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

For even k ∈ N, the matchings connectivity matrix Mk is a binary matrix indexed by perfect matchings on k vertices; the entry at (M, M') is 1 iff M ∪ M' forms a single cycle. Cygan et al. (STOC 2013) showed that the rank of Mk over Z2 is [EQUATION] and used this to give an [EQUATION] time algorithm for counting Hamiltonian cycles modulo 2 on graphs of pathwidth pw, carrying over to the decision problem via witness isolation. 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), using the representation theory of the symmetric group and various insights from algebraic combinatorics. We also show that the rank of Mk over Zp is Ω(1.57k) for any prime p ≠ 2. Combining our rank bounds with the new pattern propagation technique, we show 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*(3.57pw), indicating that the modulus can affect the complexity in intricate ways.