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So in a practical setting, if we wanted to solve large maximum independent set problems with this approach, we could imagine starting with the basic set of edge constraints, running the LP, checking to see if we have an integral solution, and if not, intelligently adding new constraints.

Do you have any ideas about good ways to choose which additional constraints to add?

(As you may know, this is essentially the problem addressed in the following paper, though the paper is about general MAP inference rather than MIS:
D. Sontag, T. Meltzer, A. Globerson, Y. Weiss, T. Jaakkola. Tightening LP Relaxations for MAP using Message Passing. Uncertainty in Artificial Intelligence (UAI) 24, July 2008.)

Look at, say, graph on (4, 5) position in the first big picture. It is not the only optimal LP solution, an off-the-shelf LP algorithm can assign 1/2 to the both nodes in the left column, and to the both nodes in the right column. So, should one use some technique that prefers integral solutions? Is it the common case when there are several optimums, and integral among them?

Roman - Good question, I actually don't know why integer solution was chosen in that case. There are quite a few cases like that, with non-unique maximum, yet the solver choosing an integer solution, perhaps it's a side effect of the simplex method

Danny - that sounds like a promising approach -- here you could see which regions of the graph produce non-integer solutions, add constraints of the last kind for those vertex sets, and rerun the LP solver

Concerning Roman's question I think it is possible to add random small epsilons (either positive or negative) to the right parts of constaints in LP relaxation. We will seem to loose the property of half-integrality but in such unlikely case when there are several solutions we will definitely converge to the integer one.

I think the reason of 1/2 solution is that the variable takes value 1 in some of the maximum independent sets and 0 in others. In easy cases of Independent-Set, e.g. on Erdos-Renyi random graphs with average connectivity smaller than 2.71828..., leaf-removal algorithm gives an optimal solution, then I guess LP will give a tight solution. In hard cases, when leaf-removal core exists, no efficient algorithm exists to solve the problem, and LP gives fraction solution.

In my example with the bowtie graph, there's no maximum independent set that contains middle node, yet it gets value 1/2

The possible sets of nonintegral values in optimal LP solutions are closed under intersection (Picard&Queyranne: On the integer-valued variables in the linear vertex packing problem, 1977, http://dx.doi.org/10.1007/BF01593772). Thus, there is a unique optimal LP solution with the minimum number of fractional values, and you can find it for example by trying to fix each variable to 1 and seeing if the objective changes.

On G(n, p)-graphs, for any p, with high probability all variables 1/2 is the only optimal LP solution (Pulleyblank: Minimum node covers and 2-bicritical graphs, 1979, http://dx.doi.org/10.1007/BF01588228).

For Roman's question, the 1/2-solutions proposed are not basic solutions since they are the convex combination of other solutions (in particular, the alternative integer solutions). So, as long as your LP algorithm returns basic solutions (for instance the simplex algorithm; interior point methods have approaches that force basic solutions but that is not required), you will not get the 1/2 solutions.

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