The.Algorithm.Design.Manual.Springer-Verlag.1998

Linear Programming
variable assignment that best fits the equations, we can replace each variable by and
solve the new system as a linear program, minimizing the sum of the error terms.
● Graph algorithms - Many of the standard graph problems described in this book, such as shortest
paths, bipartite matching, and network flow, can all be solved as special cases of linear
programming. Most of the rest, including traveling salesman, set cover, and knapsack, can be
solved using integer linear programming.
The standard algorithm for linear programming is called the simplex method. Each constraint in a linear
programming problem acts like a knife that carves away a region from the space of possible solutions.
We seek the point within the remaining region that maximizes (or minimizes) f(X). By appropriately
rotating the solution space, the optimal point can always be made to be the highest point in the region.
Since the region (simplex) formed by the intersection of a set of linear constraints is convex, we can find
the highest point by starting from any vertex of the region and walking to a higher neighboring vertex.
When there is no higher neighbor, we are at the highest point.
While the basic simplex algorithm is not too difficult to program, there is a considerable art to producing
an efficient implementation capable of solving large linear programs. For example, large programs tend
to be sparse (meaning that most inequalities use few variables), so sophisticated data structures must be
used. There are issues of numerical stability and robustness, as well as which neighbor we should walk
to next (so called pivoting rules). Finally, there exist sophisticated interior-point methods, which cut
through the interior of the simplex instead of walking along the outside, that beat simplex in many
applications.
The bottom line on linear programming is this: you are much better off using an existing LP code than
writing your own. Further, you are much better off paying money than surfing the net. Linear
programming is one algorithmic problem of such economic importance that commercial implementations
are far superior to free versions.
Issues that arise in linear programming include:
● Do any variables have integrality constraints? - It is impossible to send 6.54 airplanes from New
York to Washington each business day, even if that value maximizes profit according to your
model. Such variables often have natural integrality constraints. A linear program is called an
integer program when all its variables have integrality constraints, or a mixed integer progam if
some of them do.
Unfortunately, it is NP-complete to solve integer or mixed programs to optimality. However,
there are techniques for integer programming that work reasonably well in practice. Cutting
plane techniques solves the problem first as a linear program, and then adds extra constraints to
enforce integrality around the optimal solution point before solving it again. After a sufficient
number of iterations, the optimum point of the resulting linear program matches that of the
original integer program. As with most exponential-time algorithms, run times for integer
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