Generality of Languages

Transcription

1 Increasing generality Decreasing efficiency Generality of Languages Linear Programs Reduction Algorithm Simplex Algorithm, Interior Point Algorithms Min Cost Flow Algorithm Klein s algorithm Reduction Max (s,t)-flow Algorithm Edmonds-Karp Reduction Maximum Cardinality Matching Algorithm Hopcroft-Karp 1

2 Min Cost Flow Problem Given flow network with costs G =(V,E,c,b,k), find flow f minimizing cost(f). 2

3 Min Cost Flow Problem Given G=(V,E,c,b,k), find f: V V! R minimizing (u,v) 2 V V k(u,v) f(u,v) so that 8 (u,v): f(u,v) c(u,v). 8 (u,v): f(u,v) = f(v,u). 8 u 2 V: v 2 V f(u,v) = b(u) 3

4 Min Cost Flow Problem Find (x 1, x 2,, x 64 ) 2 R 64 minimizing k 1 x 1 + k 2 x k 64 x 64 so that x 1 3, x 2-1,.. x 1 = - x 19, x 2 = - x 27,. x 9 + x 17 + x 34 = 17, x 12 + x 11 + x 19 = 0,. 4

5 Linear Programs Find x 2 R n minimizing or maximizing a linear form hx,ci = i c i x i (the objective function) so that a given set of linear equations and inequalities are satisfied. A feasible solution to the program is a point x satisfying the equations and inequalities. 5

6 Linear Programs Linear Programs generalize the min cost flow problem. Linear Programs generalize systems of linear equations. To solve general linear programs we seem to need a common generalization of Klein s algorithm and Gaussian elimination! 6

7 Diet Problem Serving Size Energy (kcal) Protein (g) Calcium (mg) Oatmeal 28 g Chicken 100 g Eggs 2 large Whole Milk 237 cc Cherry Pie 170 g Pork with Beans 260 g Price per serving (cents) Necessary daily intake: Energy 2000 kcal, Protein 55 g, Calcium 800 mg Compose a diet minimizing price and fulfilling necessary daily intake. 7

8 Diet Problem Serving Size Energy (kcal) Protein (g) Calcium (mg) Oatmeal x 1 28 g Chicken x g Eggs x 3 2 large Whole Milk x cc Cherry Pie x g Pork with Beans x g Price per serving (cents) Necessary daily intake: Energy 2000 kcal, Protein 55 g, Calcium 800 mg Compose a diet minimizing price and fulfilling necessary daily intake. 8

9 Linear Program for Diet Problem Find (x 1, x 2, x 3, x 4, x 5, x 6 ) 2 (R + ) n minimizing 3x x x 3 + 9x x x 6 so that 110x x x x x x x x x x x x x x x x x x

10 Linear Programs, Geometric View x y = 0 10

11 x y 0 11

12 x + 2y 1 Æ x + 3 y -1 Æ x y 0 Æ 2 x - y 3 x + 2y = 1 x + 3y = -1 x y = 0 2x y = 3 12

13 x + 2y 1 Æ x + 3 y -1 Æ x y 0 Æ 2 x - y 3 13

14 The set of feasible solutions F is a convex Polyhedron. F 14

15 Minimize x-2y over F x-2y = 0 F x-2y = 1 15

16 Minimize x-2y over F x-2y = -2/3 F 16

17 Linear Programs, Geometric view The set of feasible solutions of a linear program is a polyhedron in R n. The minimal (maximal) value of the linear objective function is attained in a corner of the polyhedron. Simplex algorithm: Do a local search, walking from corner to corner of the polyhedron improving the objective function until no improvement is possible. Exceptions: The set of feasible solutions may be unbounded or empty (the linear program is then said to be infeasible). 17

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19 Linear Programs in Standard Form The objective must be to maximize a linear function. All variables are restricted to nonnegative values. All constraints have the form d 1 x 1 + d 2 x d n x n e 19

20 Linear Programs in Standard Form Find (x 1, x 2, x 3,, x n ) 2 (R + ) n maximizing c 1 x 1 + c 2 x c n x n so that a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1 + a 22 x a 2n x n b 2 a m1 x 1 + a m2 x a mn x n b m Exceptions: If no fesible solution exist, report Infeasible. If arbitrarily good feasible solutions exist, report Unbounded. 20

21 Linear Program in Standard Form Matrix Notation Given A 2 R m n, b 2 R m, c 2 R n, find x 2 (R + ) n maximizing hc,xi so that Ax b. 21

22 Reduction from LP in general form to LP in standard form Suppose we want to minimize (not maximize) c 1 x 1 + c 2 x 2 + c n x n Maximize - c 1 x 1 - c 2 x c n x n 22

23 Reduction from LP in general form to LP in standard form Suppose we have a constraint d 1 x 1 + d 2 x d n x n e Replace with - d 1 x 1 - d 2 x d n x n - e 23

24 Reduction from LP in general form to LP in standard form Suppose we have a constraint d 1 x 1 + d 2 x d n x n = e Replace with d 1 x 1 + d 2 x d n x n e - d 1 x 1 - d 2 x d n x n - e 24

25 Reduction from LP in general form to LP in standard form Suppose y is a variable for which we do not want the constraint y 0. Suppose we do want a constraint such as y -17 Introduce new variable y = y+17 and replace y with y -17 everywhere in the program. 25

26 Reduction from LP in general form to LP in standard form Suppose y is a variable for which we do not want the constraint y 0 and no other constraint y c. Introduce two new variables y +, y - with constraints y + 0, y - 0. Replace y with y + - y - everywhere in the program. Do this for every unconstrained variable in the program. 26

27 Reduction from LP in general form to LP in standard form The old program P and the new program P are related as follows: A feasible solution to P can be converted to a feasible solution to P with same value by setting y + = max(y,0), y - = min(y,0). A feasible solution to P can be converted to a feasible solution to P with same value by setting y = y + - y -. 27

28 Reduction from LP in general form to LP in standard form Conclusion: Linear Programs in general form can be converted to linear programs in standard form by doubling the number of constraints and doubling number of variables. From a practical point of view, the factor 2 2= 4 blowup in size is undesirable. Exercise: One extra constraint and one extra variable suffices. 28