A type of ‘non-traditional constituents’ motivates an extended class of Combinatory Categorial Grammars, CCGs with Generalized Type-Raised Categories (CCG-GTRC) involving variables. Although the class of standard CCGs is known to be polynomially parsable, unrestricted use of variables can destroy this essential requirement for a practical parser. This paper argues for polynomial parsability of CCG-GTRC from practical and theoretical points of view. First, we show that an experimental parser runs polynomially in practice on a realistic fragment of Japanese by eliminating spurious ambiguity and excluding genuine ambiguities. Then, we present a worst-case polynomial recognition algorithm for CCG-GTRC by extending the polynomial algorithm for the standard CCGs.