\textsf{ Holant} is an essential framework in the field of counting complexity. For over fifteen years, researchers have been clarifying the complexity classification for complex-valued \textsf{ Holant} on the Boolean domain, a challenge that remains unresolved. In this article, we prove a complexity dichotomy for complex-valued \textsf{ Holant} on Boolean domain when a non-trivial signature of odd arity exists. This dichotomy is based on the dichotomy for \textsf{ \#EO}, and consequently is an $\text{ FP}^\text{ NP}$ vs. \#P dichotomy as well, stating that each problem is either in $\text{ FP}^\text{ NP}$ or \#P-hard. Furthermore, we establish a generalized version of the decomposition lemma for complex-valued \textsf{ Holant} on Boolean domain. It asserts that each signature can be derived from its tensor product with other signatures, or conversely, the problem itself is in $\text{ FP}^\text{ NP}$. We believe that this result is a powerful method for building reductions in complex-valued \textsf{ Holant}, as it is also employed as a pivotal technique in the proof of the aforementioned dichotomy in this article.
Categories: cs.CC