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

It has long been known, since the classical work of (Arora, Karger, Karpinski, JCSS~99), that \MC\ admits a PTAS on dense graphs, and more generally, \kCSP\ admits a PTAS on “dense” instances with $\mathrm{\Omega }(n^k)$ constraints. In this paper we extend and generalize their exhaustive sampling approach, presenting a framework for $(1-\text{eps})$-approximating any \kCSP
problem in \emph{sub-exponential} time while significantly relaxing the denseness requirement on the input instance. Specifically, we prove that for any constants $\delta \in (0,1]$ and $\text{eps}>0$, we can approximate \kCSP
problems with $\mathrm{\Omega }(n^{k-1+\delta })$ constraints within a factor of $(1-\text{eps})$ in time $2^{O(n^{1-\delta }\ln n\mathrm{/}{\text{eps}}^3)}$. The framework is quite general and includes classical optimization problems, such as \MC, {\sc Max}-DICUT, \kSAT, and (with a slight extension) $k$-{\sc Densest Subgraph}, as special cases. For \MC\ in particular (where $k=2$), it gives an approximation scheme that runs in time sub-exponential in $n$ even for “almost-sparse” instances (graphs with $n^{1+\delta }$ edges). We prove that our results are essentially best possible, assuming the ETH. First, the density requirement cannot 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 approximated within a ratio better than $r$ in time $2^{O(n^{1-\delta })}$. Second, the running time of our algorithm is almost tight \emph{for all densities}. Even for \MC\ there exists $r<1$ such that for all ${\delta }^{\mathrm{\prime }}>\delta >0$, \MC
instances with $n^{1+\delta }$ edges cannot be approximated within a ratio better than $r$ in time $2^{n^{1-{\delta }^{\mathrm{\prime }}}}$.