It is known that, in general, Constraint Optimization Problems (COP) are NP-hard. Existing arithmetic circuits for secure protocols solving such problems are exponential in the number of variables, $n$. Recently a combinatorial optimization algorithm was proposed whose cost is exponential only in a parameter of the Depth First Search tree (DFS) of the constraint graph, smaller than $n$. We show how to construct an arithmetic circuit with this property and solving any COP. For forest constraint graphs, this leads to a linear cost secure solver.
Silaghi, M.C., Petcu, A., Faltings, B. (2005). Secure combinatorial optimization using DFS-based variable elimination (CS-2005-15). Melbourne, FL. Florida Institute of Technology.