In this paper, we study the packet classification problem and the filter conflict resolution problem, both motivated by applications in the routing of packets in an IP network. For the first problem, we focus on the static 2-dimensional conflict-free (i.e., nested) filters. We design a linear space data structure with O(T w (n)+(log log n)2) query time on a RAM with word size O(w) bits where n is the number of filters in the router, w is the number of bits in an IP address and $$T_{w}(n)=O\biggl(\min\biggl\{\log w,\frac{\log w\log\log n}{\log\log w},\sqrt{\frac{\log n}{\log\log n}}\biggr\}\biggr).$$ that is definable in the e-model, either $\mathrm{coUP}\subseteq\mathcal{C}$or $\mathcal{C}\subseteq\mathrm{NP}$. For the existing models, gap theorems, where they exist at all, only identify gaps for the definability by regular languages. We prove gaps for the general case, i.e., for the definability by arbitrary languages. We obtain such general gaps for NP, coNP, 1NP, and co1NP. For the regular case we prove further gap theorems for Σ2P, Π2P, and Δ2P. These are the first gap theorems for Δ2P. This work is related to former work by Bovet, Crescenzi, and Silvestri, Vereshchagin, Hertrampf et al., Burtschick and Vollmer, and Borchert et al.