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dramatically If all cities were on a circle or on a straight line, the solution would be easy But in general, an exact solution is not feasible in a sequential computer such as a desktop computer The system becomes overwhelmed with possibilities Problems that are hard can theoretically be solved quickly in a massively parallel computer using as many processors as needed Whenever a decision must be made that has several different outcomes, several new processors are started up, each chasing a possible solution Parallel processing for this purpose has never been considered realistic by practical engineers When parallel processing like this is necessary, theoreticians describe the problem in a rather nebulous way as belonging to class NP This designation is puzzling at rst sight, as is its literal meaning, nondeterministic polynomial.

TABLE 16-4

Nondeterministic is an academic way of saying that a large number of processors are necessary; polynomial means that a solution proceeds quickly only if a large number of processors are used Polynomial time refers to computing time that increases in proportion to some power l of the number of variables: for example, KPnl where KP is a constant giving an upper bound to computing time Thus, if a nondeterministic computer is available, the computational time is polynomial Unfortunately, by merely increasing the numbers of variables slightly, the number of parallel processors required becomes astronomical and impractical, and soon exceeds the number of atoms in the known universe hence the term nondeterministic Exponential time is de ned to be a computational time that increases as KE2n, where KE is a constant that gives an upper bound to computing time.

be the (finite) set of actions. The basic model of agents interacting with their environments is as follows. The environment starts in some state, and the agent begins by choosing an action to perform on that state. As a result of t h s action, the environment can respond with a number of possible states. However, only one state will actually result - though of course, the agent does not know in advance which it will be. On the basis of t h s second state, the agent again chooses an action to perform. The environment responds with one of a set of possible states, the agent then chooses another action, and so on. A run, r, of an agent in an environment is thus a sequence of interleaved environment states and actions:

The show snmp command allows for the display of the configuration of the SNMP parameters on the Nortel VPN Router. A sample output of the show snmp command options is as follows:

They really want to emphasize that NP problems cannot be solved in a reasonable way on an ordinary sequential computer Such computers are feasible only for problems in class P, those that compute in polynomial time Sequential computers are examples of Turing machines, named for Alan Turing (1912 1954), who deserves credit for his vision of sequential machines for computing in the 1930s When faced with a hard problem with an exponential explosion of work, often there are fairly good approximations to a solution However, an exact solution is sometimes desired, if only a large enough parallel processor were available Problems in class NP can be reduced mathematically to an equivalent problem that is also dif cult to solve, but whose solution can be veri ed quickly.

3DVelasco,Uribe & Garcia-Colin, Phys. Rev. E 66 (2002) 032103. "Gorban & Karlin, Physica A 190 (1992) 393; Transp. Theory Stat. Phys. 23 (1994) 559.

That is, given a proposed solution, a yes or no answer can quickly be given as to whether or not the solution is correct For example, it is easy to see if a jigsaw puzzle is put together correctly Thus, problems in class NP can be reduced to a problem in what is called NP-complete Problems in class NP whose solutions can be veri ed quickly are de ned to be in class NP-complete This is another puzzling designation It suggests problems on the boundary of the NP region in an abstract space of problems (see Figure 9-11) The boundary completes the region That is, the region NP is within the boundary; NPcomplete is on the boundary; P is everywhere outside the boundary The gure suggests that the difference between a problem in P and an NP-complete.