High levels of security often imply that the computation time should be independent of the value of involved secrets. When the expected answer of the solver is either a solution or "unsatisfiable", then the previous assumption leads to algorithms that take always the computation time of the worst case. This is particularly disturbing for NP-hard combinatorial problems. Here we start from the observation that sometimes (specially for hard problems) users find it acceptable to receive as answer either a solution, the answer "unsatisfiable" or a failure with meaning "don't know". More exactly users accept "incomplete" solvers. For certain problems privacy reasons lead users to prefer having an answer meaning "don't know" even when the secure multi-party computation could have proven "unsatisfiable" (to avoid revealing that all alternatives are infeasible). While the solution proposed previously is slower than complete algorithms, here we show secure stochastic solutions that are faster than complete solvers, allowing to address larger problem instances.
Silaghi, M.C., Friedrich, G. (2005). Secure stochastic multi-party computation for combinatorial problems (CS-2005-14). Melbourne, FL. Florida Institute of Technology.