1. Vijay V. Vazirani Georgia Tech A Postmortem of the Last Decade and Some Directions for the Future

2. Although this may seem a paradox, all exact science is dominated by the idea of approximation. Bertrand Russell (1872-1970)

3. Exact algorithms have been studied intensively for over four decades, and yet basic insights are still being obtained. Since polynomial time solvability is the exception rather than the rule, it is only reasonable to expect the theory of approximation algorithms to grow considerably over the years.

9. Beyond the list … Unique Games Conjecture Simpler proof of PCP Theorem Online algorithms for AdWords problem

10. Beyond the list … Unique Games Conjecture Simpler proof of PCP Theorem Online algorithms for AdWords problem Integrality gaps vs approximability

13. Raghevendra, 2008: Assuming UGC, for every constrained satisfaction problem: Can achieve approximation factor = integrality gap of “standard SDP” NP-hard to approximate better.

14. Future Directions Status of UGC Raghavendra-type results for LP-relaxations Randomized dual growth in primal-dual algorithms

15. Approximability: sharp thresholds For a natural problem: Can achieve approximation factor in P. If we can achieve in P => complexity-theoretic disaster

16. Conjecture There is a natural problem having sharp thresholds w.r.t. time classes

17. Group Steiner Tree Problem Chekuri & Pal, 2005: Halperin & Krauthgamer, 2003:

20. What lies at the core of approximation algorithms?

21. What lies at the core of approximation algorithms? Combinatorial optimization!

22. Combinatorial optimization Central problems have LP-relaxations that always have integer optimal solutions! ILP: Integral LP

23. Combinatorial optimization Central problems have LP-relaxations that always have integer optimal solutions! ILP: Integral LP i.e., it “behaves” like an IP!

24. Massive accident!

25. Cornerstone problems in P Matching (general graph) Network flow Shortest paths Minimum spanning tree Minimum branching

26. Is combinatorial optimization relevant today? Why design combinatorial algorithms, especially today that LP-solvers are so fast?

30. Combinatorial algorithms Very rich theory Gave field of algorithms some of its formative and fundamental notions, e.g. P Preferable in applications, since efficient and malleable.

31. Helped spawn off algorithmic areas, e.g., approximation algorithms and parallel algorithms.

32. Problems admitting ILPs Combinatorial optimization studied:

33. Problems admitting LP-relaxations with bounded integrality gaps Approximation algorithms studied:

34. Problems admitting LP-relaxations with bounded integrality gaps Problems admitting ILPs

35. Rational convex program A nonlinear convex program that always has a rational solution (if feasible), using polynomially many bits, if all parameters are rational.

36. Rational convex program Always has a rational solution (if feasible) using polynomially many bits, if all parameters are rational. i.e., it “behaves” like an LP!

37. Rational convex program Always has a rational solution (if feasible) using polynomially many bits, if all parameters are rational. i.e., it “behaves” like an LP! Do they exist??

40. KKT optimality conditions

41. Possible RCPs

42. Quadratic RCPs

43. Two opportunities for RCPs: Program A: Combinatorial, polynomial time (strongly poly.) algorithm Program B: Polynomial time (strongly poly.) algorithm, given LP-oracle.

44. Helgason, Kennington & Lall, 1980  Single constraint Minoux, 1984  Minimum quadratic cost flow Frank & Karzanov, 1992  Closest point from origin to bipartite perfect matching polytope. Karzanov & McCormick, 1997  Any totally unimodular matrix. Combinatorial Algorithms

45. Ben-Tal & Nemirovski, 1999 Polyhedral approximation of second-order cone Main technique: Solves any quadratic RCP in polynomial time, given an LP-oracle.

46. Ben-Tal & Nemirovski, 1999 Polyhedral approximation of second-order cone Main technique: Solves any quadratic RCP in polynomial time, given an LP-oracle. Strongly polynomial algorithm?

47. Logarithmic RCPs

48. Rationality is the exception to the rule, and needs to be established piece-meal.

49. Logarithmic RCPs Optimal solutions to such RCPs capture equilibria for various market models!

50. Arrow-Debreu Theorem, 1954 Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a deep theorem from topology - Kakutani fixed point theorem.

