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We derive a cut-and-paste surgery formula of Seiberg-Witten invariants for negative definite plumbed rational homology 3-spheres. It is similar to (and motivated by) Okuma's recursion formula [27, 4.5] targeting analytic invariants of splice-quotient singularities. Combining the two formulas automatically provides a proof of the equivariant version [11, 5.2(b)] of the Seiberg-Witten invariant conjecture [18] for these singularities.

We develop a framework for proving approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n 1/2−ǫ )-approximations for CLIQUE require linear programs of size 2 n Ω(ǫ) . This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem. Moreover, we establish a similar result for approximations of semidefinite programs by linear programs.Our main technical ingredient is a quantitative improvement of Razborov's rectangle corruption lemma (1992) for the high error regime, which gives strong lower bounds on the nonnegative rank of shifts of the unique disjointness matrix.

Motivated by [12], 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 3 2 − ε inapproximability for VertexCover (which is not of the CSP type) answering an open question in [12], we answer a weak version of our sparse graph conjecture posed in [6] showing an inapproximability factor of 1 2 +ε for bounded degree IndependentSet, and we establish inapproximability of Max-MULTI-k-CUT (a non-binary CSP). In the case of SDPs, we obtain relative inapproximability results for these problems.

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