The Computational Benefit of Correlated Instances

The starting point of this paper is that instances of computational problems often do not exist in isolation. Rather, multiple and correlated instances of the same problem arise naturally in the real world. The challenge is how to gain computationally from instance correlations when they exist. We will be interested in settings where significant computational gain can be made in solving a single primary instance by having access to additional auxiliary instances which are correlated to the primary instance via the solution space. We focus on Constraint Satisfaction Problems (CSPs), a very expressive class of computational problems that is well-studied both in terms of approximation algorithms and NP-hardness and in terms of average case hardness and usage for cryptography, e.g. Feige's random 3-SAT hypothesis, Goldreich's one way function proposal, learning-parity-with-noise, and others. To model correlations between instances, we consider generating processes over search problems, where a primary instance I is first selected according to some distribution D (e.g. worst case, uniform, etc); then auxiliary instances I_1,...,I_T are generated so that their underlying solutions S_1,...,S_T each are a \"perturbation\" of a primary solution S for I. For example, St may be obtained by the probabilistic process of flipping each bit of S with a small constant probability. We consider a variety of naturally occurring worst case and average case CSPs, and show how availability of a small number of auxiliary instances generated through a natural generating process, radically changes the complexity of solving the primary instance, from intractable to expected polynomial time. Indeed, at a high-level, knowing a {logarithmic} number of auxiliary instances enables a close polynomial time approximation of the primary solution, and when in addition the \"difference vector\" between the primary and the auxiliary solution is known, the primary solution can be exactly found. Furthermore, knowing even a single auxiliary instance already enables finding the exact primary solution for a large class of CSPs.