TY - JOUR
T1 - Counting thin subgraphs via packings faster than meet-in-the-middle time
AU - Björklund, Andreas
AU - Kaski, Petteri
AU - Kowalik, Lukasz
PY - 2017/9/1
Y1 - 2017/9/1
N2 - Vassilevska and Williams (STOC'09) showed how to count simple paths on k vertices and matchings on k/2 edges in ann-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 Bjorklund 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 O(n st/2) and O(nk/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 pk 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.
AB - Vassilevska and Williams (STOC'09) showed how to count simple paths on k vertices and matchings on k/2 edges in ann-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 Bjorklund 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 O(n st/2) and O(nk/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 pk 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.
KW - Counting low pathwidth graphs
KW - Counting matchings
KW - Counting packings
KW - Counting paths
KW - FPT algorithms
KW - Matrix multiplication
KW - Meet in the middle
UR - http://www.scopus.com/inward/record.url?scp=85030330604&partnerID=8YFLogxK
U2 - 10.1145/3125500
DO - 10.1145/3125500
M3 - Article
VL - 13
SP - 1
EP - 26
JO - ACM Transactions on Algorithms
JF - ACM Transactions on Algorithms
SN - 1549-6325
IS - 4
M1 - 48
ER -