Affine reductions for LPs and SDPs

Abstract: We define a reduction mechanism for LP and SDP formulations that degrades approximation factors in a controlled fashion. Our reduction mechanism is a minor restriction of classical reductions establishing inapproximability in the context of PCP theorems. As a consequence we establish strong linear programming inapproximability (for LPs with a polynomial number of constraints) for many problems. In particular we obtain a $3/2-\varepsilon$ inapproximability for VertexCover answering an open question in [arXiv:1309.0563] and we answer a weak version of our sparse graph conjecture posed in [arXiv:1311.4001] showing an inapproximability factor of $1/2+\varepsilon$ for bounded degree IndependentSet. In the case of SDPs, we obtain inapproximability results for these problems relative to the SDP-inapproximability of MaxCUT. Moreover, using our reduction framework we are able to reproduce various results for CSPs from [arXiv:1309.0563] via simple reductions from Max-2-XOR.