# A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

For even k, the matchings connectivity matrix 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 M_k over Z_2 is Θ(√(2)^k) 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 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 M_k is given; no stronger structural insights such as the existence of large permutation submatrices in 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 M_k over the rationals is 4^k / poly(k). We also show that the rank of M_k over Z_p is Ω(1.97^k) for any prime p≠ 2 and even Ω(2.15^k) 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^*(6^pw) 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^*(3.97^pw), indicating that the modulus can affect the complexity in intricate ways.

READ FULL TEXT