The notion of recoverable value was advocated in work of Feige, Immorlica, Mirrokni and Nazerzadeh [Approx 2009] as a measure of quality for approximation algorithms. There this concept was applied to facility location problems. In the current work we apply a similar framework to the maximum independent set problem (MIS). We say that an approximation algorithm has recoverable value @math , if for every graph it recovers an independent set of size at least @math , where @math is the degree of vertex @math , and @math ranges over all independent sets in @math . Hence, in a sense, from every vertex @math in the maximum independent set the algorithm recovers a value of at least @math towards the solution. This quality measure is most effective in graphs in which the maximum independent set is composed of low degree vertices. It easily follows from known results that some simple algorithms for MIS ensure @math . We design a new randomized algorithm for MIS that ensures an expected recoverable value of at least @math . In addition, we show that approximating MIS in graphs with a given @math -coloring within a ratio larger than @math is unique games hard. This rules out a natural approach for obtaining @math .
We consider a binary integer programming formulation (VP) for the weighted vertex packing problem in a simple graph. A sufficient “local” optimality condition for (VP) is given and this result is used to derive relations between (VP) and the linear program (VLP) obtained by deleting the integrality restrictions in (VP). Our most striking result is that those variables which assume binary values in an optimum (VLP) solution retain the same values in an optimum (VP) solution. This result is of interest because variables are (0, 1 2, 1). valued in basic feasible solutions to (VLP) and (VLP) can be solved by a “good” algorithm. This relationship and other optimality conditions are incorporated into an implicit enumeration algorithm for solving (VP). Some computational experience is reported.