Random solutions of random problems...are not just random
Dimitris Achlioptas, Amin Coja‐Oghlan · arXiv (Cornell University) · 2008
Abstract. Let In,m denote a uniformly random instance of some constraint satisfaction problem CSP with n variables and m constraints. Assume that the density r = m/n is small enough so that with high probability In,m has a solution, and consider the experiment of first choosing an instance I = In,m at random, and then sampling a random solution σ of I (if one exists). For many CSPs (e.g., k-SAT, k-NAE, or k-coloring), this experiment appears difficult both to implement and to analyze; in fact, for a large range of r, no efficient algorithm is known to even compute a single solution of I. In the present paper we show that for many CSPs the above experiment is essentially equivalent to first choosing a random assignment σ to the n variables, and then drawing a random instance satisfied by σ uniformly. In general, this second experiment is very easy to implement and amenable to a rigorous analysis. In fact, using this equivalence, we can analyze the solution space of random CSPs. Thus, we can achieve the longstanding goal of establishing rigorously a picture put forward by statistical physicists on the basis of sophisticated but non-rigorous techniques such as the cavity and the replica method. This picture is suggestive as to why random CSP instances seem difficult to deal with algorithmically. Furthermore, we show that the second experiment gives rise to one-way functions, if one assumes that random instances of CSP are hard for some range of densities.