Affine reductions for LPs and SDPs

TL;DR

This paper introduces a reduction framework for LPs and SDPs that controls approximation degradation, leading to new inapproximability results for problems like VertexCover and IndependentSet, and reproduces known CSP results.

Contribution

It presents a restricted reduction mechanism for LP and SDP formulations, establishing new inapproximability bounds and simplifying derivations of existing results.

Findings

Proves 3/2-ε inapproximability for VertexCover.

Shows 1/2+ε inapproximability for bounded degree IndependentSet.

Reproduces CSP inapproximability results via simple reductions.

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.