A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

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Presentation transcript:

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

Hamiltonicity

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

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!

Refined DP: Matchings Connectivity Matrix 1 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 ∗ 2+ 2 𝑝𝑤 time, use permutation submatrix to show 𝑂 ∗ 2+ 2 −𝜀 𝑝𝑤 time violates SETH.

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

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!!