

Subhash Khot and the Unique Games Conjecture - digital55
http://www.simonsfoundation.org/quanta/20140812-a-grand-vision-for-the-impossible/

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simplekoala
Feel humbled to read about him. Floored!

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krebby
Khot was one of my favorite professors at NYU and always lead on with way
better and more in depth stories than the material presented initially. So
great to read about his other successes.

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lotharbot
Further reading:
[http://en.wikipedia.org/wiki/Unique_games_conjecture](http://en.wikipedia.org/wiki/Unique_games_conjecture)

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scythe
I'm sort of curious about the scope of UGC. There are well-known approximation
algorithms for the Traveling Salesman Problem, which is NP-hard, so how much
harder does a problem have to be to invoke UGC? Is there a no-go result that
prevents "translating" TSP approximations to the approximation of other
problems?

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codeflo
> There are well-known approximation algorithms for the Traveling Salesman
> Problem, which is NP-hard

Actually, it's either not NP-hard, or not approximable, depending on your
definition of TSP. Let me clarify.

TSP, as most people define it, means "given a graph, find the shortest
roundtrip". As you rightly state, this problem can be well approximated. But
it's not a _decision problem_ (it doesn't have a yes/no answer), and thus
doesn't even fit into categories like "NP" or "NP-complete". Talking about the
NP-completess of this "Optimization TSP" is essentially a type error.

It's only when we define a "Decision TSP" problem that we can talk about
things like NP-hardness. For example, we might ask "Given a graph and a number
c, is there a roundtrip of length less than c?". And this new problem _is_ NP-
hard. Given any problem in NP, like a SAT instance for example, we can
efficiently construct a (usually very contrived) graph that will have a
roundtrip of a certain length if and only the original SAT instance is
satisfiable.

But unlike "Optimization TSP", this "Decision TSP" can't be approximated. It
doesn't even make sense to talk about approximations: a roundtrip of length <
c can't "almost exist", the answer is always either yes or no. And while an
exact solution for Optimization TSP can trivially answer Decision TSP, any
_approximation_ of Optimization TSP is essentially useless in solving Decision
TSP.

That's why TSP (in its optimization form) can be easy to approximate, while
also (in its decision form) being NP-hard.

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beagle3
Great description. Obviously codeflo knows this, but it's worth pointing out
that "roundtrip" means "visiting every city _exactly_ once". Euclidean problem
geometry guarantees it (it's always faster to go directly) but many problems,
even though emerging from seemingly euclidean space, do not have this property
- e.g. if you're using a car, the road network might induce non-euclidean
distance geometry that would make the fast roundtrip include more than one
visit to a node.

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sssilver
Comic Sans killed it

