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

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**Counting thin subgraphs via packings faster than meet-in-the-middle time.** / Björklund, Andreas; Kaski, Petteri; Kowalik, Lukasz.

Research output: Contribution to journal › Article

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*ACM Transactions on Algorithms*, vol. 13, no. 4, 48, pp. 1-26. https://doi.org/10.1145/3125500

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*ACM Transactions on Algorithms*,

*13*(4), 1-26. [48]. https://doi.org/10.1145/3125500

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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 -

ID: 15782919