What are some good ways of estimating 'approximate' semantic similarity between sentences?

I have been looking at the nlp tag on SO for the past couple of hours and am confident I did not miss anything but if I did, please do point me to the question.

In the mean time though, I will describe what I am trying to do. A common notion that I observed on many posts is that semantic similarity is difficult. For instance, from this post, the accepted solution suggests the following:

First of all, neither from the perspective of computational 
linguistics nor of theoretical linguistics is it clear what 
the term 'semantic similarity' means exactly. .... 
Consider these examples:

Pete and Rob have found a dog near the station.
Pete and Rob have never found a dog near the station.
Pete and Rob both like programming a lot.
Patricia fo开发者_开发百科und a dog near the station.
It was a dog who found Pete and Rob under the snow.

Which of the sentences 2-4 are similar to 1? 2 is the exact 
opposite of 1, still it is about Pete and Rob (not) finding a 
dog.

My high-level requirement is to utilize k-means clustering and categorize the text based on semantic similarity so all I need to know is whether they are an approximate match. For instance, in the above example, I am OK with classifying 1,2,4,5 into one category and 3 into another (of course, 3 will be backed up with some more similar sentences). Something like, find related articles, but they don't have to be 100% related.

I am thinking I need to ultimately construct vector representations of each sentence, sort of like its fingerprint but exactly what this vector should contain is still an open question for me. Is it n-grams, or something from the wordnet or just the individual stemmed words or something else altogether?

This thread did a fantastic job of enumerating all related techniques but unfortunately stopped just when the post got to what I wanted. Any suggestions on what is the latest state-of-the-art in this area?

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Latent Semantic Modeling could be useful. It's basically just yet another application of the Singular Value Decomposition. The SVDLIBC is a pretty nice C implementation of this approach, which is an oldie but a goodie, and there are even python binding in the form of sparsesvd.

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I suggest you try a topic modelling framework such as Latent Dirichlet Allocation (LDA). The idea there is that documents (in your case sentences, which might prove to be a problem) are generated from a set of latent (hidden) topics; LDA retrieves those topics, representing them by word clusters.

An implementation of LDA in Python is available as part of the free Gensim package. You could try to apply it to your sentences, then run k-means on its output.