Algorithms for circuits and circuits for algorithms: Connecting the tractable and intractable

The title of this paper highlights an emerging duality between two basic topics in algorithms and complexity theory. Algorithms for circuits refers to the design of algorithms which can analyze finite logical circuits or Boolean functions as input, checking a simple property about the complexity of the underlying function. For instance, an algorithm determining if a given logical circuit C has an input that makes C output true would solve the NP-complete Circuit-SAT problem. Such an algorithm is unlikely to run in polynomial time, but could possibly be more efficient than exhaustively trying all possible inputs to the circuit. Circuits for algorithms refers to the modeling of "complex" uniform algorithms with "simple" Boolean circuit families, or proving that such modeling is impossible. For example, can every exponential-time algorithm be simulated using Boolean circuit families of only polynomial size? It is widely conjectured that the answer is no, but the present mathematical tools available are still too crude to resolve this kind of separation problem. This paper surveys these two generic subjects and the connections that have been developed between them, focusing on connections between non-trivial circuit-analysis algorithms and proofs of circuit complexity lower bounds.