51. Arrow-Debreu Theorem, 1954 Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem. Highly non-constructive!

52. Arrow-Debreu Theorem, 1954 Celebrated theorem in Mathematical Economics Established existence of market equilibrium under very general conditions using a theorem from topology - Kakutani fixed point theorem. Continuous, quasiconcave, satisfying non-satiation.

53. Complexity-theoretic question For “reasonable” utility fns., can market equilibrium be computed in P? If not, what is its complexity?

54. Short summary So far, all markets whose equilibria can be computed efficiently admit convex or quasiconvex programs, many of which are RCPs!

55. Combinatorial Algorithm for Linear Case of Fisher’s Model Devanur, Papadimitriou, Saberi & V., 2002 By extending primal-dual paradigm to setting of convex programs & KKT conditions

56. Eisenberg-Gale Program, 1959

57. prices p j

58. KKT conditions

59. Proof of rationality Guess positive allocation variables (say k). Substitute 1/p j by a new variable. LP with (k + g) equations and non-negativity constraint for each variable.

60. Auction for Google’s TV ads N. Nisan et. al, 2009: Used market equilibrium based approach. Combinatorial algorithms for linear case provided “inspiration”.

61. Long-standing open problem Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, piecewise-linear, concave utilities?

62. utility Piecewise linear, concave amount of j Additively separable over goods

63. Long-standing open problem Complexity of finding an equilibrium for Fisher and Arrow-Debreu models under separable, piecewise-linear, concave utilities? Equilibrium is rational!

64. Markets with separable, plc utilities are PPAD-complete Chen, Dai, Du, Teng, 2009 Chen & Teng, 2009 V. & Yannakakis, 2009

65. Markets with separable, plc utilities are PPAD-complete Chen, Dai, Du, Teng, 2009 Chen & Teng, 2009 V. & Yannakakis, 2009 (Building on combinatorial insights from DPSV)

66. Theorem (V., 2002): Generalized linear Fisher market to Spending constraint utilities. Polynomial time algorithm for computing equilibrium.

67. Is there a convex program for this model? “We believe the answer to this question should be ‘yes’. In our experience, non-trivial polynomial time algorithms for problems are rare and happen for a good reason – a deep mathematical structure intimately connected to the problem.”

68. EG convex program = Devanur’s program Price disc. Market Goel & V. Spending constraint market V., 2005 Nash Bargaining V., 2008 Eisenberg-Gale Markets Jain & V., 2007 EG[2] Markets Chakrabarty, Devanur & V. 2008

69. V., 2010: Assuming perfect price discrimination, can handle: Continuously differentiable, quasiconcave (non-separable) utilities, satisfying non-satiation.

70. V., 2010: Continuously differentiable, quasiconcave (non-separable) utilities, satisfying non-satiation. Compare with Arrow-Debreu utilities!! continuous, quasiconcave, satisfying non-satiation.

71. A new development Orlin, 2009: Strongly polynomial algorithm for Fisher’s linear case, using scaling. Open: For rest

72. Are there other classes of RCPs?

73. Sturmfels & Uhler, 2009:

75. EG convex program = Devanur’s program Price disc. Market Goel & V. Spending constraint market V., 2005 Nash Bargaining V., 2008 Eisenberg-Gale Markets Jain & V., 2007 EG[2] Markets Chakrabarty, Devanur & V. 2008

76. Building on Karzanov & McCormick, 1997: Combinatorial algorithm for min cost flow under concave cost functions on edges.

77. EG convex program = Devanur’s program Price disc. Market Goel & V. Spending constraint market V., 2005 Nash Bargaining V., 2008 Eisenberg-Gale Markets Jain & V., 2007 EG[2] Markets Chakrabarty, Devanur & V. 2008

78. Rational (combinatorial) approximations to convex programs Problems admitting RCPs

79. EG convex program = Devanur’s program Price disc. Market Goel & V. Spending constraint market V., 2005 Nash Bargaining V., 2008 Eisenberg-Gale Markets Jain & V., 2007 EG[2] Markets Chakrabarty, Devanur & V. 2008

80. Rational (combinatorial) approximations to convex programs Problems admitting RCPs