Inapproximability of Combinatorial Problems via Small LPs and SDPs

Motivated by [arXiv:1309.0563], we provide a framework for studying the size of linear programming formulations as well as semidefinite programming formulations of combinatorial optimization problems without encoding them first as linear programs. This is done via a factorization theorem for the optimization problem itself (and not a specific encoding of such). As a result we define a consistent reduction mechanism that degrades approximation factors in a controlled fashion and which, at the same time, is compatible with approximate linear and semidefinite programming formulations. Moreover, 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 several problems that are not $0/1$-CSPs: we obtain a $\frac{3}{2}-\varepsilon$ inapproximability for VertexCover (which is not of the CSP type) answering an open question in [arXiv:1309.0563], we answer a weak version of our sparse graph conjecture posed in [arXiv:1311.4001] showing an inapproximability factor of $\frac{1}{2}+\varepsilon$ for bounded degree Max-IndependentSet, and we establish inapproximability of Max-k-MULTICUT (a non-binary CSP). In the case of SDPs, while an SDP-inapproximable base problem is unknown, we obtain relative inapproximability results for those problems.