NP-hardness of 2-to-2 Games

In fact, stronger conditions hold: (a) the YES cases: there is a set X of 1 − δ fraction of the vertices such that all constraints inside X are satisfied, and (b) the NO case: any set containing δ fraction of the vertices contains Ω(δ2) fraction of the edges of the graph. These conditions are necessary toward certain applications, to the vertex cover and the independent set problems. For further implications, see Chapter 1. The discussion herein focuses on the presentation of techniques and ideas that go into the reduction. Following the proof of the PCP theorem, a general framework for proving hard­ ness of approximation results has been developed [Arora and Safra 1998; Arora et al. 1998; Bellare et al. 1998; Raz 1998, Håstad 2001]. Using this framework, a PCP con­ struction for a problem P is a composition of two separate modules: “Inner PCP” and “Outer PCP.”