  A widely used method for solving SOS (Sum Of Squares) decomposition problem
is to reduce it to the problem of semi-definite programs (SDPs) which can be
efficiently solved in theory. In practice, although many SDP solvers can work
out some problems of big scale, the efficiency and reliability of such method
decrease greatly while the input size increases. Recently, by exploiting the
sparsity of the input SOS decomposition problem, some preprocessing algorithms
were proposed [5,17], which first divide the input problem satisfying special
definitions or properties into smaller SDP problems and then pass the smaller
ones to SDP solvers to obtain reliable results efficiently. A natural question
is that to what extent the above mentioned preprocessing algorithms work. That
is, how many polynomials satisfying those definitions or properties are there
in the SOS polynomials? In this paper, we define a concept of block SOS
decomposable polynomials which is a generalization of those special classes in
[5] and [17]. Roughly speaking, it is a class of polynomials whose SOS
decomposition problem can be transformed into smaller ones (in other words, the
corresponding SDP matrices can be block-diagnolized) by considering their
supports only (coefficients are not considered). Then we prove that the set of
block SOS decomposable polynomials has measure zero in the set of SOS
polynomials. That means if we only consider supports (not with coefficients) of
polynomials, such algorithms decreasing the size of SDPs for those SDP-based
SOS solvers can only work on very few polynomials. As a result, this shows that
the SOS decomposition problems that can be optimized by the above mentioned
preprocessing algorithms are very few.
