1. 1 Combinatorial Dominance Analysis Keywords: Combinatorial Optimization (CO) Approximation Algorithms (AA) Approximation Ratio (a.r) Combinatorial Dominance (CD) Domination number/ratio (domn, domr) DOM -good approximation DOM -easy problem by: Yochai Twitto

2. 2 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. maximum Cut Problem: maximum Cut Summary

3. 3 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. maximum Cut Problem: maximum Cut Summary

4. 4 Background NP complexity class. AA and quality of approximations. The classical approximation ratio analysis. Example: Approximation for TSP.

5. 5 NP If P ≠ NP, then finding the optimum of NP-hard problem is difficult. If P = NP, P would encompass the NP and NP-Complete areas.

6. 6 Approximations So we are satisfied with an approximate solution. Question: How can we measure the solution quality ? Solutions quality line OPT Infeasible Near optimal

7. 7 Solution Quality Most of the time, naturally derived from the problem definition. If not, it should be given as external information.

8. 8 The classical Approximation Ratio (For maximization problem) Assume 0 ≤ β ≤ 1. A.r. ≥ β if the solution quality is greater than β·OPT Solutions quality line OPT Infeasible Near optimal ½ OPT

9. 9 Example: The Traveling Salesman Problem Given a weighted complete graph G, find the optimal tour. We will assume the graph is metric. We will see: The MST approximation. MST approximation ratio analysis.

10. 10 MST Approximation for TSP Find a minimum spanning tree for G. DFS the tree. Make shortcuts.

11. 11 MST Approx. ratio analysis Observation: If you remove an edge from a tour then you get a spanning tree! This means that Tour cost more than a minimum spanning tree.

12. 12 MST Approx. ratio analysis Thus, DFSing the MST is of cost No more than twice MST cost. I.e. no more than twice OPT. After shortcuts we get a tour with cost at most twice the optimum Since the graph is metric.

13. 13 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. maximum Cut Problem: maximum Cut Summary

14. 14 Combinatorial Dominance What is a “ combinatorial dominance guarantee ” ? Why do we need such guarantees ? min partition problem Example: the min partition problem. Definitions and notations.

15. 15 What is a “ combinatorial dominance guarantee ” ? A letter of reference: “ She is half as good as I am, but I am the best in the world …” “ she finished first in my class of 75 students …” The former is akin to an approximation ratio. The latter to combinatorial dominance guarantee.

16. 16 What is a “ combinatorial dominance guarantee ” ? (cont.) We saw that MST provides a 2-factor approximation. We can ask: Is the returned solution guaranteed to be always in the top O(n) best solutions ? Solutions quality line OPT Infeasible Near optimal top O(n)

17. 17 Why do we need that ? Let us take another look Let us take another look at the MST approximation for TSP. All other edges of weight 1+ε (not shown)

18. 18 Why do we need that ? The spanning tree here is a star. DFS + Shortcuts yields OPT = 6 + 4ε ≈ 6 MST tour size: 10 In general: OPT: (n-2)(1+ε) + 2 MST: 2(n-2) + 2 OPT MST tour

19. 19 Why do we need that ? But this is the worst possible tour! Such kind of analysis is called blackball analysis. Blackball instance

20. 20 Corollary The approximation ratio analysis gives us only a partial insight of the performance of the algorithm. Dominance analysis makes the picture fuller.

21. 21 Simple example of dominance analysis The minimum partition problem. Greedy-type algorithm. Combinatorial dominance analysis of the algorithm.

22. 22 Example: The minimum partition problem Given is a set of n numbers V = { a 1, a 2, …, a n } Find a bipartition (X,Y ) of the indices such that is minimal.

23. 23 Greedy-type algorithm Without loss of generality assume a 1 ≥ a 2 ≥ … ≥ a n. Initiate X = { }, Y = { }. For j = 1, …, n Add j to X if, Otherwise add j to Y.

24. 24 Combinatorial dominance analysis of the greedy-type algorithm Observation: Any solution produced by the alg. satisfies. Assume (X ’,Y ’ ) is any solution for min partition for {a 2, a 3, …, a n }. Now, add a 1 to Y ’ if, Otherwise add a 1 to X ’.

