  We consider polymers in which M randomly selected pairs of monomers are
restricted to be in contact. Analytical arguments and numerical simulations
show that an ideal (Gaussian) chain of N monomers remains expanded as long as
M<<N; its mean squared end to end distance growing as r^2 ~ M/N. A possible
collapse transition (to a region of order unity) is related to percolation in a
one dimensional model with long--ranged connections. A directed version of the
model is also solved exactly. Based on these results, we conjecture that the
typical size of a self-avoiding polymer is reduced by the links to R >
(N/M)^(nu). The number of links needed to collapse a polymer in three
dimensions thus scales as N^(phi), with (phi) > 0.43.
