Computational Complexity for Physicists

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.Comment: references updated, reprint available from http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm

Physical portrayal of computational complexity

Computational complexity is examined using the principle of increasing entropy. To consider computation as a physical process from an initial instance to the final acceptance is motivated because many natural processes have been recognized to complete in non-polynomial time (NP). The irreversible process with three or more degrees of freedom is found intractable because, in terms of physics, flows of energy are inseparable from their driving forces. In computational terms, when solving problems in the class NP, decisions will affect subsequently available sets of decisions. The state space of a non-deterministic finite automaton is evolving due to the computation itself hence it cannot be efficiently contracted using a deterministic finite automaton that will arrive at a solution in super-polynomial time. The solution of the NP problem itself is verifiable in polynomial time (P) because the corresponding state is stationary. Likewise the class P set of states does not depend on computational history hence it can be efficiently contracted to the accepting state by a deterministic sequence of dissipative transformations. Thus it is concluded that the class P set of states is inherently smaller than the set of class NP. Since the computational time to contract a given set is proportional to dissipation, the computational complexity class P is a subset of NP.Comment: 16, pages, 7 figure

EASe : integrating search with learned episodes

Weak methods are insufficient to solve complex problems. Constrained weak methods, like hill-climbing, search too little of the problem space. Unconstrained weak methods, like breadth-first search, are intractable. Fortunately, through the integration of multiple weak methods more powerful problem solvers can be created. We demonstrate that augmenting a weak constrained search method with episodes provides a tractable method for solving a large class of problems. We demonstrate that these episodes can be generated using an unconstrained weak method while solving simple problems from a domain. We provide an analytical model of our approach and empirical results from the logic synthesis domain of VLSI design as well as the classic tile-sliding domain