Monday, April 21, 2008

Hardness of Independent Set

I wanted to say a few more things about the hardness of Max-Independent-Set (equivalently, Min-Vertex-Cover). Recall that in Lecture 25, we sketched the proof of factor $1/n^{1-\epsilon}$ NP-hardness for this problem; specifically, Hastad's result shows that distinguishing $1/n^{\epsilon}$ vs. $1/n^{1-\epsilon}$ is NP-hard for all $\epsilon > 0$.

This is a great hardness factor, but (as they used to say) "the gap is in the wrong place" -- or at least, it's not in the strongest possible place. It is of great interest to try to obtain a hardness result of the form $\Omega(1)$ vs. $o(1)$. This will give a worse "factor", but the result is essentially stronger. Note that we can't hope for $\Omega(1)$ vs. $1/n^{1-\epsilon}$ NP-hardness; Boppana and Halldorsson showed, roughly, that $1/k$ vs. $1/n^{(k-2)/(k-1)}$ is in P (using Ramsey theory) and this was improved by Alon and Kahale to, roughly, $1/k$ vs. $1/n^{(k-2)/(k+1)}$, using the Karger-Motwani-Sudan SDP coloring algorithm.

One reason hardness results like $\Omega(1)$ vs. "small" are better is that they better "factor" hardness for Min-Vertex-Cover. Recall that, trivially, a $c$ vs. $s$ hardness result for Min-Independent-Set is a $1-c$ vs. $1 - s$ hardness result for Max-Vertex-Cover, and thus a $(1-s)/(1-c)$-factor hardness for Max-Vertex-Cover. For example, Hastad's hardness gives only $1+o(1)$ factor hardness for Vertex-Cover, but a result like $1/3$ vs. $o(1)$ hardness for Independent-Set would give factor $3/2 - o(1)$ hardness for Vertex-Cover.

For a long time, the best NP-hardness result like this we knew for Independent-Set came simply from taking Hastad's $1 - \epsilon$ vs. $1/2 + \epsilon$ hardness result for Max-E3Lin and applying the FGLSS reduction to it. You should (easily) check for yourself that this gives $1/4 - \epsilon$ vs. $1/8 + \epsilon$ NP-hardness for Independent-Set. Hence it gives $3/4 + \epsilon$ vs. $7/8 - \epsilon$ NP-hardness for Vertex-Cover; i.e., factor $7/6 + \epsilon$.

In 2002, in the same conference as Khot's Unique Games Conjecture paper, Dinur and Safra published a stronger NP-hardness result for Independent-Set. This was a pretty amazing paper, because the outline of the proof was:

a) Prove a super-weak variant of the Unique Games Conjecture;
b) Do a very difficult Dictator Test based on the combinatorics of $2$-intersecting families.

Part (a), as mentioned, was done contemporaneously with the UGC paper, while part (b) was done before the work on Ek-Independent-Set described in our Lectures 8 and 9.

I like to remember their result as $1/3$ vs. $1/9$ NP-hardness for Independent-Set (omitting the $\epsilon$'s), and hence factor $(8/9)/(2/3) = 4/3$ hardness for Vertex-Cover. In fact, they pushed it a little further. Specifically, for $p = (3 - \sqrt{5})/2$, they showed $p$ vs. $4p^3 - 3p^4$ NP-hardness for Independent-Set. This is, like, .382 vs. .159. This yields factor 1.36 or so NP-hardness for Vertex-Cover.

That stands as the best NP-hardness result known. Relatively soon after the UGC paper, Khot and Regev showed showed that assuming UGC, $1/2 - \epsilon$ vs. $\epsilon$ hardness holds for Min-Independent-Set, and thus factor $2-\epsilon$ hardness for Vertex-Cover. Indeed, they showed UGC-hardness of $1/k - \epsilon$ vs. $\epsilon$ for Min-Ek-Independent-Set.

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