Counting thin subgraphs via packings faster than meet-in-the-middle time

Abstract: 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 $n<sup>{k/2+O(1)}$. In the same year, two different algorithms with the same runtime were given by Koutis and Williams~(ICALP 2009), and Bj\"orklund \emph{et al.} (ESA 2009), via $n<sup>{st/2+O(1)}$-time algorithms for counting $t$-tuples of pairwise disjoint sets drawn from a given family of $s$-sized subsets of an $n$-element universe. Shortly afterwards, Alon and Gutner (TALG 2010) showed that these problems have $\Omega(n^{\lfloor st/2\rfloor})$ and $\Omega(n^{\lfloor k/2\rfloor})$ 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 $n^{0.45470382 st + O(1)}$ for $t$ a multiple of three. This implies algorithms for counting occurrences of a fixed subgraph on $k$ vertices and pathwidth $p\ll k$ in an $n$-vertex graph in $n<sup>{0.45470382k+2p+O(1)}$ 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.