Reductions

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Sometimes a computational task is suf ciently general that any subroutine for it can also be used to solve a variety of other tasks, which at rst glance might seem unrelated For instance, we saw in 6 how an algorithm for nding the longest path in a dag can, surprisingly, also be used for nding longest increasing subsequences We describe this phenomenon by saying that the longest increasing subsequence problem reduces to the longest path problem in a dag In turn, the longest path in a dag reduces to the shortest path in a dag; here s how a subroutine for the latter can be used to solve the former: function LONGEST PATH(G) negate all edge weights of G return SHORTEST PATH(G) Let s step back and take a slightly more formal view of reductions If any subroutine for task Q can also be used to solve P , we say P reduces to Q Often, P is solvable by a single call to Q s subroutine, which means any instance x of P can be transformed into an instance y of Q such that P (x) can be deduced from Q(y):

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Algorithm for P

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Preprocess

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Algorithm Q(y) for Q

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Postprocess

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P (x)

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(Do you see that the reduction from P = LONGEST PATH to Q = SHORTEST PATH follows this schema ) If the pre- and postprocessing procedures are ef ciently computable then this creates an ef cient algorithm for P out of any ef cient algorithm for Q! Reductions enhance the power of algorithms: Once we have an algorithm for problem Q (which could be shortest path, for example) we can use it to solve other problems In fact, most of the computational tasks we study in this book are considered core computer science problems precisely because they arise in so many different applications, which is another way of saying that many problems reduce to them This is especially true of linear programming

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Variants of linear programming

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As evidenced in our examples, a general linear program has many degrees of freedom 1 It can be either a maximization or a minimization problem 2 Its constraints can be equations and/or inequalities 3 The variables are often restricted to be nonnegative, but they can also be unrestricted in sign 193

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We will now show that these various LP options can all be reduced to one another via simple transformations Here s how 1 To turn a maximization problem into a minimization (or vice versa), just multiply the coef cients of the objective function by 1 2a To turn an inequality constraint like variable s and use

n n i=1 ai xi

b into an equation, introduce a new

ai xi + s = b

s 0 This s is called the slack variable for the inequality As justi cation, observe that a vector (x1 , , xn ) satis es the original inequality constraint if and only if there is some s 0 for which it satis es the new equality constraint 2b To change an equality constraint into inequalities is easy: rewrite ax = b as the equivalent pair of constraints ax b and ax b 3 Finally, to deal with a variable x that is unrestricted in sign, do the following: Introduce two nonnegative variables, x + , x 0

Replace x, wherever it occurs in the constraints or the objective function, by x + x This way, x can take on any real value by appropriately adjusting the new variables More precisely, any feasible solution to the original LP involving x can be mapped to a feasible solution of the new LP involving x + , x , and vice versa By applying these transformations we can reduce any LP (maximization or minimization, with both inequalities and equations, and with both nonnegative and unrestricted variables) into an LP of a much more constrained kind that we call the standard form, in which the variables are all nonnegative, the constraints are all equations, and the objective function is to be minimized For example, our rst linear program gets rewritten thus: max x1 + 6x2 x1 200 x1 + x2 400 x1 , x 2 0 x2 300 min x1 6x2