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Firefighting on Trees Beyond Integrality Gaps David Adjiashvili, - PowerPoint PPT Presentation

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. Department of Mathematics ETH Zrich Aussois 2016 Firefighting on Trees Beyond Integrality Gaps David Adjiashvili, Andrea Baggio , Rico Zenklusen . Introduction 3 . . . . . . . . . . . . . . . . . . . . . r . . r .

  1. 4 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t 4 . t 5 . t 6 . t 1 t t 2 . t 3 . t 4 . t 5 1 . 3 . . . . . . . . . . . . . . . . 2 . . . . . . . . t 0 . . t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FireFighter Problem - FFP . . at t = 2 t = 1

  2. 4 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t 4 . t 5 . t 6 . t 1 t t 2 . t 3 . t 4 . t 5 1 . 3 . . . . . . . . . . . . . . . . . . . . . . . . . t 0 . t 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FireFighter Problem - FFP . . at t = 2 t = 2

  3. 4 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t 4 . t 5 . t 6 . t 1 t t 2 . t 3 . t 4 . t 5 1 . 3 . . . . . . . . . . . . . . . . . . . . . . . . . t 0 . t 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FireFighter Problem - FFP . . at t = 3 t = 2

  4. 4 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t 4 . t 5 . t 6 . t 1 t 2 2 . t 3 . t 4 . t 5 1 . . t . . . . . . . . . . . . . . . . . . . . . . . . t 0 . t 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FireFighter Problem - FFP . . at t = 3 t = 3

  5. 4 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t 3 . . t 5 . t 6 . t 1 t 2 2 . t 3 . t 4 . t 5 1 . . t . . . . . . . . . . . . . . . . . . . . . . . . t 0 . t 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FireFighter Problem - FFP . . at t = 4 t = 4

  6. 4 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t 3 . t 4 . . t 6 . t 1 t 2 2 . t 3 . t 4 . t 5 1 . . t . . . . . . . . . . . . . . . . . . . . . . . . t 0 . t 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FireFighter Problem - FFP . . at t = 5 t = 5

  7. 4 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t 3 . t 4 . t 5 . . t 1 t 2 2 . t 3 . t 4 . t 5 1 . . t . . . . . . . . . . . . . . . . . . . . . . . . t 0 . t 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FireFighter Problem - FFP . . at t = 6 t = 6

  8. 4 . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t 0 . t 1 . t 2 . t 3 t . 4 . t 5 . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FireFighter Problem - FFP . . × time t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

  9. 4 . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1 . t 2 . t 3 . t 4 . t . . . . . . . 1 . . . . Allocate one . t 0 . . . . . . . . . . . . . . . t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FireFighter Problem - FFP . . × time t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 GOAL . . × time as to maximize saved nodes.

  10. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resource Minimization for Fire Containment - RMFC t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

  11. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . terminals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resource Minimization for Fire Containment - RMFC t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

  12. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . save all . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resource Minimization for Fire Containment - RMFC . ! t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

  13. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Using only one . save all . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resource Minimization for Fire Containment - RMFC . ! t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 . . × time, impossible!

  14. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . With two . save all . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resource Minimization for Fire Containment - RMFC . ! t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 . . × time, saving all . is possible!

  15. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimize number of . . . save all . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resource Minimization for Fire Containment - RMFC . ! t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 GOAL . . × time as to save all . .

  16. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Minimize number of . . . save all . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resource Minimization for Fire Containment - RMFC . ! t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 GOAL . . × time as to save all . .

  17. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) Approximation - Hartnell and Li (2000) Simple greedy algorithm achieves a 1 / 2 -approximation. Approximation - Cai, Verbin, and Yang (2008) . 1 e -approximation (LP based!). Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012) e -approx. via monotone submodular function 6 maximization subject to a partition matroid constraint. 1 / 1 . . . . 1 / . . . . . The problem is NP-hard. . . Previous Results - FireFighter Problem On Trees

  18. Approximation - Hartnell and Li (2000) Simple greedy algorithm achieves a 1 / 2 -approximation. Approximation - Cai, Verbin, and Yang (2008) . 1 e -approximation (LP based!). Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012) e -approx. via monotone submodular function 6 maximization subject to a partition matroid constraint. 1 / 1 . . . . 1 / . . . . . The problem is NP-hard. . . Previous Results - FireFighter Problem On Trees Hardness - Finbow, King, MacGillivray, and Rizzi (2007)

  19. Approximation - Cai, Verbin, and Yang (2008) . 1 e -approximation (LP based!). Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012) e -approx. via monotone submodular function 6 maximization subject to a partition matroid constraint. 1 / 1 . . . 1 / . . . . . . The problem is NP-hard. . . Previous Results - FireFighter Problem On Trees Hardness - Finbow, King, MacGillivray, and Rizzi (2007) Approximation - Hartnell and Li (2000) Simple greedy algorithm achieves a 1 / 2 -approximation.

