A tight lower bound for counting Hamiltonian cycles via matrix rank

R. Curticapean, N. Lindzey, J. Nederlof

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Uittreksel

For even $k$, the matchings connectivity matrix $\mathbf{M}_k$ encodes which pairs of perfect matchings on $k$ vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_2$ is $\Theta(\sqrt 2^k)$ and used this to give an $O^*((2+\sqrt{2})^{\mathsf{pw}})$ time algorithm for counting Hamiltonian cycles modulo $2$ on graphs of pathwidth $\mathsf{pw}$. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within $\mathbf{M}_k$, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of $\mathbf{M}_k$ is given; no stronger structural insights such as the existence of large permutation submatrices in $\mathbf{M}_k$ are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes $p$) parameterized by pathwidth. To apply this technique, we prove that the rank of $\mathbf{M}_k$ over the rationals is $4^k / \mathrm{poly}(k)$. We also show that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_p$ is $\Omega(1.97^k)$ for any prime $p\neq 2$ and even $\Omega(2.15^k)$ for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time $O^*((6-\epsilon)^{\mathsf{pw}})$ for any $\epsilon>0$ unless SETH fails. This bound is tight due to a $O^*(6^{\mathsf{pw}})$ time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes $p\neq 2$ in time $O^*(3.97^\mathsf{pw})$, indicating that the modulus can affect the complexity in intricate ways.
TaalEngels
Artikelnummer1709.02311
Aantal pagina's32
TijdschriftarXiv.org, e-Print Archive, Physics
StatusGepubliceerd - 7 sep 2017

Vingerafdruk

Hamiltonian circuit
Counting
Lower bound
Exponential time
Pathwidth
Modulo
Permutation
Propagation
Perfect Matching
Black Box
Modulus
Connectivity
Cycle
Graph in graph theory

Bibliografische nota

improved lower bounds modulo primes, improved figures, to appear in SODA 2018

Trefwoorden

  • cs.DS
  • cs.CC
  • math.CO
  • math.RT

Citeer dit

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A tight lower bound for counting Hamiltonian cycles via matrix rank. / Curticapean, R.; Lindzey, N.; Nederlof, J.

In: arXiv.org, e-Print Archive, Physics, 07.09.2017.

Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademic

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