We study the subclass of $\mathrm{prAM}$ problems such that derandomizing the corresponding Arthur--Merlin verifier (i.e. to an $\mathrm{NP}$ verifier) reduces to the Range Avoidance problem ($\mathrm{Avoid}$), the total search problem of finding a string outside the range of an expanding circuit.

We prove a new characterization of this subclass by showing a natural complete problem for it. The problem, which we call Structured Set Lower Bound ($\mathrm{SSLB}$), can be seen as a restriction of the classical Set Lower Bound problem by Goldwasser and Sipser~(STOC 1986), in which the (small) set is well-structured and can be compactly represented in the NO case. In more detail, we show that any problem in $\mathrm{prAM}$ admits a many-one reduction to $\mathrm{SSLB}$ if and only if derandomizing its verifier (via a targeted hitting set) reduces to $\mathrm{Avoid}$, thereby providing a full characterization of the subclass.

We then show that this class constitutes rich and interesting fragments of $\mathrm{prAM}$. First of all, we show that it contains all of $\mathrm{prMA}$. Moreover, two important problems from complexity theory and cryptography---Graph Non-Isomorphism ($\mathrm{GNI}$) and the complement of the approximate Shortest Vector Problem ($\overline{\mathrm{GapSVP}}_{\gamma}$, with approximation factor $\gamma = \Theta(\sqrt{n/\log n})$)---also fall into this class. These are all well-studied problems residing in $\mathrm{prAM}$. As a corollary, if the corresponding $\mathrm{Avoid}$ families are solvable in $\mathrm{SearchNP}$, then $\mathrm{GNI}$ and $\mathrm{GapSVP}_{\gamma}$ are in $\mathrm{NP\cap coNP}$, respectively.