  20. Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012) e -approx. via monotone submodular function 6 . maximization subject to a partition matroid constraint. 1 / 1 . . . -approximation (LP based!). e . . . . . . The problem is NP-hard. . . Previous Results - FireFighter Problem On Trees Hardness - Finbow, King, MacGillivray, and Rizzi (2007) Approximation - Hartnell and Li (2000) Simple greedy algorithm achieves a 1 / 2 -approximation. Approximation - Cai, Verbin, and Yang (2008) ( ) 1 − 1 /

  21. 6 e e . . . . . . . -approximation (LP based!). . -approx. via monotone submodular function . The problem is NP-hard. . . maximization subject to a partition matroid constraint. . Previous Results - FireFighter Problem On Trees Hardness - Finbow, King, MacGillivray, and Rizzi (2007) Approximation - Hartnell and Li (2000) Simple greedy algorithm achieves a 1 / 2 -approximation. Approximation - Cai, Verbin, and Yang (2008) ( ) 1 − 1 / Approx. - Anshelevich, Chakrabarty, Hate, and Swamy (2012) ( ) 1 − 1 /

  22. Hardness - Finbow, King, MacGillivray, and Rizzi (2007) Approximation - Chalermsook and Chuzhoy (2010) . . . NP-hard to approximate within any factor better than 2. O log n -approx. via constant factor approximation for FFP. . . . O log n -approximation (LP based!). 7 Previous Results - RMFC On Trees

  23. Approximation - Chalermsook and Chuzhoy (2010) . . . NP-hard to approximate within any factor better than 2. O log n -approx. via constant factor approximation for FFP. . . . O log n -approximation (LP based!). 7 Previous Results - RMFC On Trees Hardness - Finbow, King, MacGillivray, and Rizzi (2007)

  24. Approximation - Chalermsook and Chuzhoy (2010) . . . NP-hard to approximate within any factor better than 2. . . . O log n -approximation (LP based!). 7 Previous Results - RMFC On Trees Hardness - Finbow, King, MacGillivray, and Rizzi (2007) O ( log n ) -approx. via constant factor approximation for FFP.

  25. . . . NP-hard to approximate within any factor better than 2. . . . 7 Previous Results - RMFC On Trees Hardness - Finbow, King, MacGillivray, and Rizzi (2007) O ( log n ) -approx. via constant factor approximation for FFP. Approximation - Chalermsook and Chuzhoy (2010) O ( log ∗ n ) -approximation (LP based!).

  26. Do Integrality Gaps Reflect Approximation Hardness? e matches integr. gap. partition matroid constraint (no o log n -approx!). k -center problem Similar to the Asymmetric RMFC -approx! ) . e 1 / ( no 1 Seen as monotone submodu- lar function max. subject to a Current best algorithms for FFP and RMFC are LP based. FFP Answer not clear before! up to constant-factor. O log n matches integr. gap RMFC 1 / 1 FFP 8

  27. Do Integrality Gaps Reflect Approximation Hardness? Seen as monotone submodu- (no o log n -approx!). k -center problem Similar to the Asymmetric RMFC -approx! ) . e 1 / ( no 1 partition matroid constraint lar function max. subject to a FFP Current best algorithms for FFP and RMFC are LP based. Answer not clear before! up to constant-factor. RMFC matches integr. gap. e FFP 8 O ( log ∗ n ) matches integr. gap ( ) 1 − 1 /

  28. 8 Seen as monotone submodu- (no o log n -approx!). k -center problem Similar to the Asymmetric RMFC -approx! ) . e 1 / ( no 1 partition matroid constraint lar function max. subject to a FFP Current best algorithms for FFP and RMFC are LP based. Answer not clear before! up to constant-factor. RMFC matches integr. gap. e FFP Do Integrality Gaps Reflect Approximation Hardness? O ( log ∗ n ) matches integr. gap ( ) 1 − 1 /

  29. 8 Seen as monotone submodu- (no o log n -approx!). k -center problem Similar to the Asymmetric RMFC -approx! ) . e 1 / ( no 1 partition matroid constraint lar function max. subject to a FFP Current best algorithms for FFP and RMFC are LP based. Answer not clear before! up to constant-factor. RMFC matches integr. gap. e FFP Do Integrality Gaps Reflect Approximation Hardness? O ( log ∗ n ) matches integr. gap ( ) 1 − 1 /

  30. 8 FFP (no o log n -approx!). k -center problem Similar to the Asymmetric RMFC -approx! ) . ( no partition matroid constraint lar function max. subject to a Current best algorithms for FFP and RMFC are LP based. Seen as monotone submodu- Answer not clear before! up to constant-factor. RMFC matches integr. gap. e FFP Do Integrality Gaps Reflect Approximation Hardness? O ( log ∗ n ) matches integr. gap ( ) 1 − 1 / ( ) 1 − 1 / e + ϵ

  31. 8 FFP k -center problem Similar to the Asymmetric RMFC -approx! ) . ( no partition matroid constraint lar function max. subject to a Current best algorithms for FFP and RMFC are LP based. Seen as monotone submodu- Answer not clear before! up to constant-factor. RMFC matches integr. gap. e FFP Do Integrality Gaps Reflect Approximation Hardness? O ( log ∗ n ) matches integr. gap ( ) 1 − 1 / (no o ( log ∗ n ) -approx!). ( ) 1 − 1 / e + ϵ

  32. Theorem - Adjiashvili, Baggio, and Zenklusen (2015) Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . . PTAS for the FireFighter Problem on trees. . . . O 1 -approximation for RMFC on trees. Both algorithms are guided by the same known LPs! We introduce techniques to go beyond integrality gaps. 9 Our Contributions

  33. Theorem - Adjiashvili, Baggio, and Zenklusen (2015) . . . PTAS for the FireFighter Problem on trees. . . . O 1 -approximation for RMFC on trees. Both algorithms are guided by the same known LPs! We introduce techniques to go beyond integrality gaps. 9 Our Contributions Theorem - Adjiashvili, Baggio, and Zenklusen (2015)

  34. . . . PTAS for the FireFighter Problem on trees. . . . Both algorithms are guided by the same known LPs! We introduce techniques to go beyond integrality gaps. 9 Our Contributions Theorem - Adjiashvili, Baggio, and Zenklusen (2015) Theorem - Adjiashvili, Baggio, and Zenklusen (2015) O ( 1 ) -approximation for RMFC on trees.

