1. A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
Presented at
SODA 2018
Radu Curticapean Hungarian Academy of Sciences
Nathan Lindzey University of Waterloo
Jesper Nederlof Technische Universiteit Eindhoven
Slides mostly by Radu

2. Hamiltonicity

3. Hamiltonicity under treewidth
treewidth 𝒕𝒘 𝑮 =𝒌: hierarchy of 𝑘-sized separators
enables dynamic programming
tree 1
𝑲 𝒏
𝑛−1
SPG 2
planar
~ 𝑛
deg-𝟑
~𝑛/6

4. Hamiltonicity under treewidth
The complexity of NP-hard problems on
small-treewidth instances often depends
on the rank of problem-related matrices.
Refined DP
standard
𝑂 ∗ 𝑡𝑤!
refined DP
𝑂 ∗ ( 𝑐 𝑡𝑤 )
decision
𝑐=2+ 2
optimal under SETH!
counting
𝑐=6 (assuming 𝜔=2)
this paper:
optimal under SETH!

5. Refined DP: Matchings Connectivity Matrix1
matchings connectivity matrix 𝑴 𝒃
over all perfect matchings on 𝑏 ,,
𝑴 𝒃 𝑁, 𝑁 ′ =1 iff 𝑁∪𝑁 ′ is cycle
CKN[STOC13, JACM18]:
𝑟 𝑘 𝔽 2 𝑀 𝑏 = 2 𝑏/2−1 , and 𝑀 𝑏 contains equally large permutation submatrix,
decide the existence and count HC‘s mod 2 in 𝑂 ∗ 𝑝𝑤 time,
use permutation submatrix to show 𝑂 ∗ −𝜀 𝑝𝑤 time violates SETH.

6. 1+2 1 2 Our main contributions
Determine rk ℝ ( 𝑴 𝒃 ) using representation theory of the symmetric group 1 2
New reduction idea
turns any rank lower bound
into SETH lower bound
BCKN[Inf. Compt‘15],
W[IPEC16]
Counting HCs
rk ℝ 𝑀 𝑏 ≈ 4 𝑏
𝑂 ∗ ( 6 𝑡𝑤 ) time
optimal base
under SETH

7. Thank you!! ? Decide/count HCs mod 2 Counting HCs mod p≠2 Counting HCs
rk ℤ 2 𝑀 𝑏 = 2 𝑏/2−1
𝑂 ∗ ( 3.41 𝑡𝑤 ) time
optimal base
under SETH
CKN[STOC13, JACM18]
r 𝑘 ℤ 𝑝 𝑀 𝑏 ≥ 1.97 𝑏
Counting HCs mod p≠2 ?
Ω ∗ ( 3.97 𝑡𝑤 ) time under SETH
[this paper]
[this paper]
BCKN[Inf. Compt‘15],
W[IPEC16]
Counting HCs
𝑂 ∗ ( 6 𝑡𝑤 ) time
rk ℝ 𝑀 𝑏 ≈ 4 𝑏
optimal base
under SETH
Thank you!!