Hard Optimization Problems: Practical Approach

1. Hard Optimization Problems: Practical Approach DORIT RON Tel. 08 934 2141 Ziskind room #303 dorit.ron@weizmann.ac.il

2. Course outline • 1st lecture: Introduction and motivation

3. INTRODUCTION • What is an optimization problem?

4. An optimization problem consist of: • Variables: • Energy functional to be minimized/maximized: min / max

5. Unconstrained minimizationFind the global minimum

6. An optimization problem consist of: • Variables: • Energy functional to minimized/maximized: min / max Possibly subject to: • Equality constraints: • Inequality constraints:

7. Constrained minimizationsubject to

8. INTRODUCTION • What is an optimization problem? • Examples

9. Example 1: 2D Ising spins • Discrete (combinatorial) optimization min -S <i,j>si sj si = { +1 , -1}

10. 3D Ising model • Each spin represents a tiny magnet • The spins tend to align below a certain Tc • Ferromagnet – Iron at room temperature magnet ------ Tc------non-magnet ----|------------------------|---------------------------|--> T room tempferromagnetismmelting 770oC1538oC • At T=0 the system settles at its ground states

11. Example 2: 1D graph ordering • Given a graph G=(V,E), find a permutation of the vertices that minimizes E()=i jwi j| (i) -(j) |p where i , jare in V and wi j is theedge weight between i and j (wi j =0 if ij is not in E) • p=1 : Linear arrangement • p=2 : Quadratic energy • p= : The Bandwidth

12. Minimum Linear Arrangement Problem j i

13. Minimum Linear Arrangement Problem j i 1 2 3 4 5

14. Minimum Linear Arrangement Problem j i 1 2 3 4 5

15. Minimum Linear Arrangement Problem j i 1 2 3 4 5

16. Minimum Linear Arrangement Problem j i 1 2 3 4 5

17. Minimum Linear Arrangement Problem j i 1 2 3 4 5

18. General Linear Arrangement Problem • Variable nodes sizes E(x)=i jwi j| xi -xj |p xi = vi /2 + k:p(k)<p(i) vk i j xi xj

19. Other graph ordering problems • Minimize various functionals: envelope size, cutwidth, profile of graph, etc. • Traveling salesman problem – TSP

20. The Traveling Salesman Problem

21. The Traveling Salesman Problem

22. Other graph ordering problems • Minimize various functionals: envelope size, cutwidth, profile of graph, etc. • Traveling salesman problem – TSP • Graph bisectioning • Graph partitioning • Graph coloring • Graph drawing

23. Drawing Graphs

24. Drawing Graphs

25. Drawing Graphs

26. Example 3: 2D circuit placement • Bottleneck in the microchip industry • Given a hypergraph, find the discrete placement of each module (gate) while minimizing the wirelength

27. The hypergraph for a microchip

28. Placement on a grid of pins

29. Placement on a grid of pins

30. Routing over the placement

31. Example 3: 2D circuit placement • Bottleneck in the microchip industry • Given a hypergraph, find the discrete placement of each module (gate) while minimizing the wirelength • No overlap is allowed • No overflow is allowed • Critical paths must be shorter • Leave white space for routing • Typical IBM chip ~270 meters on 1cm2

32. Place and route

33. Exponential growth of transistors for Intel processors

34. INTRODUCTION • What is an optimization problem? • Examples • Summary of difficulties

35. Difficulties: • Many variables: 106 , 107 … • Many constraints: 106 , 107 … • Multitude of local optima • Discrete nature • Conflicting objectives • Reasonable running time

36. INTRODUCTION • What is an optimization problem? • Examples • Summary of difficulties • Is the global optimum really needed / obtainable?

37. PEKO=PLACEMENT EXAMPLE WITH KNOWN OPTIMUM • Place the nodes – this is the solution • Create the net list locally and compactly • The optimum wire length – the sum of all the edges between the nodes, is known and can be proven to be minimal

38. SOLUTION QUALITY

39. INTRODUCTION • What is an optimization problem? • Examples • Summary of difficulties • Is the global optimum really needed / obtainable? • What is expected of a “good approximate” solution?

40. “Good approximate” solution • As optimal as possible: high quality solution • Achievable in linear time • Scalable in the problem size

41. RUNTIME

42. RealityCheck

43. INTRODUCTION • What is an optimization problem? • Examples • Summary of difficulties • Is the global optimum really needed / obtainable? • What is expected of a “good approximate” solution? • Multilevel philosophy

44. MULTILEVEL APPROACH • PARTIAL DIFFERENTIAL EQUATIONS (Achi Brandt since the early 70’s) • STATISTICAL PHYSICS • CHEMISTRY • IMAGE SEGMENTATION • TOMOGRAPHY • GRAPH OPTIMIZATION PROBLEMS

46. SOLUTION QUALITY

47. SOLUTION QUALITY

48. ORIGINAL PICTURE

49. ORIGINAL FENG SHUI (1) FENG SHUI (2) mPL KRAFTWERK CAPO DRAGON OURS

50. OUR PLACEMENT