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Simplex is not a polynomial time algorithm Certain rare kinds of linear programs cause it to go from one corner of the feasible region to a better corner and then to a still better one, and so on for an exponential number of steps For a long time, linear programming was considered a paradox, a problem that can be solved in practice, but not in theory! Then, in 1979, a young Soviet mathematician called Leonid Khachiyan came up with the ellipsoid algorithm, one that is very different from simplex, extremely simple in its conception (but sophisticated in its proof) and yet one that solves any linear program in polynomial time Instead of chasing the solution from one corner of the polyhedron to the next, Khachiyan s algorithm con nes it to smaller and smaller ellipsoids (skewed highdimensional balls) When this algorithm was announced, it became a kind of mathematical Sputnik, a splashy achievement that had the US establishment worried, in the height of the Cold War, about the possible scienti c superiority of the Soviet Union The ellipsoid algorithm turned out to be an important theoretical advance, but did not compete well with simplex in practice The paradox of linear programming deepened: A problem with two algorithms, one that is ef cient in theory, and one that is ef cient in practice! A few years later Narendra Karmarkar, a graduate student at UC Berkeley, came up with a completely different idea, which led to another provably polynomial algorithm for linear programming Karmarkar s algorithm is known as the interior point method, because it does just that: it dashes to the optimum corner not by hopping from corner to corner on the surface of the polyhedron like simplex does, but by cutting a clever path in the interior of the polyhedron And it does perform well in practice But perhaps the greatest advance in linear programming algorithms was not Khachiyan s theoretical breakthrough or Karmarkar s novel approach, but an unexpected consequence of the latter: the erce competition between the two approaches, simplex and interior point, resulted in the development of very fast code for linear programming.

hr@aos>CREATE OR REPLACE PACKAGE hremployee_utility IS FUNCTION get_user_department_id ( user_name IN VARCHAR2 DEFAULT SYS_CONTEXT('USERENV', 'SESSION_USER') ) RETURN NUMBER; FUNCTION get_user_department_name ( user_name IN VARCHAR2 DEFAULT SYS_CONTEXT('USERENV', 'SESSION_USER') ) RETURN VARCHAR2; FUNCTION is_department_manager ( user_name IN VARCHAR2 DEFAULT SYS_CONTEXT('USERENV', 'SESSION_USER') ) RETURN NUMBER; END; / PL/SQL procedure successfully completed

To make your script easy to run, you can turn it into a usable application and put an icon for it somewhere handy for example, on the Dock AppleScript Editor enables you to save a script in the five different formats explained in Table 3-2

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The importance of linear programming stems from the astounding variety of problems that reduce to it and thereby bear witness to its expressive power In a sense, this next one is the ultimate application We are given a Boolean circuit, that is, a dag of gates of the following types Input gates have indegree zero, with value true or false

This function includes a set of queries similar to those we issued above to assert the fact that the logged-in user is a sales department manager With this function in place, we can create an identity management related factor for the department manager condition as follows:

This is the basic format for scripts you run in AppleScript Editor or from the Scripts menu This creates an executable application that you can run on any Mac You can include a startup screen showing the script s description, make the script readonly, and choose to leave it open after it finishes running

diego_dbvmgr@aos>BEGIN dbms_macadmcreate_factor( factor_name => 'Is_Department_Manager' , factor_type_name => 'User', description =>

The CIRCUIT VALUE problem is the following: when the laws of Boolean logic are applied to the gates in topological order, does the output evaluate to true There is a simple, automatic way of translating this problem into a linear program Create a variable xg for each gate g, with constraints 0 x g 1 Add additional constraints for each type of gate:

You re creating a script, or you have a finished script that you want to run from the Scripts menu You ve created a script that doesn t use any external components (such as documents or graphics) and are ready to distribute it

Part II:

These constraints force all the gates to take on exactly the right values 0 for false, and 1 for true We don t need to maximize or minimize anything, and we can read the answer off from the variable xo corresponding to the output gate This is a straightforward reduction to linear programming, from a problem that may not seem very interesting at rst However, the CIRCUIT VALUE problem is in a sense the most general problem solvable in polynomial time! After all, any algorithm will eventually run on a computer, and the computer is ultimately a Boolean combinational circuit implemented on a chip If the algorithm runs in polynomial time, it can be rendered as a Boolean circuit consisting of polynomially many copies of the computer s circuit, one per unit of time, with the values of the gates in one layer used to compute the values for the next Hence, the fact that CIRCUIT VALUE reduces to linear programming means that all problems that can be solved in polynomial time do! In our next topic, NP-completeness, we shall see that many hard problems reduce, much the same way, to integer programming, linear programming s dif cult twin 218

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