  35. . . . PTAS for the FireFighter Problem on trees. . . . We introduce techniques to go beyond integrality gaps. 9 Our Contributions Theorem - Adjiashvili, Baggio, and Zenklusen (2015) Theorem - Adjiashvili, Baggio, and Zenklusen (2015) O ( 1 ) -approximation for RMFC on trees. Both algorithms are guided by the same known LPs!

  36. . . . PTAS for the FireFighter Problem on trees. . . . We introduce techniques to go beyond integrality gaps. 9 Our Contributions Theorem - Adjiashvili, Baggio, and Zenklusen (2015) Theorem - Adjiashvili, Baggio, and Zenklusen (2015) O ( 1 ) -approximation for RMFC on trees. Both algorithms are guided by the same known LPs!

  37. . PTAS For The FireFighter Problem

  38. u V c u x u w P w V t x w 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v x w t c u subtree of u P v path r v LP max s t 1 V t v leaves t t 1 L x u 0 1 u V layers ow. . . . . . . . . . . r . . . 0 . . V nodes r x u 1 if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program t = 1 t = 2 t = 3 t = 4 t = 5 t = 6

  39. u V c u x u w P w V t x w 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v x w t c u subtree of u P v path r v LP max s t 1 V t v leaves t t 1 L x u 0 1 u V layers ow. . . . . . . . . . . r . . . 0 . . V nodes r x u 1 if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program t = 1 t = 2 t = 3 t = 4 t = 5 t = L

  40. u V c u x u w P w V t x w 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v x w t c u subtree of u P v path r v LP max s t 1 V t v leaves t t 1 L x u 0 1 u V layers ow. . r . . . . . . . . . . . 0 . . . . . x u 1 if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program V = nodes \ { r } t = 1 t = 2 t = 3 t = 4 t = 5 t = L

  41. u V c u x u w P w V t x w 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v x w t c u subtree of u P v path r v LP max s t 1 V t v leaves t t 1 L x u 0 1 u V layers ow. . . . . . . . . . . . r . 0 . . . . . u 1 if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program V = nodes \ { r } { t = 1 x u = t = 2 t = 3 t = 4 t = 5 t = L

  42. u V c u x u w P w V t x w 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 0 ow. V t layers t c u subtree of u P v path r LP . max v x w 1 v leaves t t 1 L . . if . . . . . . . . . . . . 1 . . r . . . . . . . u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program V = nodes \ { r } { t = 1 x u = t = 2 t = 3 t = 4 t = 5 t = L s . t . x u ∈ { 0 , 1 } ∀ u ∈ V

  43. u V c u x u w P 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . c u . . u 1 if . . . 0 ow. subtree of u . P v path r v LP max v x w 1 v leaves u . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program V = nodes \ { r } { t = 1 x u = t = 2 V t = layers ≤ t t = 3 t = 4 t = 5 t = L s . t . ∑ V t x w ≤ t ∀ t = 1 , . . . , L w ∈ x u ∈ { 0 , 1 } ∀ u ∈ V

  44. w P 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u . . u . u 1 if . 0 . ow. P v path r v LP v x w 1 v leaves . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program V = nodes \ { r } { t = 1 x u = t = 2 V t = layers ≤ t t = 3 t = 4 c u = | subtree of u | t = 5 t = L max ∑ u ∈ V c u x u s . t . ∑ V t x w ≤ t ∀ t = 1 , . . . , L w ∈ x u ∈ { 0 , 1 } ∀ u ∈ V

  45. 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . u . . u u . . v 1 if . . . 0 ow. LP . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program V = nodes \ { r } { t = 1 x u = t = 2 V t = layers ≤ t t = 3 t = 4 c u = | subtree of u | t = 5 P v = path r → v t = L max ∑ u ∈ V c u x u s . t . ∑ v x w ≤ 1 ∀ v ∈ leaves w ∈ P ∑ V t x w ≤ t ∀ t = 1 , . . . , L w ∈ x u ∈ { 0 , 1 } ∀ u ∈ V

  46. 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . 1 if . . . 0 ow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program V = nodes \ { r } { t = 1 x u = t = 2 V t = layers ≤ t t = 3 t = 4 c u = | subtree of u | t = 5 P v = path r → v t = L max ∑ u ∈ V c u x u s . t . ∑ v x w ≤ 1 ∀ v ∈ leaves w ∈ P ( LP ) ∑ V t x w ≤ t ∀ t = 1 , . . . , L w ∈ x u ∈ [ 0 , 1 ] ∀ u ∈ V

