorithms for paths and matchings, and circumventing the color-coding lower bound. We also give improved bounds for counting t-tuples of disjoint s-sets for s = 2,3,4. Our algorithms use fast matrix multiplication. We show an argument that this is necessary to go below the meet-in-the-middle barrier.