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Generality of Languages

Generality of Languages. Linear Programs. Algorithm. Simplex Algorithm , Interior Point Algorithms. Reduction. Klein’s algorithm. Min Cost Flow. Algorithm. Reduction. Decreasing efficiency. Algorithm. Max (s,t)-Flow. Edmonds-Karp. Increasing generality. Reduction. Maximum

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Generality of Languages

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  1. Generality of Languages Linear Programs Algorithm Simplex Algorithm, Interior Point Algorithms Reduction Klein’s algorithm Min Cost Flow Algorithm Reduction Decreasing efficiency Algorithm Max (s,t)-Flow Edmonds-Karp Increasing generality Reduction Maximum Cardinality Matching Algorithm Hopcroft-Karp

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

  3. Min Cost Flow Problem • Given G=(V,E,c,b,k), find f: V£V!R minimizing (u,v) 2V£Vk(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). 8u2V: v2Vf(u,v) = b(u)

  4. Min Cost Flow Problem Find (x1, x2,…, x64) 2R64 minimizing k1x1 + k2x2 +  + k64x64 so that x1· 3, x2· -1, ….. x1 = - x19, x2 = - x27, …. x9 + x17 + x34 = 17, x12 + x11 + x19 = 0, ….

  5. Linear Programs Find x2Rn minimizing or maximizing a linear formhx,ci = i ci xi (the objective function) so that a given set of linear equations and inequalitiesare satisfied. A feasible solution to the program is a point xsatisfying the equations and inequalities.

  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!

  7. Diet Problem Necessary daily intake: Energy 2000 kcal, Protein 55 g, Calcium 800 mg Compose a diet minimizing price and fulfilling necessary daily intake.

  8. Diet Problem Necessary daily intake: Energy 2000 kcal, Protein 55 g, Calcium 800 mg Compose a diet minimizing price and fulfilling necessary daily intake.

  9. Linear Program for Diet Problem Find (x1, x2, x3, x4, x5, x6) 2 (R+)n minimizing 3x1 + 24x2 + 13x3 + 9x4 + 20x5 + 19x6 so that 110x1 + 205x2 + 160 x3 + 160 x4 + 420 x5 + 260 x6¸ 2000 4 x1 + 32 x2 + 13 x3 + 8 x4 + 4 x5 + 14 x6¸ 55 2 x1 + 12 x2 + 54 x3 + 285 x4 + 22 x5 + 80 x6¸ 800

  10. Linear Programs, Geometric View x – y = 0

  11. x – y ¸ 0

  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

  13. x + 2y · 1 Æ x + 3 y ¸ -1 Æ x – y ¸ 0 Æ 2 x - y · 3

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

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

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

  17. Linear Programs, Geometric view • The set of feasible solutions of a linear program is a polyhedronin Rn. • 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 unboundedor empty (the linear program is then said to be infeasible).

  18. Linear Programs in Standard Form • The objective must be to maximize a linear function. • All variables are restricted to non-negative values. • All constraints have the form d1x1 + d2x2 +  + dn xn·e

  19. Linear Programs in Standard Form Find (x1, x2, x3, …, xn) 2 (R+)n maximizing c1 x1+ c2 x2 +  + cn xn so that a11x1 + a12x2 +  + a1nxn·b1 a21x1 + a22x2 +  + a2nxn· b2 … am1 x1 + am2x2 +  + amnxn·bm Exceptions: If no fesible solution exist, report “Infeasible”. If arbitrarily good feasible solutions exist, report “Unbounded”.

  20. Linear Program in Standard FormMatrix Notation Given A 2Rm£n, b2Rm, c2Rn, find x2 (R+)n maximizing hc,xi so that Ax·b.

  21. Reduction from LP in general form to LP in standard form Suppose we want to minimize (not maximize) c1x1 + c2x2 + cn xn Maximize - c1x1 - c2x2 -  - cn xn

  22. Reduction from LP in general form to LP in standard form Suppose we have a constraint d1x1 + d2x2 +  + dnxn¸e Replace with - d1x1 - d2x2 -  - dn xn· - e

  23. Reduction from LP in general form to LP in standard form Suppose we have a constraint d1x1 + d2x2 +  + dn xn = e Replace with d1x1 + d2x2 +  + dn xn·e - d1x1 - d2x2 -  - dn xn· - e

  24. Reduction from LP in general form to LP in standard form • Suppose yis 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. 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. 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. 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.

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