  47. 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 if . . . 0 ow. . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program V = nodes \ { r } { t = 1 x u = t = 2 V t = layers ≤ t t = 3 t = 4 c u = | subtree of u | t = 5 P v = path r → v t = L max ∑ u ∈ V c u x u s . t . ∑ v x w ≤ 1 ∀ v ∈ leaves w ∈ P ( LP ) ∑ V t x w ≤ t ∀ t = 1 , . . . , L w ∈ x u ∈ [ 0 , 1 ] ∀ u ∈ V

  48. 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 if . . . 0 ow. . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Linear Program V = nodes \ { r } { t = 1 x u = t = 2 V t = layers ≤ t t = 3 t = 4 c u = | subtree of u | t = 5 P v = path r → v t = L max ∑ u ∈ V c u x u s . t . ∑ v x w ≤ 1 ∀ v ∈ leaves w ∈ P ( LP ) ∑ V t x w ≤ t ∀ t = 1 , . . . , L w ∈ x u ∈ [ 0 , 1 ] ∀ u ∈ V

  49. u x w u x w . and discard fractions PLAN: reallocate . : . . A node u is loose w.r.t. x if . . Let x be a vertex sol. of LP . r . . . . . . . . 0 . . . . . . . • x u 12 • . 2. return integral part . . push fraction down to some do nothing . 1. for each . . . w P 1 w P • 0 • x u A node u is tight w.r.t. x if . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loose and Tight Nodes

  50. u x w u x w . and discard fractions PLAN: reallocate . : 0 . 0 . 12 . . . 0 . 0 . 0 . . . . . . r . . . . . . . . 0 A node u is loose w.r.t. x if 0 w P 2. return integral part . . push fraction down to some do nothing . 1. for each . . . . 1 • . 0 • x u A node u is tight w.r.t. x if . . 1 w P • 0 • x u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loose and Tight Nodes Let x ∗ be a vertex sol. of ( LP ) . . 25 . 75 . 25 . 5 . 25 . 25

  51. u x w u x w . and discard fractions PLAN: reallocate . : 0 . 0 . 12 . . . 0 . 0 . 0 . . . . . . r . . . . . . . . 0 . 0 1 2. return integral part . . push fraction down to some do nothing . 1. for each . . . . w P . • 0 • x u A node u is tight w.r.t. x if . . 1 w P • 0 • x u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loose and Tight Nodes Let x ∗ be a vertex sol. of ( LP ) . A node u is loose w.r.t. x ∗ if . 25 . 75 . 25 . 5 . 25 . 25

  52. u x w u x w . and discard fractions PLAN: reallocate . : . . . . . . r . . . . 12 . . . . . . . . . . . . . w P . . 2. return integral part . . push fraction down to some do nothing . 1. for each . . . • 1 w P • 0 • x u A node u is tight w.r.t. x if . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loose and Tight Nodes Let x ∗ be a vertex sol. of ( LP ) . A node u is loose w.r.t. x ∗ if . 25 • x ∗ u > 0 . 75 . 25 . 5 . 25 . 25

  53. u x w . and discard fractions PLAN: reallocate . : . . . . r . . . . . . 12 . . . . . . . . . . . . . . . . 2. return integral part . . push fraction down to some do nothing . 1. for each . . . . 1 w P • 0 • x u A node u is tight w.r.t. x if . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loose and Tight Nodes Let x ∗ be a vertex sol. of ( LP ) . A node u is loose w.r.t. x ∗ if . 25 • x ∗ u > 0 . 75 • ∑ u x w < 1 w ∈ P . 25 . 5 . 25 . 25

  54. u x w . and discard fractions PLAN: reallocate . : 12 . . . . . . 0 . 0 . 0 . . . r 0 . . . . . . . . . . . . . 1 2. return integral part . . push fraction down to some do nothing . 1. for each . . . . w P 0 • 0 • x u . . . . 0 . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loose and Tight Nodes Let x ∗ be a vertex sol. of ( LP ) . A node u is loose w.r.t. x ∗ if . 25 • x ∗ u > 0 . 75 • ∑ u x w < 1 w ∈ P . 25 A node u is tight w.r.t. x ∗ if . 5 . 25 . 25

  55. u x w . and discard fractions PLAN: reallocate . : 12 r . . . . . . . . . . . . . . . . . . . . . . . . . 2. return integral part . . push fraction down to some do nothing . 1. for each . . . . 1 w P • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loose and Tight Nodes Let x ∗ be a vertex sol. of ( LP ) . A node u is loose w.r.t. x ∗ if . 25 • x ∗ u > 0 . 75 • ∑ u x w < 1 w ∈ P . 25 A node u is tight w.r.t. x ∗ if . 5 . 25 • x ∗ u > 0 . 25

  56. . and discard fractions PLAN: reallocate . : . . . . . . . . . 12 . r . . . . . . . . . . . . . . 2. return integral part . . push fraction down to some do nothing . 1. for each . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loose and Tight Nodes Let x ∗ be a vertex sol. of ( LP ) . A node u is loose w.r.t. x ∗ if . 25 • x ∗ u > 0 . 75 • ∑ u x w < 1 w ∈ P . 25 A node u is tight w.r.t. x ∗ if . 5 . 25 • x ∗ u > 0 . 25 • ∑ u x w = 1 w ∈ P

