Guest column: inapproximability results via Long Code based PCPs

Summer has come again. And what better way is there to spend a summer than to relax on a sandy beach, on a mountain top, or at a park's picnic tables, and... think theory! Summer is a particularly good time to attack the big questions whose openness just plain annoys you. In light of Reingold's L = SL result, does L-vs.-RL tempt you? If so, take it on! (But perhaps peek first at Reingold--Trevisan--Vadhan's ECCC TR05-022.) Are you convinced that UP = NP should imply the collapse of the polynomial hierarchy? Make it so! (But if you hope to do so via showing that UP is contained in the low hierarchy, peek first at Long--Sheu's 1996 MST article.) You know that S2 ⊆ SNP∩coNP2 ⊆ ZPPNP and SNP2 ⊆ ZPPNPNP, but believe that some of those "⊆"s are "=" s or are (gasp!) provably strict containments (thus giving some insight into whether recent improvements (Cai in FOCS 2001; Cai et al. in Information and Computation 2005) in the collapses of the polynomial hierarchy from assumptions that NP is in P/poly, (NP ∩ coNP)/poly, or coNP/poly are strict improvements)? Well, prove an "=" or find evidence for a strict containment... or to really make this a summer for us all to remember, prove a strict containment! Wishing you happy theorems.This article gives an overview of recent PCP constructions based on the Long Code and the inapproximability results implied by these constructions. We cover the inapproximability results for (1) MAX-3SAT and CLIQUE (2) Results for SPARSEST CUT, VERTEX COVER and MAX-CUT implied by the Unique Games Conjecture and (3) HYPERGRAPH VERTEX COVER and coloring problems on hypergraphs. The article aims at explaining the general framework that, loosely speaking, incorporates all the above results.