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Linear Programming and Approximation

Linear Programming and Approximation. Seminar in Approximation Algorithms. Linear Programming. Linear objective function and Linear constraints Canonical form Standard form. Linear Programming – Example. x=(2,1,3) is a feasible solution 7 * 2 + 1 + 5 * 3 = 30 is an upper bound the optimum.

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Linear Programming and Approximation

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  1. Linear Programming and Approximation Seminar in Approximation Algorithms

  2. Linear Programming • Linear objective function and Linear constraints • Canonical form • Standard form

  3. Linear Programming – Example • x=(2,1,3) is a feasible solution • 7 * 2 + 1 + 5 * 3 = 30 is an upper bound the optimum

  4. Lower Bound • How can we find a lower bound? • For example,

  5. Lower Bound Another example:

  6. Lower Bound • Assign a non-negative coefficient yito every primal inequality such that • Lower bound 10y1+6y2

  7. LP Duality • The problem of finding the best lower bound is a linear program

  8. LP Duality • For every x and y: cTx  bTy • Thus, Opt(primal)  Opt(dual) • The dual of the dual is the primal

  9. Example x=(7/4,0,11/4) and y=(2,1) are feasible solutions 7*7/4 + 0 + 5*11/4 = 10*2 + 6*1 = 26 Thus, x and y are optimal

  10. Max Flow vs. Min s,t-cut • Path(s,t) - the set of directed paths from s to t. • Max Flow: • Dual: • Implicit • Min s,t-cut • Opt(Dual) = Opt(Min s,t-cut)

  11. LP-duality Theorem Theorem: • Opt(primal) is finite  Opt(dual) is finite • If x* and y* are optimal Conclusion: LP  NPcoNP

  12. Weak Duality Theorem If x and y are feasible Proof:

  13. Complementary Slackness Conditions x and y are optimal  Primal conditions: Dual conditions:

  14. Complementary Slackness Conditions - Proof • By the duality theorem • Similar arguments work for y • For other direction read the slide upwards

  15. Algorithms for Solving LP • Simplex [Dantzig 47] • Ellipsoid [Khachian 79] • Interior point [Karmakar 84] • Open question: Is there a strongly polynomial time algorithm for LP?

  16. Integer Programming • NP-hard • Branch and Bound • LP relaxation • Opt(LP)  Opt(IP) • Integrality Gap – Opt(IP)/Opt(LP)

  17. Using LP for Approximation • Finding a good lower bound - Opt(LP)  Opt(IP) • Integrality gap is the best we can hope for Techniques: • Rounding • Solve LP-relaxation and then round solution. • Primal Dual • Find a feasible dual solution y, and a feasible integral primal solution x such that cTx  r * bTy

  18. Minimum Vertex Cover LP-relaxation and its dual: • (P) • (D)

  19. Solving the Dual Algorithm: • Solve (D) • Define: Analysis: CD is a cover:

  20. Solving the Dual CD is r-approx: • Weak duality The.

  21. Rounding Algorithm: • Solve (P) • Define: Analysis:  x*is not feasible • Due to the complementary conditions: • Conclusion: x(CP) is 2-approx

  22. Rounding Algorithm: • Solve (P) • Define: Analysis:  x*is not feasible • Clearly, • Conclusion: x(C½) is 2-approx

  23. The Geometry of LP • Claim: Let x1,x2 be feasible solutions Then, x1+(1-)x2 is a feasible solution • Definition: x is a vertex if • Theorem: If a finite solution exists, there exists an optimal solution x* which is a vertex

  24. Half Integrality Theorem: If x is a vertex then x{0,1/2,1}n Proof: • Let x be a vertex such that x{0,1/2,1}n • Both solutions are feasible •  = shortest distance to {0,1/2,1} Conclusion:

  25. Half Integrality Algorithm: • Construct the following bipartite graph G’: • Find a vertex cover C’ in G’ (This can be done by using max-flow.)

  26. Primal Dual • Construct an integral feasible solution x and a feasible solution y such that cTx  r bTy • Weak duality theorem  bTy  Opt(LP) • cTx  r Opt(LP)  r Opt(IP) • x is r-approx

  27. Greedy H()-Approximation Algorithm • While there exists an uncovered edge: • Remove u and incident edges

  28. Primal Dual Schema • Modified version of the primal dual method for solving LP • Construct integral primal solution and feasible dual solution simultaneously. • We impose the primal complementary slackness conditions, while relaxing the dual conditions

  29. Relaxed Dual Conditions • Primal: • Relaxed dual: Claim: If x,y satisfy the above conditions Proof:

  30. Vertex Cover Algorithm: • For every edge e=(u,v) do: • Output C (We can construct C at the end.)

  31. Vertex Cover • There are no uncovered edges, or over packed vertices at the end. So x=x(C) and y are feasible. • Since x and y satisfy the relaxed complementary slackness conditions with =2, x is a 2-approx solution. • Primal: • Relaxed Dual:

  32. Generalized Vertex Cover LP-relaxation and its dual: • (P) • (D)

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