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 $Ω(n^k)$ constraints. In this paper we extend and generalize their exhaustive sampling approach, presenting a framework for $(1-\eps)$-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 $δ\in (0, 1]$ and $\eps > 0$, we can approximate \kCSP\ problems with $Ω(n^{k-1+δ})$ constraints within a factor of $(1-\eps)$ in time $2^{O(n^{1-δ}\ln n /\eps^3)}$. 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 $n^{1+δ}$ 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 $δ> 0$, \kSAT\ instances with $O(n^{k-1})$ clauses cannot be approximated within a ratio better than $r$ in time $2^{O(n^{1-δ})}$. 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 $δ' > δ>0$, \MC\ instances with $n^{1+δ}$ edges cannot be approximated within a ratio better than $r$ in time $2^{n^{1-δ'}}$.

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