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It has long been known, since the classical work of (Arora, Karger,Karpinski, JCSS~99), that \MC\ admits a PTAS on dense graphs, and moregenerally, \kCSP\ admits a PTAS on "dense" instances with $\Omega(n^k)$constraints. In this paper we extend and generalize their exhaustive samplingapproach, presenting a framework for $(1-\eps)$-approximating any \kCSP\problem in \emph{sub-exponential} time while significantly relaxing thedenseness requirement on the input instance. Specifically, we prove that forany constants $\delta \in (0, 1]$ and $\eps > 0$, we can approximate \kCSP\problems with $\Omega(n^{k-1+\delta})$ constraints within a factor of$(1-\eps)$ in time $2^{O(n^{1-\delta}\ln n /\eps^3)}$. The framework is quitegeneral and includes classical optimization problems, such as \MC, {\scMax}-DICUT, \kSAT, and (with a slight extension) $k$-{\sc Densest Subgraph}, asspecial cases. For \MC\ in particular (where $k=2$), it gives an approximationscheme that runs in time sub-exponential in $n$ even for "almost-sparse"instances (graphs with $n^{1+\delta}$ edges). We prove that our results areessentially best possible, assuming the ETH. First, the density requirementcannot be relaxed further: there exists a constant $r < 1$ such that for all$\delta > 0$, \kSAT\ instances with $O(n^{k-1})$ clauses cannot be approximatedwithin a ratio better than $r$ in time $2^{O(n^{1-\delta})}$. Second, therunning time of our algorithm is almost tight \emph{for all densities}. Evenfor \MC\ there exists $r<1$ such that for all $\delta' > \delta >0$, \MC\instances with $n^{1+\delta}$ edges cannot be approximated within a ratiobetter than $r$ in time $2^{n^{1-\delta'}}$.

Sub-exponential Approximation Schemes for CSPs: from Dense to Almost Sparse