A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
(1709.02311)Abstract
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 $\Theta(\sqrt 2k)$ 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 $4k / \mathrm{poly}(k)$. We also show that the rank of $\mathbf{M}k$ over $\mathbb{Z}_p$ is $\Omega(1.97k)$ for any prime $p\neq 2$ and even $\Omega(2.15k)$ 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.
We're not able to analyze this paper right now due to high demand.
Please check back later (sorry!).
Generate a summary of this paper on our Pro plan:
We ran into a problem analyzing this paper.