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Vijay V. Vazirani Georgia Tech

A Postmortem of the Last Decade and Some Directions for the Future. Vijay V. Vazirani Georgia Tech. Although this may seem a paradox, all exact science is dominated by the idea of approximation. Bertrand Russell (1872-1970). Exact algorithms have been studied

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Vijay V. Vazirani Georgia Tech

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

  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.

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

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

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

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

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

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

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

  11. What lies at the core of approximation algorithms?

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

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

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

  15. Massive accident!

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

  17. Is combinatorial optimizationrelevant today? Why design combinatorial algorithms, especially today that LP-solvers are so fast?

  18. 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.

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

  20. Combinatorial optimization studied: Problems admitting ILPs

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

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

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

  24. 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!

  25. 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??

  26. KKT optimality conditions

  27. Possible RCPs

  28. Quadratic RCPs

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

  30. Combinatorial Algorithms • 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.

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

  32. 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?

  33. Logarithmic RCPs

  34. Logarithmic RCPs Rationality is the exception to the rule, and needs to be established piece-meal.

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

  36. 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.

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