  57. 12 . 0 . . . . . . . r . . . . 0 . . . . . . . . . . . . . . . 2. return integral part . . push fraction down to some do nothing . 1. for each . . . . . 0 . . . 0 . 0 . 0 . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loose and Tight Nodes Let x ∗ be a vertex sol. of ( LP ) . A node u is loose w.r.t. x ∗ if . 25 • x ∗ u > 0 . 75 • ∑ u x w < 1 w ∈ P . 25 A node u is tight w.r.t. x ∗ if . 5 . 25 • x ∗ u > 0 . 25 • ∑ u x w = 1 w ∈ P . and discard fractions PLAN: reallocate { . :

  58. . at most OPT . . have to become few . : Sparsity Lemma 13 We need to reduce L ! Number of loose nodes at most L (depth of tree). . . . Bound on number of . We want loss from reallocating • ad-hoc guessing • tree transformations and reallocation light! . . . PLAN - Reallocate Loose Nodes

  59. . have to become few . : Sparsity Lemma 13 . We need to reduce L ! Number of loose nodes at most L (depth of tree). . . . Bound on number of We want loss from reallocating • ad-hoc guessing • tree transformations and reallocation light! . . . PLAN - Reallocate Loose Nodes . at most ϵ OPT .

  60. . : Sparsity Lemma 13 . We need to reduce L ! Number of loose nodes at most L (depth of tree). . . . Bound on number of We want loss from reallocating • ad-hoc guessing • tree transformations and reallocation light! . . . PLAN - Reallocate Loose Nodes . at most ϵ OPT . . have to become few

  61. . : Sparsity Lemma 13 Bound on number of We need to reduce L ! Number of loose nodes at most L (depth of tree). . . . . • ad-hoc guessing We want loss from reallocating • tree transformations and reallocation light! . . . PLAN - Reallocate Loose Nodes . at most ϵ OPT . ⇐ . have to become few

  62. . : Sparsity Lemma 13 Bound on number of We need to reduce L ! Number of loose nodes at most L (depth of tree). . . . . • ad-hoc guessing We want loss from reallocating • tree transformations and reallocation light! . . . PLAN - Reallocate Loose Nodes . at most ϵ OPT . ⇐ . have to become few

  63. Sparsity Lemma 13 Bound on number of We need to reduce L ! Number of loose nodes at most L (depth of tree). . . . . • ad-hoc guessing We want loss from reallocating • tree transformations and reallocation light! . . . PLAN - Reallocate Loose Nodes . at most ϵ OPT . ⇐ . have to become few . :

  64. We need to reduce L ! 13 We want loss from reallocating Number of loose nodes at most L (depth of tree). . . . . Bound on number of • ad-hoc guessing • tree transformations and reallocation light! . . . PLAN - Reallocate Loose Nodes . at most ϵ OPT . ⇐ . have to become few . : Sparsity Lemma

  65. 13 We want loss from reallocating We need to reduce L ! Number of loose nodes at most L (depth of tree). . . . . Bound on number of • ad-hoc guessing • tree transformations and reallocation light! . . . PLAN - Reallocate Loose Nodes . at most ϵ OPT . ⇐ . have to become few . : Sparsity Lemma

  66. • Poly-time transformation. 1 / 2 optimal value LP orig . 14 . . . . 1 . . . 2 . . . 1 . . . 4 . . 1 . . . . . 2 . . . 3 . . 6 15 . . . 16 . . . 1 . . . . . . 2 . . . 1 . . 1 4 . . . 1 . . . . . 1 . . 7 . . . 25 . . 1 . . . . 1 . . . 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 2 i , i log L layers. • Compressed tree has • Optimal value LP comp . . connect erase 0 1 l . . . . push down c u 4 . 2 . . . r . . . 1 . . . 3 . . 8 . . . . 9 . . . 11 . . 1 . . 1 1 . . . 1 . . . . . . . 3 . . . 1 . . 1 . . . . 1 . . . 10 . . 1 . . . . 12 . . . 1 . 7 . 1 1 . . . 3 . . . . . . . 5 . . . 1 . . . . . 10 1 . . . 9 . . . . . . . 23 . . . 28 . . . . 2 . . . . 1 . . . 1 . 7 . 3 . . . 6 . . . . . . . . 13 . . . 1 . . 1 . . . . 2 . . . 4 . . 12 . 1 2 . . . 1 . . . . . . . 3 . . . 6 . . 1 . . . . 18 . 1 . . . 1 . . . 2 . . . 1 . . 22 . 4 . . . . . . . 1 19 . . . . . . 1 . . . 2 . Compression - Reducing Depth L t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8

  67. • Poly-time transformation. 1 / 2 optimal value LP orig . 14 2 1 . . . 1 . . . . . . . 1 . . . 4 . . 1 . . . . . 2 . . . 3 . . . 15 . . . 16 . . . . . 6 . . . . 2 . . . 1 . . . 4 . . . 1 . . 1 . . . . . 7 . . . 25 . . . 1 . . . 1 . . . 2 1 . . . . . . . . . . . . . . . . . . . . . . . . 2 i , i log L layers. • Compressed tree has • Optimal value LP comp . . connect erase 0 1 l . . . . push down c u 4 . 2 . . . r 1 . . 1 . . . 3 . . 8 . . . . 9 . . . 11 . 1 . . 1 1 . . . 1 . . . . . . . 3 . . . 1 . . . . . . . 1 . . . 10 . . 1 . . . . 12 . . . 1 . 7 . 1 1 . . . 3 . . . . . . . 5 . . . 1 . . . . . 10 1 . . . 9 . . . . . . . 23 . . . 28 . . . . 2 . . . . 1 . . . 1 . 7 . 3 . . . 6 . . . . . . . . . 13 . . . 1 . . 12 1 . . . 2 . . . 4 . . . . . 2 . . . 1 . . 1 . . . . 3 . . . 6 . 1 . . 1 22 . . . 1 . . . . . . . 2 . . . 1 . . . 4 . . 18 . . . . 1 . 19 2 . . . . . 1 . . . . Compression - Reducing Depth L t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8

