Chapter 10: Iterative Improvement

1. The Design and Analysis of Algorithms Chapter 10:Iterative Improvement Simplex Method

2. Iterative Improvement • Introduction • Linear Programming • The Simplex Method • Standard Form of LP Problem • Basic Feasible Solutions • Outline of the Simplex Method • Example • Notes on the Simplex Method • Improvements

3. Introduction Algorithm design technique for solving optimization problems • Start with a feasible solution • Repeat the following step until no improvement can be found: • change the current feasible solution to a feasible solution with a better value of the objective function • Return the last feasible solution as optimal

4. Introduction • Note: Typically, a change in a current solution is “small” (local search) • Major difficulty: Local optimum vs. global optimum

5. Important Examples • Simplex method • Ford-Fulkerson algorithm for maximum flow problem • Maximum matching of graph vertices • Gale-Shapley algorithm for the stable marriage problem

6. Linear Programming • Linear programming (LP) problem is to optimize a linear function of several variables subject to linear constraints: maximize (or minimize) c1x1+ ...+ cnxn subject to ai1x1+ ...+ ainxn ≤ (or ≥ or =) bi , i = 1,...,m , x1≥ 0, ... , xn ≥ 0 The function z = c1x1+ ...+ cnxnis called the objective function; constraints x1≥ 0, ... , xn ≥ 0 are called non-negativity constraints

7. Example maximize 3x + 5y subject to x + y ≤ 4 x + 3y ≤ 6 x ≥ 0, y ≥ 0 Feasible region is the set of points defined by the constraints

8. Geometric solution maximize 3x + 5y subject to x + y ≤ 4 x + 3y ≤ 6 x ≥ 0, y ≥ 0 Extreme Point Theorem Any LP problem with a nonempty bounded feasible region has an optimal solution; moreover, an optimal solution can always be found at an extreme point of the problem's feasible region.

9. Possible outcomes in solving an LP problem • has a finite optimal solution, which may not be unique • unbounded: the objective function of maximization (minimization) LP problem is unbounded from above (below) on its feasible region • infeasible: there are no points satisfying all the constraints, i.e. the constraints are contradictory

10. The Simplex Method • Simplex method is the classic method for solving LP problems, one of the most important algorithms ever invented • Invented by George Dantzig in 1947 (Stanford University) • Based on the iterative improvement idea: Generates a sequence of adjacent points of the problem’s feasible region with improving values of the objective function untilno further improvement is possible

11. Standard form of LP problem • must be a maximization problem • all constraints (except the non-negativity constraints) must be in the form of linear equations • all the variables must be required to be nonnegative • Thus, the general linear programming problem in standard form with m constraints and n unknowns (n ≥ m) is • maximize c1x1+ ...+ cnxn • subject toai1x1+ ...+ ainxn= bi ,, i = 1,...,m, • x1≥ 0, ... , xn ≥ 0 • Every LP problem can be represented in such form

12. Example maximize 3x + 5y maximize 3x + 5y + 0u + 0v subject to subject to x + y ≤ 4 x + y + u = 4 x + 3y ≤ 6 x + 3y + v = 6 x≥0, y≥0 x≥0, y≥0, u≥0, v≥0 Variables u and v, transforming inequality constraints into equality constrains, are called slack variables

13. Basic feasible solutions A basic solution to a system of m linear equations in nunknowns (n ≥ m) is obtained by setting n – mvariables to 0 and solving the resulting system to get the values of the other m variables. The variables set to 0 are called nonbasic; the variables obtained by solving the system are called basic. A basic solution is called feasibleif all its (basic) variables are nonnegative. Example x + y + u = 4 x + 3y + v = 6 (0, 0, 4, 6) is basic feasible solution (x, y are nonbasic; u, v are basic)

14. Simplex Tableau maximize z = 3x + 5y + 0u + 0v subject to x + y + u = 4 x + 3y + v = 6 x≥0, y≥0, u≥0, v≥0

15. Outline of the Simplex Method Step 0 [Initialization] Present a given LP problem in standard form and set up initial tableau. Step 1 [Optimality test] If all entries in the objective row are nonnegative — stop: the tableau represents an optimal solution. Step 2 [Find entering variable] Select (the most) negative entry in the objective row. Mark its column to indicate the entering variable and the pivot column.

16. Outline of the Simplex Method • Step 3 [Find departing variable] • For each positive entry in the pivot column, calculate the θ-ratio by dividing that row's entry in the rightmost column by its entry in the pivot column. • (If there are no positive entries in the pivot column — stop: the problem is unbounded.) • Find the row with the smallest θ-ratio, mark this row to indicate the departing variable and the pivot row. • Step 4 [Form the next tableau] • Divide all the entries in the pivot row by its entry in the pivot column. • Subtract from each of the other rows, including the objective row, the new pivot row multiplied by the entry in the pivot column of the row in question. • Replace the label of the pivot row by the variable's name of the pivot column and go back to Step 1.

