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Efficient Approaches for Optimization Problems: A Comprehensive Study

This research explores optimization problems including Vertex Cover, Set Cover, and Knapsack, proposing approximation algorithms and solutions for minimizing costs. Learn about complexities, feasible solutions, and algorithmic strategies.

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Efficient Approaches for Optimization Problems: A Comprehensive Study

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  1. Optimization problems INSTANCE FEASIBLE SOLUTIONS COST

  2. Vertex Cover problem INSTANCE graph G FEASIBLE SOLUTIONS SV, such that (eE) Se0 COST c(S) = |S|

  3. Set Cover problem INSTANCE family of sets A1,...,An   FEASIBLE SOLUTIONS S[n], such that Ai =  COST c(S) = |S| iS

  4. Vertex Cover problem Set Cover problem INSTANCE family of sets A1,...,An FEASIBLE SOLUTIONS S[n], such that  Ai= COST c(S) = |S| INSTANCE graph G FEASIBLE SOLUTIONS SV, such that (eE) Se0 COST c(S) = |S| iS

  5. Vertex Cover problem Set Cover problem INSTANCE family of sets A1,...,An FEASIBLE SOLUTIONS S[n], such that  Ai= COST c(S) = |S| INSTANCE graph G FEASIBLE SOLUTIONS SV, such that (eE) Se0 COST c(S) = |S| iS = E Ai E is the set of edges adjacent to i V

  6. Optimization problems INSTANCE FEASIBLE SOLUTIONS COST OPTIMAL SOLUTION OPT= min c(T) T FEASIBLE SOLUTIONS

  7. -approximation algorithm INSTANCE  T such that c(T)  OPT

  8. Last Class: 2-approximation algorithm for Vertex-Cover 2-approximation algorithm for Metric TSP 1.5-approximation algorithm for Metric TSP

  9. This Class: (1+)-approximation algorithm for Knapsack O(log n)-approximation algorithm for Set Cover

  10. Knapsack INSTANCE: value vi, weight wi, for i {1,...,n} weight limit W FEASIBLE SOLUTION: collection of items S {1,...,n} with total weight  W COST (MAXIMIZE): sum of the values of items in S

  11. Knapsack INSTANCE: value vi, weight wi, for i {1,...,n} weight limit W FEASIBLE SOLUTION: collection of items S {1,...,n} with total weight  W COST (MAXIMIZE): sum of the values of items in S We had: pseudo-polynomial algorithm, time = O(Wn) pseudo-polynomial algorithm, time = O(Vn), where V = v1 + ... +vn

  12. Knapsack INSTANCE: value vi, weight wi, for i {1,...,n} weight limit W FEASIBLE SOLUTION: collection of items S {1,...,n} with total weight  W COST (MAXIMIZE): sum of the values of items in S GOAL convert pseudo-polynomial algorithm, time = O(Vn), where V = v1 + ... +vn into an approximation algorithm IDEA = rounding

  13. Knapsack wlog all wi W M = maximum of vi vi vi’ :=  n vi / (M)  OPT’  n2/ S = optimal solution in original S’ = optimal solution in modified Will show: optimal solution in modified is an approximately optimal solution in original

  14. Knapsack vi vi’ :=  n vi / (M)  S = optimal solution in original S’ = optimal solution in modified (n/(M)) vi vi’  vi’  ( nvi / (M) -1 ) i S’ i S’ i S i S Will show: optimal solution in modified is an approximately optimal solution in original

  15. Knapsack vi vi’ :=  n vi / (M)  S = optimal solution in original S’ = optimal solution in modified (n/(M)) vi vi’  vi’  ( nvi / (M) -1 ) i S’ i S’ i S i S vi ( vi - ) 1 OPT – M  OPT(1– ) n/(M) i S’ i S Will show: optimal solution in modified is an approximately optimal solution in original

  16. Running time? pseudo-polynomial algorithm, time = O(V’n), where V’ = v’1 + ... +v’n M = maximum of vi vi vi’ :=  n vi / (M) 

  17. Running time? pseudo-polynomial algorithm, time = O(V’n), where V’ = v’1 + ... +v’n M = maximum of vi vi vi’ :=  n vi / (M)  v’i n/ V’ n2/ running time = O(n3/)

  18. We have an algorithm for the Knapsack problem, which outputs a solution with value  (1-) OPT and runs in time O(n3/) FPTAS Fully polynomial-time approximation scheme (1+)-approximation algorithm running in time poly(INPUT,1/)

  19. Weighted set cover problem INSTANCE: A1,...,Am, weights w1,...,wm FEASIBLE SOLUTION: collection S of the Ai covering  OBJECTIVE (minimize): the cost of the collection (in the unweighted version we have wi =1)

  20. Weighted set cover problem Greedy algorithm: pick Ai with minimal wi / |Ai| remove elements in Ai from  repeat

  21. Negative example (last class) approximation ratio = (log n)

  22. Weighted set cover problem Greedy algorithm: pick Ai with minimal wi / |Ai| remove elements in Ai from  repeat Theorem: O(log n) approximation algorithm.

