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Faster Deterministic Volume Estimation in the Oracle Model via Thin Lattice Coverings

This paper presents a nearly optimal deterministic algorithm for volume estimation in the oracle model. It introduces a new construction for "easy-to-enumerate" thin-covering lattices. The algorithm reduces volume estimation to counting lattice points in a well-calibrated lattice.

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Faster Deterministic Volume Estimation in the Oracle Model via Thin Lattice Coverings

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  1. Faster Deterministic Volume Estimation in the Oracle ModelviaThin Lattice Coverings Daniel Dadush Centrum Wiskunde & Informatica

  2. Volume Estimation Problem Given convex body and factor , compute s.t. . Answers: is in ? given by a membership oracle.

  3. Volume Estimation Problem Given convex body and factor , compute s.t. . Assume appropriate sandwiching guarantees.

  4. Volume Estimation: Known Bounds Bárány, Füredi 86+88: Any deterministic-approximation uses at least membership queries for . Dyer, Frieze, Kannan 91: Randomized polynomial time-approximation algorithm.

  5. Volume Estimation: Known Bounds Bárány, Füredi 86+88: Any deterministic-approximation uses at least membership queries for . Dyer, Frieze, Kannan 91: Randomized polynomial time-approximation algorithm. Open Problem: Deterministic PTAS for explicit polytopes?

  6. Volume Estimation: Known Bounds Bárány, Füredi 86+88: Any deterministic-approximation uses at least membership queries for . Dyer, Frieze, Kannan 91: Randomized polynomial time-approximation algorithm. More modest question: Can we match the oracle model lower bounds?

  7. Main Result Theorem[D. 14+]: Deterministic -approximation using -time and -space. Close toBárány & Füredilower bound.

  8. Deterministic -approximations symmetric asymmetric

  9. Lattices A lattice is all integral combinations of a basis .

  10. Lattices A lattice is all integral combinations of a basis . The determinant of is.

  11. Lattices A lattice is all integral combinations of a basis . Every tiling domain has volume .

  12. High Level Idea Reduce approximating to counting lattice points in a ``well-calibrated’’ lattice . volume is

  13. High Level Algorithm Input:convex body, approx factor .

  14. High Level Algorithm Compute point about which is ``approximately symmetric’’: Treat above as optimization problem.

  15. High Level Algorithm Compute point about which is ``approximately symmetric’’: Build using an “evolving net” over , evaluatingthe objective using volume algorithm for symmetric bodies.

  16. High Level Algorithm Build “thin” lattice which covers space w.r.t. : b. a.

  17. High Level Algorithm Build “thin” lattice which covers space w.r.t. : c.Enumeration over requires-space & -time.

  18. High Level Algorithm Compute by enumeration. Return .

  19. Algorithm Recap Find approximate center of symmetry . Build easy to enumerate thin covering lattice . Enumerate lattice points in blowup of .

  20. Algorithm Recap Find approximate center of symmetry . Build easy to enumerate thin covering lattice . Enumerate lattice points in blowup of .

  21. How to build Want easy to enumerate thin covering lattice .

  22. How to build A good start:Pick basis of to be axes of an M-ellipsoid of .

  23. Milman’s Ellipsoid is an M-Ellipsoid of if translates of cover and vice versa.

  24. Milman’s Ellipsoid translates of suffice to cover .

  25. Milman’s Ellipsoid translates of suffice to cover .

  26. Milman’s Ellipsoid Theorem[D., Vempala 13]:Can construct M-ellipsoid in deterministic-time and -space.

  27. How to build A good start:Pick basis of to be axes of an M-ellipsoid of . Get:& “easy to enumerate” .

  28. How to build A good start:Pick basis of to be axes of an M-ellipsoid of . Don’t get: covering property.

  29. How to build A good start:Pick basis of to be axes of an M-ellipsoid of . Idea: Find lattice “close” to satisfying covering property.

  30. How to build Sparsify : Take “random” sublattice to remove vectors in .

  31. How to build Sparsify : Take “random” sublattice to remove vectors in .

  32. How to build Densify Greedily add cosets of that miss until no longer possible [Rogers 50].

  33. How to build Densify Greedily add cosets of that miss until no longer possible [Rogers 50].

  34. How to build Densify Greedily add cosets of that miss until no longer possible [Rogers 50].

  35. How to build Return as final lattice.

  36. Summary Nearly optimal deterministic algorithm for volume estimation in the oracle model. New construction of “easy to enumerate” thin-covering lattices. Open Questions Ignoring efficiency, can we match query lower bound? Other applications of thin-covering lattices?

  37. -Net Barrier: Lemma: For any convex body and s.t. we have . Theorem [Barvinok 12]: Take small enough, then for a symmetric convex body , there exists a polytope with at most vertices. Question: Can we compute a as above using only queries?

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