2 CSPs All Are Approximable Within a Constant Differential Factor

Abstract

Only a few facts are known regarding the approximability of optimization CSPs with respect to the differential approximation measure, which compares the gain of a given solution over the worst solution value to the instance diameter. Notably, the question whether $\mathsf {k\,CSP\!-\!q}$ is approximable within any constant factor is open in case when $q \ge 3$ or $k\ge 4$. Using a family of combinatorial designs we introduce for our purpose, we show that, given any three constant integers $k\ge 2$, $p\ge k$ and $q >p$, $\mathsf {k\,CSP\!-\!q}$ reduces to $\mathsf {k\,CSP\!-\!p}$ with an expansion of $1/(q~-~p~+~k/2)^k$ on the approximation guarantee. When $p =k =2$, this implies together with the result of Nesterov as regards $\mathsf {2\,CSP\!-\!2}$ [1] that for all constant integers $q\ge 2$, $\mathsf {2\,CSP\!-\!q}$ is approximable within factor $(2~-~\pi /2)/(q~-~1)^2$.