For even k, the matchings connectivity matrix Mk encodes which
pairs of perfect matchings on k vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of Mk over Z2 is
Θ(2k) and used this to give an O∗((2+2)pw)
time algorithm for counting Hamiltonian cycles modulo 2 on graphs of
pathwidth 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
Mk, 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 Mk is given; no
stronger structural insights such as the existence of large permutation
submatrices in Mk 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 Mk over the
rationals is 4k/poly(k). We also show that the rank of
Mk over Zp is Ω(1.97k) for any prime p=2
and even Ω(2.15k) for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
O∗((6−ϵ)pw) for any ϵ>0 unless SETH fails. This
bound is tight due to a O∗(6pw) time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes p=2 in time O∗(3.97pw), indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201