If I were a UGC skeptic…

Inapproximability in an alternate universe

1992: PCP Theorem proven; Max-3Sat is hard, 1 vs. .9999

1994: 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.

2003 – 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 − 1.25000001ε 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.