In this thesis we prove intractability results for several well studied problems in combinatorial optimization.

Closest Vector Problem with Pre-processing (CVPP): We show that the pre-processing version of the well known CLOSEST VECTOR PROBLEM is hard to approximate to an almost polynomial factor unless NP is in quasi polynomial time. The approximability of CVPP is closely related to the security of lattice based cryptosystems.

Pricing Loss Leaders: We show hardness of approximation results for the problem of maximizing profit from buyers with single minded valuations where each buyer is interested in bundles of at most k items, and the items are allowed to have negative prices ("Loss Leaders"). For k = 2, we show that assuming the UNIQUE GAMES CONJECTURE, it is hard to approximate the profit to any constant factor. For k ≥ 2, we show the same result assuming P ≠ NP.

Integrality gaps: We show Semi-Definite Programming (SDP) integrality gaps for UNIQUE GAMES and 2-to-1 GAMES. Inapproximability results for these problems imply inapproximability results for many fundamental optimization problems. For the first problem, we show "approximate" integrality gaps for super constant rounds of the powerful Lasserre hierarchy. For the second problem we show integrality gaps for the basic SDP relaxation with perfect completeness.