The Unique Label Cover problem (ULC for short) is the problem that the Unique Games Conjecture is about: The task is to label each vertex of a given graph with a letter from an alphabet, given constraints on edges. Each edge of the graph comes with a bijection that tells how a label on one endpoint should be translated into a label on the second endpoint. Clearly, it's trivial to verify whether *all* constraints may be satisfied at once; the true goal is to violate as few edges as possible. The Unique Games Conjecture treats about hardness of determining the *fraction* of edges that may be violated; in this post I'm going to focus on instances where we would violate only very few constraints. In other words, I'm going to study fixed-parameter algorithms for ULC parameterized by the number of violated constraints (denoted k).

The main claim of our FOCS'12 paper is that ULC problem can be solved in O(s^{O(k2)} n^{O(1)}) time for n-vertex graph and alphabet of size s. This gives that ULC is polynomial-time solvable for constant alphabet and k=O(sqrt(log(n))), improving the obvious k=O(1) algorithm. And the main technique used is very simple: If you don't know what to do, randomly contract (equivalently, make undeletable) each edge independently with probability 0.5 - maybe something interesting will be highlighted in the graph with sufficiently high probability.

What does it mean sufficiently high probability? As we are aiming at time FPT in s and k, probability of the form 1/f(s,k) is fine for us.

So, what can nice happen with such a step? First, we can observe that this is a very easy way to find good separators to recurse in Kawarabayashi-Thorup way. Without going into technical details, if we find a small (of size f(s,k)) cut that separates two large (of size at least g(s,k)) connected subgraphs, we can recurse on one of the sides. With 2^{-f(s,k)-2g(s,k)} probability we won't contract edges of the cut, while contracting a large (of size g(s,k)) connected subgraph of each side of the cut. After such operation, if two vertices of "weight" at least g(s,k) can be separated with a cut of size f(s,k), we've found a good separation.

This step needs to augment the problem with border terminals and pay attention to a few details, but the overall message is: from now on we can assume that each small (in the ULC case if suffices to say here "of size at most f(s,k)=2k") separator leaves one side of small size (in the ULC case of size roughly g(s,k)=s^{k}).

Let X be a solution we're looking for. We know that in G-X there is one huge connected component, and at most k (as X is of size at most k) small connected components. What do we do now? Apply the trick again! With sufficiently high probability:

- each edge of X is not contracted;
- each small connected component is contracted onto one vertex;
- for each edge e in X and each its endpoint v that is contained in the large connected component of G-X, there are at least g(s,k) vertices contracted onto v.

The core observation now is as follows: assume the above, and let v and w be two *heavy* vertices: that is, on both v and w we have contracted at least g(s,k) other vertices. Since we cannot recurse, there are at least 2k+1 edge-disjoint paths between v and w; each such path gives exactly one way to translate a label of v into a label of w. With a budget of k violated constraints, we can cut only k such paths; thus, for each label of v, there is at most one reasonable label of w - the one that is yielded by a majority of paths. Thus, there are only s ways to label all heavy vertices. The rest of the algorithm is just a matter of a few small observations which we won't describe in details here.

Summing up, we found the above trick interesting, with one - not so often in the fixed-parameter tractability world - property that it is capable of handling weighted versions of the problem: it is not hard to generalize the above idea to find minimum cost set of violated constraints of size at most k. Naturally, there are a few technical devils hidden in this approach - that were sufficient to blow up the paper to 58 pages on arxiv - but still, it enabled us also to improve the k-Way Cut algorithm of Kawarabayashi and Thorup, as well as give an alternative FPT algorithm for Multiway Cut-Uncut.

We conclude with a summary of what properties are necessary to apply this approach - in case you would like to use it to you favourite problem.

- First, this approach needs to be some sort of graph deletion problem, with a possibility to bypass (or make undeletable) vertices or edges

that are not to be deleted.
- To make a recursive step, we need that there are only limited number of ways a solution can behave on a separator. In our algorithm we've used the fact that, in a recursive step, we need only to take care of the labeling of the separator (s
^{p} choices for a separator of p vertices). This is closely related to the notion of *finite integer index*.
- Finally, we need to make use of the knowledge that in the final step, the graph has very good connectivity - in the sense that a small separator can remove only a small part of the graph. In the ULC case we've observed that parts with pairwise large (of size 2k+1) flow between them have a very limited number of ways they can be labeled. This part is problem-specific, and, from our experience, is the hardest part of the algorithm.

Let's make an example of the vertex k-way cut problem, which is known to be W[1]-hard, and see what breaks in the above approach. In this problem we're given a graph and integers k and p, and we're to delete p vertices to obtain at least k connected components (we parameterize by both k and p). It is not hard to see that the recursion works perfectly well: we need to remember only how does the border vertices split into components, and how many connected components were already created. However, the last step is hard; if you look at the hardness proof, the constructed instance has already strong connectivity - you can remove only isolated vertices by small separators - but it is still capable of encoding a clique instance.

A reverse situation happens in the Bisection problem, where we're to delete k edges (k is the parameter) to obtain two halves of n/2 vertices. Here the final step seems easy, and boils down to a variant of the knapsack problem. However, in the recursive step we need to remember the balance between the parts of the graph - which gives a linear in n number of choices - and the approach breaks down. To the best of our knowledge, fixed-parameter tractability of Bisection remains open.