  68. • Poly-time transformation. 1 / 2 optimal value LP orig . 14 2 1 . . . 1 . . . . . . . 1 . . . 4 . . 1 . . . . . 2 . . . 3 . . . 15 . . . 16 . . . . . 6 . . . . 2 . . . 1 . . . 4 . . . 1 . . 1 . . . . . 7 . . . 25 . . . 1 . . . 1 . . . 2 1 . . . . . . . . . . . . . . . . . . . . . . . . 2 i , i log L layers. • Compressed tree has • Optimal value LP comp . . connect erase 0 1 l . . . . push down c u 4 . 2 . . . r 1 . . 1 . . . 3 . . 8 . . . . 9 . . . 11 . 1 . . 1 1 . . . 1 . . . . . . . 3 . . . 1 . . . . . . . 1 . . . 10 . . 1 . . . . 12 . . . 1 . 7 . 1 1 . . . 3 . . . . . . . 5 . . . 1 . . . . . 10 1 . . . 9 . . . . . . . 23 . . . 28 . . . . 2 . . . . 1 . . . 1 . 7 . 3 . . . 6 . . . . . . . . . 13 . . . 1 . . 12 1 . . . 2 . . . 4 . . . . . 2 . . . 1 . . 1 . . . . 3 . . . 6 . 1 . . 1 22 . . . 1 . . . . . . . 2 . . . 1 . . . 4 . . 18 . . . . 1 . 19 2 . . . . . 1 . . . . Compression - Reducing Depth L t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7 t = 8

  69. • Poly-time transformation. 1 / 2 optimal value LP orig . 14 . . . 1 . . . 2 . 1 . 1 . . . 4 . . . . . . 15 2 . . . 3 . . . . . . . 16 . . . 1 . 6 . . . . 2 . . . 1 . . 4 . . . . 1 . . . 1 . 1 . 1 7 . . . 25 . . . . . . . 1 . . . 2 . . . 1 . . . . . . . . . . . . . . . . . . . r . . 1 . log L layers. • Compressed tree has • Optimal value LP comp . . connect erase . . push down . c u . . . . . . . . . . . . . . 3 . . . 8 . . 1 9 . . . 11 . . . . . . . . 1 . . . 1 . . 3 . . . . 1 . . . 1 . . . . 1 1 . . . 10 . . . . . . . 12 . . . 1 . . . . . . . 2 3 . . . 1 . 7 . 5 . . . 1 . . . . 2 . . . . . 9 . . . 10 . . . 23 . . . 28 . . 1 . . . . . 1 . . . 1 . . . 3 . . . 6 . . . 7 1 . . 1 13 . . . 1 . . . . . . . 2 . . . . . . . . . . . . 1 . . . 1 . 12 . 3 . . . 6 . . . . 4 18 . . . . 1 . . . 2 . . 1 . . . 4 . . . 1 . . . 1 . 1 . . 1 . . . 19 . . . . 2 . . . 22 . Compression - Reducing Depth L t = 1 t = 2 t = 3 l = 2 i , i = 0 , 1 ,... × 2 t = 4 t = 5 t = 6 t = 7 × 4 t = 8

  70. • Poly-time transformation. 1 / 2 optimal value LP orig . 14 . . . 1 . . . 2 . 1 . 1 . . . 4 . . . . . . 15 2 . . . 3 . . . . . . . 16 . . . 1 . 6 . . . . 2 . . . 1 . . 4 . . . . 1 . . . 1 . 1 . 1 7 . . . 25 . . . . . . . 1 . . . 2 . . . 1 . . . . . . . . . . . . . . . . . . . r . . 1 . log L layers. • Compressed tree has • Optimal value LP comp . . connect erase . . push down . c u . . . . . . . . . . . . . . 3 . . . 8 . . 1 9 . . . 11 . . . . . . . . 1 . . . 1 . . 3 . . . . 1 . . . 1 . . . . 1 1 . . . 10 . . . . . . . 12 . . . 1 . . . . . . . 2 3 . . . 1 . 7 . 5 . . . 1 . . . . 2 . . . . . 9 . . . 10 . . . 23 . . . 28 . . 1 . . . . . 1 . . . 1 . . . 3 . . . 6 . . . 7 1 . . 1 13 . . . 1 . . . . . . . 2 . . . . . . . . . . . . 1 . . . 1 . 12 . 3 . . . 6 . . . . 4 18 . . . . 1 . . . 2 . . 1 . . . 4 . . . 1 . . . 1 . 1 . . 1 . . . 19 . . . . 2 . . . 22 . Compression - Reducing Depth L t = 1 t = 2 t = 3 l = 2 i , i = 0 , 1 ,... × 2 t = 4 t = 5 t = 6 t = 7 × 4 t = 8

