I decided to look at the award-winning Arora-Boaz-Steurer paper giving a subexponential algorithm for unique games. I first wanted to take a look at the algorithm itself, temporarily ignoring the analysis and focusing on the algorithmic techniques only.
According to the plan in the intro, the algorithm is in section 5. There, first, without loss of generality they assume that the graph is regular -- for that, the reader can simply check Appendix A. I go there and review the construction. Going back to section 5, they then make the graph lazy by adding self-loops - I skim the construction.
We are then sent to section 4 and Theorem 4.5. Going there, we are redirected to Theorem 4.1 "using Theorem 3.2 instead of 2.3". Making a mental note of that and going to Theorem 4.1, I see that the proof starts with a Lemma, 4.2, whose proof is a mere call to Theorem 2.3 for some particular value of a parameter. Thus it is time to go to Theorem 3.2. Going there, I am redirected to Theorem 2.3. Going there, I am redirected to Theorem 3.1. Going to its proof, I see that the lemmas are just doing an analysis, without algorithmic contents, until I get to a comment "... moreover, following the proof we see that if the condition is violated we can get a set S ...". Aaargh!
Giving up on that for now and going back to the thread of the proof of Theorem 4.1, I skim the next couple of lemmas, that seem to do some combination of greedy partitioning and of iteratively re-partitioning parts into smaller pieces a few times. Going back to section 5, the next step, in my impression, assigns labels independently for each part of the partition. More precisely, for each label-extended part of the partition, it redirects us to the algorithm of Theorem 2.2. Miraculously, the proof of Theorem 2.2 is basically self-contained: take the subspace spanned by eigenvectors of the large eigenvalues, put an epsilon-net in its unit ball, and, for each vector v in that epsilon-net, use its "large enough" coordinates to define a subset S(v) of vertices of the graph. Going back to section 5, apparently for each set S(v) there will be a mysterious assignment, and the algorithm exhaustively enumerates the v's and picks the assignment satisfying the most constraints.
What have I learned? Basically nothing so far. Frustrating! If the result was not so good, I would give up on that paper. Instead, I may consider setting aside a few days solely for deciphering it...