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We consider the algorithmic problem of generating each subset of$[n]:=\{1,2,\ldots,n\}$ whose size is in some interval $[k,l]$, $0\leq k\leql\leq n$, exactly once (cyclically) by repeatedly adding or removing a singleelement, or by exchanging a single element. For $k=0$ and $l=n$ this is theclassical problem of generating all $2^n$ subsets of $[n]$ by elementadditions/removals, and for $k=l$ this is the classical problem of generatingall $\binom{n}{k}$ subsets of $[n]$ by element exchanges. We prove theexistence of such cyclic minimum-change enumerations for a large range ofvalues $n$, $k$, and $l$, improving upon and generalizing several previousresults. For all these existential results we provide optimal algorithms tocompute the corresponding Gray codes in constant $\mathcal{O}(1)$ time pergenerated set and $\mathcal{O}(n)$ space. Rephrased in terms of graph theory,our results establish the existence of (almost) Hamilton cycles in the subgraphof the $n$-dimensional cube $Q_n$ induced by all levels $[k,l]$. We reduce allremaining open cases to a generalized version of the middle levels conjecture,which asserts that the subgraph of $Q_{2k+1}$ induced by all levels$[k-c,k+1+c]$, $c\in\{0,1,\ldots,k\}$, has a Hamilton cycle. We also prove anapproximate version of this generalized conjecture, showing that this graph hasa cycle that visits a $(1-o(1))$-fraction of all vertices.

Trimming and gluing Gray codes