  71. • Poly-time transformation. 1 / 2 optimal value LP orig . 14 . . . 1 . . . 2 . 1 . 1 . . . 4 . . . . . . 15 2 . . . 3 . . . . . . . 16 . . . 1 . 6 . . . . 2 . . . 1 . . 4 . . . . 1 . . . 1 . 1 . 1 7 . . . 25 . . . . . . . 1 . . . 2 . . . 1 . . . . . . . . . . . . . . . . . . . r . . 1 . log L layers. • Compressed tree has • Optimal value LP comp . . connect erase . . push down . c u . . . . . . . . . . . . . . 3 . . . 8 . . 1 9 . . . 11 . . . . . . . . 1 . . . 1 . . 3 . . . . 1 . . . 1 . . . . 1 1 . . . 10 . . . . . . . 12 . . . 1 . . . . . . . 2 3 . . . 1 . 7 . 5 . . . 1 . . . . 2 . . . . . 9 . . . 10 . . . 23 . . . 28 . . 1 . . . . . 1 . . . 1 . . . 3 . . . 6 . . . 7 1 . . 1 13 . . . 1 . . . . . . . 2 . . . . . . . . . . . . 1 . . . 1 . 12 . 3 . . . 6 . . . . 4 18 . . . . 1 . . . 2 . . 1 . . . 4 . . . 1 . . . 1 . 1 . . 1 . . . 19 . . . . 2 . . . 22 . Compression - Reducing Depth L t = 1 t = 2 t = 3 l = 2 i , i = 0 , 1 ,... × 2 t = 4 t = 5 t = 6 t = 7 × 4 t = 8

  72. • Poly-time transformation. 14 . . 1 . . . 2 . . 1 . . . . 4 . . . 6 . 1 . . . . . 3 . . . 15 . . . 16 . . . 1 . . . 7 . 4 2 . . . 1 . . . . . . . 1 . . . 1 . . . . . . . . 25 . . . 1 . 1 . 1 . . . 2 . . . 2 . . . . . . . . . . . . . . . . . . . r . . . . . push down log L layers. • Compressed tree has . . connect erase . . . c u . . . . . . . . . 1 . . . . 3 . . . 8 . . 9 . . . . 11 . . . 1 . 1 . 3 1 . . . 1 . . . . . . . 1 . . . 1 . . . . 2 1 1 . . . 10 . . . . . . . 12 . . . 1 . . . . . . . . . . . . 1 . 7 . 5 . . . 1 . . . . 2 . . . . . 9 . . . 10 . . . 23 . . . 28 . . 1 . . . . . 1 . . . 1 . . . 3 . . . 6 . . . 7 1 3 . 1 1 . . . . . . 2 . . . . . . . 4 . . 1 . 18 . . . . . . 6 . . 12 . . . . 13 . . . 1 . . . . . . 2 . . . 1 . 4 1 . . . 1 . . . 1 . . . . . 19 . . . 1 . . 2 . . . . 22 . . . 1 . 3 Compression - Reducing Depth L t = 1 t = 2 t = 3 l = 2 i , i = 0 , 1 ,... × 2 t = 4 t = 5 t = 6 t = 7 × 4 t = 8 • Optimal value ( LP comp ) ≥ 1 / 2 optimal value ( LP orig ) .

  73. • Poly-time transformation. 14 . . . . 2 . . . 1 . . . 4 . . . 6 . . 1 . 7 . . 3 . . . 15 . . 16 . . . . 1 . . . 1 . . . . . . . 1 . . . 4 . . . 1 . . . 1 . . 2 . . . . 25 . . . 1 . . 1 . . . . 2 . . . 1 . . 2 . . . . . . . . . . . . . . . r . . 1 . . . . . . . connect erase . . . push down c u . . . . . . . . . . . 1 2 . . . . 8 . . . 9 . . . 11 . . . 1 . . 3 . . . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 . . . 1 1 . . . 10 . . . . . . . 12 . . . 1 . . . . . . . 3 . . . 1 . . 7 5 . . . 1 . . . . 2 1 . . . . 9 . . . 10 . . . 23 . . . 28 . . 1 . . . . . 1 . . . 1 . . . 3 . . . 6 . . . 7 . . . . . . 1 . . . 1 . . 13 2 . . . 4 . . . . . . . . 1 . . . 1 . . 3 . . . . 6 . . . 12 . 18 . . . . 1 . . . 2 . . . 1 . . 4 . . . 1 . . . 1 . . . 1 . . . 1 2 . . . . . . 19 22 . . . Compression - Reducing Depth L t = 1 t = 2 t = 3 l = 2 i , i = 0 , 1 ,... × 2 t = 4 t = 5 t = 6 t = 7 × 4 t = 8 • Optimal value ( LP comp ) ≥ 1 / 2 optimal value ( LP orig ) . • Compressed tree has ≈ log ( L ) layers.

