If I were a UGC skeptic…

Inapproximability in an alternate universe 1992: PCP Theorem proven; Max-3Sat is hard, 1 vs : Ran Raz takes up painting, Feige and Kilian never meet, no one proves the Parallel Repetition Theorem. (OR) 1997: Johan Håstad takes up fishing, no one writes Some Optimal Inapproximability Results. 2001: People bemoan lack of sharp inapproximability results.

Inapproximability in an alternate universe 2002: Feige publishes [Fei02], shows that “Hypothesis 1” implies sharp inapproximability ratios for Max-3Sat, Max-3And, Max-3Lin; some hardness for Min-Bisection, Densest Subgraph, etc. 2003: Misha Alekhnovich publishes [Ale03], shows that “Conjecture 1” implies Feige’s “Hypothesis 1”. Focuses attention on the following problem: Given a random 3Lin instance with O(n) equations and a planted 1 − ε solution, find a 1/2 + ε solution. In particular, Misha conjectures that w.h.p. over the instance, not doable in poly time – Alekhnovich Conjecture fever spreads across complexity theory…

Comparison with UGC We currently have a similar situation: Contentious conjecture  many strong inapproximability results. But the situation in the alternate universe is far more compelling. Why? Because we can generate hard-seeming instances.

UGC on average As far as I know, no one knows a way to (randomly) generate UGC instances that “seem harder” than known NP-hardness bounds ([Feige-Reichman]). As far as I know, no one knows a way to (randomly) generate 2Lin instances that “seem harder” than known NP-hardness bounds. Puts UGC True Believers in a bit of a difficult spot.

Challenge Problem Come up with a distribution on 2Lin instances with 1 − ε solutions such that: Neither you nor, say, Amin Coja-Oghlan can give a polynomial-time algorithm finding 1 − ε solutions. (If you believe UGC, even finding 1 − c ε 1/2 solutions should be hard.)

Or maybe UGC is easyish on feasibly generated inputs.