25. 25 Combinatorial dominance analysis of the greedy-type algorithm (cont.) Obtained solution: (X ’’,Y ’’ ). (X ’’, Y ’’ ) is a solution of the original problem. We have Conclusion: The solution provided by the algorithm dominates at least 2 n-1 solutions.

26. 26 Definitions & Notations Domination number: domn Domination ratio: domr DOM -good approximation DOM -easy problem

27. 27 Domination Number: domn Let P be a CO problem. Let A be an approximation for P. For an instance I of P, the domination number domn( I, A ) of A on I is the number of feasible solutions of I that are not better than the solution found by A.

28. 28 domn (example) STSP on 5 vertices. There exist 12 tours If A returns a tour of length 7 then domn( I, A ) = 8 4, 5, 5, 6, 7, 9, 9, 11, 11, 12, 14, 14 (tours lengths)

29. 29 Domination Number: domn Let P be a CO problem. Let A be an approximation for P. minimum For any size n of P, the domination number domn( P, n, A ) of an approximation A for P is the minimum of domn( I, A ) over all instances I of P of size n.

30. 30 Domination Ratio: domr Let P be a CO problem. Let A be an approximation for P. sol( I ) Denote by sol( I ) the number of all feasible solutions of I. minimum For any size n of P, the domination ratio domn( P, n, A ) of an approximation A for P is the minimum of domn( I, A ) / sol( I ) taken over all instances I of P of size n.

31. 31 DOM -good approximation A is a DOM -good approximation algorithm for P, if It is a polynomial time complexity alg. There exists a polynomial p(n) in the size of P, such that The domination ratio of A is at least 1/p(n) for any size n of P.

32. 32 DOM -easy problem A CO problem is a DOM -easy problem if it admits a DOM -good approximation. Problems not having this property are DOM -hard. Corollary: Minimum Partition is DOM -easy. Furthermore, p(n) is a constant.

33. 33 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. Maximum Cut Problem: Maximum Cut Summary

34. 34 Maximum Cut The problem. Simple greedy algorithm. Combinatorial dominance of the algorithm. We ’ ll see … Maximum Cut is DOM -easy.

35. 35 Problem: Maximum Cut Input: weighted complete graph G=(V, E, w) Find a bipartition (X, Y) of V maximizing the sum Denote n = |V|. Let W be the sum of weights of all edges.

36. 36 Problem: Maximum Cut Denote the average weight of a cut by Notice that. Next: We ’ ll see a simple algorithm which produces solutions that are always better than. We ’ ll show it is a DOM -good approximation for maxCut.

37. 37 Algorithm: greedy maxCut Algrorithm: Initiate X = {}, Y = {} For each j = 1 … n Add v j to X or Y so as to maximize its marginal value. Theorem: The above algorithm is a 2-factor approximation for maxCut. Moreover, it produces a cut of weight at least.

38. 38 CD analysis We will show that the number of cuts of weight at most is at least a polynomial part of all cuts Call them “ bad ” cuts Note that this is a general analysis technique. Can be applied to another algs./problems

39. 39 CD analysis A k-cut is a cut (X, Y) for which |X| = k. A fixed edge crosses k-cuts. Hence the average weight of a k-cut is

40. 40 CD analysis Let b k be the number of bad k-cuts. i.e. k-cuts of weight less than. Then

41. 41 CD analysis Solving for b k we get

42. 42 CD analysis Hence the number of bad cuts in G is at least (by DeMoivre-Laplace theorem)

43. 43 CD analysis Thus, G has more than bad cuts. Corollary: Maximum Cut is DOM -easy.

44. 44 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. maximum Cut Problem: maximum Cut Summary

45. 45 Summary Solutions quality line OPT Infeasible Near optimal ½ OPT Solutions quality line OPT Infeasible Near optimal top O(n)

46. 46 Summary MST tour OPT

47. 47 Summary Domination number: domn Domination ratio: domr DOM -good approximation DOM -easy problem

48. 48 Summary Domn(MST, TSP) = 1 Minimum Partition is DOM -easy. Maximum Cut is DOM -easy. Clique is DOM -hard unless P=NP. blackball

49. 49 Combinatorial Dominance Analysis