  74. 14 . . . . 2 . . . 1 . . . 4 . . . 6 . . 1 . 7 . . 3 . . . 15 . . 16 . . . . 1 . . . 1 . . . . . . 1 . . . 4 . . 2 1 . . . 1 . . . . . . . . 25 . . . 1 . . 1 . . . . 2 . . . 1 . . 2 . . . . . . . . . . . . . . . . r . . 1 . . . . . . . connect erase . . . push down c u . . . . . . . . . . . 1 . . . . . 8 . . . 9 . . . 11 . . . 1 . . 3 . . . . . 1 . . . 3 . . . 1 . . . 1 . . . 1 2 . . 1 1 . . . 10 . . . . . . . 12 . . . 1 . . . . . . . 3 . . . 1 . . 7 5 . . . 1 . . . . 2 1 . . . . 9 . . . 10 . . . 23 . . . 28 . . 1 . . . . . 1 . . . 1 . . . 3 . . . 6 . . . 7 . . . . . . 1 . . . 1 . . 13 2 . . . 4 . . . . . . 3 1 . . . 1 . . . . . . . 6 . . . 12 . 18 . . . . 1 . . . 2 . . . 1 . . 4 . . . 1 . . . 1 . . . 1 . . . 2 1 . . . . . . 22 19 . . . Compression - Reducing Depth L t = 1 t = 2 t = 3 l = 2 i , i = 0 , 1 ,... × 2 t = 4 t = 5 t = 6 t = 7 × 4 t = 8 • Optimal value ( LP comp ) ≥ 1 / 2 optimal value ( LP orig ) . • Compressed tree has ≈ log ( L ) layers. • Poly-time transformation.

  75. n , n Lemma optimal value LP orig . • Compressed tree has O log L / 15 • Poly-time transformation. layers. 1 • Optimal value LP comp . . . Previously, we selected l 0 1 ? , 0 1 1 What if we select l 0 1 2 i , i δ -compression

  76. n , n Lemma optimal value LP orig . • Compressed tree has O log L / . • Poly-time transformation. layers. 1 • Optimal value LP comp 15 . . 0 1 ? , 0 1 1 What if we select l δ -compression Previously, we selected l = 2 i , i = 0 , 1 , . . .

  77. Lemma optimal value LP orig . • Compressed tree has O log L / . . . • Optimal value LP comp 1 layers. • Poly-time transformation. 15 δ -compression Previously, we selected l = 2 i , i = 0 , 1 , . . . What if we select l = ⌊ ( 1 + δ ) n ⌋ , n = 0 , 1 , . . . , δ ∈ ] 0 , 1 [ ?

  78. • Compressed tree has O log L / . . . layers. • Poly-time transformation. 15 δ -compression Previously, we selected l = 2 i , i = 0 , 1 , . . . What if we select l = ⌊ ( 1 + δ ) n ⌋ , n = 0 , 1 , . . . , δ ∈ ] 0 , 1 [ ? Lemma • Optimal value ( LP comp ) ≥ ( 1 − δ ) optimal value ( LP orig ) .

  79. . . . layers. • Poly-time transformation. 15 δ -compression Previously, we selected l = 2 i , i = 0 , 1 , . . . What if we select l = ⌊ ( 1 + δ ) n ⌋ , n = 0 , 1 , . . . , δ ∈ ] 0 , 1 [ ? Lemma • Optimal value ( LP comp ) ≥ ( 1 − δ ) optimal value ( LP orig ) . ( log L / δ ) • Compressed tree has O

  80. . . . layers. 15 δ -compression Previously, we selected l = 2 i , i = 0 , 1 , . . . What if we select l = ⌊ ( 1 + δ ) n ⌋ , n = 0 , 1 , . . . , δ ∈ ] 0 , 1 [ ? Lemma • Optimal value ( LP comp ) ≥ ( 1 − δ ) optimal value ( LP orig ) . ( log L / δ ) • Compressed tree has O • Poly-time transformation.

  81. 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . loose 2 . 0 . 1 . . 1 . . 8 . tight . . loss . critical . . . . . . . . . . . . . . . . . . . . . . . r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Nodes - Bounding Loss From Singular Reallocation . 9 . 8 . 1 . 1 . 2

  82. and discard fractions 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tight . . loss . critical . . . . . . . 1. for each . do nothing push fraction down to some . . 2. return integral part loose 1 . . . . . . . . . . r . . . . . . 8 . 2 . 0 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Nodes - Bounding Loss From Singular Reallocation . 9 . 8 . 1 . 1 . 2 PLAN: reallocate { . :

  83. and discard fractions 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tight . . loss . critical . . . . . . . 1. for each . do nothing push fraction down to some . . 2. return integral part loose 1 . . . . . . . . . . r . . . . . . 8 . 2 . 0 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Nodes - Bounding Loss From Singular Reallocation . 9 . 8 . 1 . 1 . 2 PLAN: reallocate { . :

  84. and discard fractions 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tight . . loss . critical . . . . . . . 1. for each . do nothing push fraction down to some . . 2. return integral part loose 1 . . . . . . . . . . r . . . . . . 8 . 2 . 0 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Nodes - Bounding Loss From Singular Reallocation . 9 . 8 . 1 . 1 . 2 PLAN: reallocate { . :

  85. and discard fractions 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tight . . loss . critical . . . . . . . 1. for each . do nothing push fraction down to some . . 2. return integral part loose . . . . . . . . . r . 8 . 2 9 1 . 1 . 1 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Nodes - Bounding Loss From Singular Reallocation . 8 . 1 . 2 PLAN: reallocate { . :

  86. 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . loose . . tight . . loss . critical . . . . . . 1. for each . do nothing push fraction down to some . . 2. return integral part . 1 . 2 . . . . . . . r . 8 . . . 9 . 1 . 1 . . . 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Nodes - Bounding Loss From Singular Reallocation . 8 . 1 . 2 . and discard fractions PLAN: reallocate { . :

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