We initiate a study of when the value of mathematical relaxations such as
linear and semidefinite programs for constraint satisfaction problems (CSPs) is
approximately preserved when restricting the instance to a sub-instance induced
by a small random subsample of the variables. Let C be a family of CSPs such
as 3SAT, Max-Cut, etc., and let Î be a relaxation for C, in the sense
that for every instance P∈C, Π(P) is an upper bound the maximum
fraction of satisfiable constraints of P. Loosely speaking, we say that
subsampling holds for C and Πif for every sufficiently dense instance P∈C and every ϵ>0, if we let P′ be the instance obtained by
restricting P to a sufficiently large constant number of variables, then
Π(P′)∈(1±ϵ)Π(P). We say that weak subsampling holds if the
above guarantee is replaced with Π(P′)=1−Θ(γ) whenever
Π(P)=1−γ. We show: 1. Subsampling holds for the BasicLP and BasicSDP
programs. BasicSDP is a variant of the relaxation considered by Raghavendra
(2008), who showed it gives an optimal approximation factor for every CSP under
the unique games conjecture. BasicLP is the linear programming analog of
BasicSDP. 2. For tighter versions of BasicSDP obtained by adding additional
constraints from the Lasserre hierarchy, weak subsampling holds for CSPs of
unique games type. 3. There are non-unique CSPs for which even weak subsampling
fails for the above tighter semidefinite programs. Also there are unique CSPs
for which subsampling fails for the Sherali-Adams linear programming hierarchy.
As a corollary of our weak subsampling for strong semidefinite programs, we
obtain a polynomial-time algorithm to certify that random geometric graphs (of
the type considered by Feige and Schechtman, 2002) of max-cut value 1−γ
have a cut value at most 1−γ/10.Comment: Includes several more general results that subsume the previous
version of the paper