29 Np-hard Problems 29.1 'efficient' Problems

A generally-accepted minimum requirement for an algorithm to be considered ‘efficient’ is that its running time is polynomial: O(nc) for some constant c, where n is the size of the input.1 Researchers recognized early on that not all problems can be solved this quickly, but we had a hard time figuring out exactly which ones could and which ones couldn’t. there are several so-called NP-hard problems, which most people believe cannot be solved in polynomial time, even though nobody can prove a super-polynomial lower bound. Circuit satisfiability is a good example of a problem that we don’t know how to solve in polynomial time. In this problem, the input is a boolean circuit: a collection of AND, OR, and NOT gates connected by wires. We will assume that there are no loops in the circuit (so no delay lines or flip-flops). The input to the circuit is a set of m boolean (TRUE/FALSE) values x1, . . . , xm. The output is a single boolean value. Given specific input values, we can calculate the output of the circuit in polynomial (actually, linear) time using depth-first-search, since we can compute the output of a k-input gate in O(k) time.