17. basic feasible sol. (0, 0, 4, 6) z = 0 basic feasible sol. (0, 2, 2, 0) z = 10 basic feasible sol. (3, 1, 0, 0) z = 14 Example of Simplex Method maximize z = 3x + 5y + 0u + 0v subject to x + y + u = 4 x + 3y + v = 6 x≥0, y≥0, u≥0, v≥0

18. Notes on the Simplex Method • Finding an initial basic feasible solution may pose a problem • Theoretical possibility of cycling • Typical number of iterations is between m and 3m, where m is the number of equality constraints in the standard form. Number of operations per iteration: O(nm) • Worse-case efficiency is exponential

19. Improvements • L. G. Khachian introduced an ellipsoid method (1979) that seemed to overcome some of the simplex method's limitations. O(n6). Disadvantage – runs with the same complexity on all problems • Narendra K. Karmarkar of AT&T Bell Laboratories proposed in1984 a new very efficient interior-point algorithm - O(n 3.5). In empirical tests it performs competitively with the simplex method.

20. Unweighted Bipartite Matching

21. Unweighted Bipartite Matching

22. Definitions Matching Free Vertex

23. Definitions • Maximum Matching: matching with the largest number of edges

24. Definition • Note that maximum matching is not unique.

25. Intuition • Let the top set of vertices be men • Let the bottom set of vertices be women • Suppose each edge represents a pair of man and woman who like each other • Maximum matching tries to maximize the number of couples!

26. Alternating Path • Alternating between matching and non-matching edges. a c d e b f h i j g d-h-e: alternating path a-f-b-h-d-i: alternating path starts and ends with free vertices f-b-h-e: not alternating path e-j: alternating path starts and ends with free vertices

27. Idea • “Flip” augmenting path to get better matching • Note: After flipping, the number of matched edges will increase by 1! 

28. Idea • Theorem (Berge 1975): A matching M in G is maximum iffThere is no augmenting path Proof: • () If there is an augmenting path, clearly not maximum. (Flip matching and non-matching edges in that path to get a “better” matching!)

29. Idea of Algorithm • Start with an arbitrary matching • While we still can find an augmenting path • Find the augmenting path P • Flip the edges in P

30. Labelling Algorithm • Start with arbitrary matching

31. Labelling Algorithm • Pick a free vertex in the bottom

32. Labelling Algorithm • Run BFS

33. Labelling Algorithm • Alternate unmatched/matched edges

34. Labelling Algorithm • Until a augmenting path is found

35. Augmenting Tree

36. Flip!

37. Repeat • Pick another free vertex in the bottom

38. Repeat • Run BFS

39. Repeat • Flip

40. Answer • Since we cannot find any augmenting path, stop!

41. Overall algorithm • Start with an arbitrary matching (e.g., empty matching) • Repeat forever • For all free vertices in the bottom, • do bfs to find augmenting paths • If found, then flip the edges • If fail to find, stop and report the maximum matching.

42. Time analysis • We can find at most |V| augmenting paths • To find an augmenting path, we use bfs! Time required = O( |V| + |E| ) • Total time: O(|V|2 + |V| |E|)

43. Improvement • We can try to find augmenting paths in parallel for all free nodes in every iteration. • Using such approach, the time complexity is improved to O(|V|0.5 |E|)

44. The Stable Marriage Problem • Consider the following information: • a collection of eligible men, each with a rank ordered list of to be husband • a collection of eligible women, each with a rank ordered list of to be wife • Assume that: • the number of men and women are equal • all men rank all women, and vice versa • rank ordered lists are in decreasing order of preference "The Match Maker"

45. The Stable Marriage Problem (cont’d) • We define a marriage arrangement to be a pairing of the men and women such that: • each man is paired with one woman and • every person appears in exactly one pair. • We say that a marriage arrangement is stable if there does not exist a woman X and man Y such that: • X prefers Y over X's husband, and • Y prefers X over Y's wife. • Note: If there would exist such people, then they would want to run off together, creating an unstable situation. "The Match Maker"

46. The Stable Marriage Problem (cont’d) • The problem is: • “Given the preference rankings of the men and women, construct a stable marriage arrangement.” "The Match Maker"

47. The Stable Marriage Problem • The method For solving this problem works something like this: • The first man will propose to the first woman on his list. • She having no better offer at this point, will accept. • The second man will then propose to his first choice, and so on. • Eventually it may happen that a man proposes to a woman who already has a partner. "The Match Maker"

48. The Stable Marriage Problem (cont’d) • She will compare the new offer to her current partner and will accept whoever is higher on her list. • The man she rejects will then go back to his list and propose to his second choice, third choice, and so forth until he comes to a woman who accepts his offer. • If this woman already had a partner, her old partner gets rejected and he in turn starts proposing to women further down his list. • Eventually everything gets sorted out. "The Match Maker"

49. The Stable Marriage Problem (cont’d) • Algorithm: • Each person starts with no people "canceled" from his or her list. • People will be canceled from lists as the algorithm progresses. • For each man m, do propose(m), as defined below. "The Match Maker"

50. The Stable Marriage Problem (cont’d) • propose(m): • Let W be the first uncanceled woman on m's preference list. • Do: refuse(W,m), as defined in the next slide. • refuse(W,m): • Let m' be W 's current partner (if any). • If W prefers m' to m, then she rejects m, in which case: • cancel m off W 's list and W off m's list; • do: propose(m). (Now m must propose to someone else.) • Otherwise, W accepts m as her new partner, in which case: • cancel m' off W 's list and W off m' 's list; • do: propose(m'). (Now m' 's must propose to someone else.) "The Match Maker"