  23. Weighted set cover problem Theorem: O(log n) approximation algorithm. Greedy algorithm: pick Ai with minimal wi / |Ai| remove elements in Ai from  repeat when Ai picked, cost of the solution increases by wi Ai everybody pays wi / |Ai|

  24. Weighted set cover problem Let B be a set of weight w. How much did the guys in B pay? pick me! cost=w/B B

  25. Weighted set cover problem Theorem: O(log n) approximation algorithm. Greedy algorithm: pick Ai with minimal wi / |Ai| remove elements in Ai from  repeat sorry Ai was cheaper pick me! cost=w/B B paid less than w/B Ai

  26. Weighted set cover problem Theorem: O(log n) approximation algorithm. Greedy algorithm: pick Ai with minimal wi / |Ai| remove elements in Ai from  repeat B

  27. Weighted set cover problem continue, size of B went down by 1 pick me! cost=w/(B-1) B

  28. Weighted set cover problem Theorem: O(log n) approximation algorithm. Greedy algorithm: pick Ai with minimal wi / |Ai| remove elements in Ai from  repeat sorry Aj was cheaper pick me! cost=w/(B-1) B paid less than w/(B-1) Aj

  29. Weighted set cover problem Theorem: O(log n) approximation algorithm. Greedy algorithm: pick Ai with minimal wi / |Ai| remove elements in Ai from  repeat B

  30. Weighted set cover problem continue, size of B went down by 1 pick me! cost=w/(B-2) B

  31. Weighted set cover problem B paidw/B paidw/(B-1) paidw/2 paidw/(B-2) paidw vertices in order they are covered by greedy

  32. Weighted set cover problem B paidw/B paidw/(B-1) paidw/2 paidw/(B-2) paidw TOTAL PAID  w (1/B + 1/(B-1) + ... +1/2 + 1) = w O(ln B) = w O(ln n)

  33. Weighted set cover problem INSTANCE: A1,...,Am, weights w1,...,wm FEASIBLE SOLUTION: collection S of the Ai covering  OBJECTIVE (minimize): the cost of the collection Greedy algorithm: pick Ai with minimal wi / |Ai| remove elements in Ai from  repeat Theorem: O(log n) approximation algorithm.

  34. Clustering n points in Rm d(i,j) = distance between points i,j partition the points into k clusters of small diameter diam(C) = max d(i,j) i,jC

  35. Clustering k = 3

  36. Clustering k = 3

  37. Clustering k = 2

  38. Clustering k = 2

  39. k-Clustering INSTANCE n points in Rm FEASIBLE SOLUTION partition of [n] into C1,...,Ck COST max diam(Ci) i[k] diam(C) = max d(i,j) i,jC

  40. k-Clustering GREEDY ALGORITHM pick s1 [n] for i from 2 to k do pick si the farthest point from s1,...,si-1 Ci = {x [n] whose closest center is si}

  41. k-Clustering GREEDY ALGORITHM pick s1 [n] for i from 2 to k do pick si the farthest point from s1,...,si-1 Ci = {x [n] whose closest center is si} s1

  42. k-Clustering GREEDY ALGORITHM pick s1 [n] for i from 2 to k do pick si the farthest point from s1,...,si-1 Ci = {x [n] whose closest center is si} s1 s2

  43. k-Clustering GREEDY ALGORITHM pick s1 [n] for i from 2 to k do pick si the farthest point from s1,...,si-1 Ci = {x [n] whose closest center is si} s1 s2 s3

  44. k-Clustering GREEDY ALGORITHM pick s1 [n] for i from 2 to k do pick si the farthest point from s1,...,si-1 Ci = {x [n] whose closest center is si} s1 s2 s3

  45. k-Clustering GREEDY ALGORITHM pick s1 [n] for i from 2 to k do pick si the farthest point from s1,...,si-1 Ci = {x [n] whose closest center is si} Theorem: GREEDY ALGORITHM IS A 2-APPROXIMATION ALGORITHM

  46. Theorem: GREEDY ALGORITHM IS A 2-APPROXIMATION ALGORITHM s2 s1 sk sk+1 d(si,sj)  d(sk+1,{s1,...,sk}) = r OPT  r cost of greedy  2r

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