  We investigate the use of message-passing algorithms for the problem of
finding the max-weight independent set (MWIS) in a graph. First, we study the
performance of the classical loopy max-product belief propagation. We show that
each fixed point estimate of max-product can be mapped in a natural way to an
extreme point of the LP polytope associated with the MWIS problem. However,
this extreme point may not be the one that maximizes the value of node weights;
the particular extreme point at final convergence depends on the initialization
of max-product. We then show that if max-product is started from the natural
initialization of uninformative messages, it always solves the correct LP -- if
it converges. This result is obtained via a direct analysis of the iterative
algorithm, and cannot be obtained by looking only at fixed points.
  The tightness of the LP relaxation is thus necessary for max-product
optimality, but it is not sufficient. Motivated by this observation, we show
that a simple modification of max-product becomes gradient descent on (a
convexified version of) the dual of the LP, and converges to the dual optimum.
We also develop a message-passing algorithm that recovers the primal MWIS
solution from the output of the descent algorithm. We show that the MWIS
estimate obtained using these two algorithms in conjunction is correct when the
graph is bipartite and the MWIS is unique.
  Finally, we show that any problem of MAP estimation for probability
distributions over finite domains can be reduced to an MWIS problem. We believe
this reduction will yield new insights and algorithms for MAP estimation.
