MITSUBISHI ELECTRIC RESEARCH LABORATORIES Cambridge, Massachusetts. Introduction to Matroids and Applications. Srikumar Ramalingam

Transcription

1 Cmrige, Msshusetts Introution to Mtrois n Applitions Srikumr Rmlingm MERL mm//yy

2 Liner Alger (,0,0) (0,,0) Liner inepenene in vetors: v, v2,..., For ll non-trivil we hve s v s v n s, s2,..., s n 2v2... s n vn 0. (0,0,) (0,,) Ientify susets of linerly inepenent vetors. MERL mm//yy 2

3 Grph Theory MITSUBISHI ELECTRIC RESEARCH LABORATORIES Choose susets of eges without yles (lso known s forests olletion of trees). MERL mm//yy 3

4 Assignment of Jos Person tkes Jo. Every jo hs only opening. 2 3 Applints Jos Ientify possile ssignments etween pplints n jos D E D E D E MERL mm//yy 4

5 Buget onstrints E { } E {,, }. 2 2 Ientify susets of the set E (,,, ) suh tht mximum of element is tken from suset E { } n mximum of 2 elements re tken from suset E {,, } MERL mm//yy 5

6 Vertex isjoint pths S T Fin ll vertex isjoint pths from verties in S to verties in T. S T S T S T MERL mm//yy 6

7 Common properties All five prolems hve the sme solutions.,,,,,,,,,,, The size of the lrgest set: 3 The empty set is lwys solution. All susets of given solution is lso solution. All these senrios n e represente using mtrois! MERL mm//yy 7

8 Mtrois re everywhere, if only we knew how to look. MERL mm//yy 8

9 Mtroi Definition (introue y Whitney in 935) A mtroi is pir (E,Ι ) where E is finite set. Ι is fmily of susets of E suh tht: (I) Ι (I2) If A B n B Ι then A Ι. (I3) If A, B Ι n A B, then there exists e B suh tht ( A e) I. E is lle the groun set n olletion of inepenent sets. Ι is referre to s the MERL mm//yy 9

10 Grphi Mtroi Let G ( V, E ) e grph n let Ι e olletion of ege sets (susets of E) without yles, then (E,Ι ) is mtroi. E {,,, } Inepenent sets MERL

11 Mtroi Definition (using n Exmple) (I) is n inepenent set. (2) Sine is inepenent, then ll its suset {,,,,,, } re lso inepenent. (I3) If I, J Ι n I J there exists, e J suh tht ( I e) I.,,,,,,,,,, MERL mm//yy

12 Liner Mtroi Let E e finite set of vetors in vetor spe V n let Ι e the olletion of linerly epenent sets in E then (E,Ι ) is mtroi. (,0,0) (0,,0) (0,,) (0,0,) E {,,, } MERL

13 Prtition Mtroi Let E e finite set of vetors n let E, E2,..., E N e isjoint susets of E. Given N positive integers k,..., k, let Ι e the olletion of susets of eh suset hs tmost from (E,Ι ) is mtroi. N E k. i E i E, k E 2, k2 2 Inepenent sets MERL

14 Trnsversl Mtroi Let G e iprtite grph with iprtition ( D, E). Let Ι e the olletion of susets of E whih n e mthe to (E,Ι ) D. is mtroi. 2 3 D E D E D E D E MERL mm//yy 4

15 Bsis of Mtroi MITSUBISHI ELECTRIC RESEARCH LABORATORIES A sis of mtroi is mximl inepenent set. Exmple: Bses:,, All the ses of mtroi hve the sme size. MERL mm//yy 5

16 Rnk Let (E,Ι ) e mtroi. The rnk of suset of E is given y the size of the lrgest inepenent set ontine in it. The rnk of set A, A E : rnk(a) rgmx B A,B I B rnk({, rnk({,,,}) }) 2 3 rnk({,,}) 2 MERL mm//yy 6

17 Dul of mtroi is mtroi If the ul mtroi of M (E,Ι ) * * * is M ( E,Ι ) n A Ι then E A hs se of M (E,Ι ). M (E,Ι ), * M ( E,Ι ) *,,,,,,,,,,,,,,, MERL mm//yy 7

18 Greey Algorithm Given mtroi (E,Ι ) n weights w : E R, fin sis of minimum weight.. Strt with A {}. 2. A to A the smllest e s.t A e I. 3. Repet until you hve sis. (Greey lgorithm gurntees n optiml soln.) (The unerlying struture is mtroi) MERL mm//yy 8

19 Greey Algorithm Given mtroi (E,Ι ) n weights w : E R, fin sis of minimum weight. min A E s. t. e A i A A I, M w e i i rnk( E) p ( E, I) Miniml spnning lgorithm is very simple n useful! MERL mm//yy 9

20 Mtrois in other omins physil relizility They lso pper in severl geometry prolems: rrngements of hyperplnes, onfigurtions of points, et. Sugihr s pproh lifts line rwings to 3D spe for triherl rwings. Chek whether line rwing is physilly relizle or not. The lifting proeure where the 3D points re ompute on the projetion rys stisfying ll the onstrints from projetions n line lels. For generl line rwings, Whiteley extene Sugihr s work using mtrois in 989. MERL mm//yy 20

21 Line-Lifting Given single imge pture y your moile phone or other evies: Imge Re, Blue n Green enote lines in orthogonl iretions VRML moel of the line reonstrution [Rmlingm n Brn, 203] MERL mm//yy 2

22 Our Min Ie G H C F D Orthogonl input lines from rel imge B E A 9 intersetions y onneting nery lines. H is wrong intersetion All intersetions n not simultneously stisfy mer projetion, orthogonlity n prllelism onstrints F E G D A 7 intersetions when we remove H B C F E Only 6 intersetions when we inlue H Our min ssumption is tht the mximum rinlity suset tht stisfies ll the onstrints will onsist of orret intersetions. MERL mm//yy 22 D A H B F E G D A H B C

23 Miniml Spnning Tree (MST) for line-lifting ll intersetions Intersetions in the MST Two perspetive views of the line reonstrution Using gp etween the line segments s the ege osts, we ompute MST to ientify the lest numer of onstrints to lift the lines to 3D spe. MERL mm//yy 23

24 Qulittive Evlution imge etete lines Two perspetive views of the line reonstrution MERL mm//yy 24

25 Greey Algorithms for sumoulr ojetive funtions uner mtroi onstrint We n lso fin suset tht mximizes sumoulr funtion uner the onstrint tht the suset is n inepenent set of mtroi. The solution omes with some optimlity gurntees. sumoulr [Nemhuser, Fisher & Wolsey 78] MERL mm//yy 25

26 Mximize monotoni sumoulr funtions uner one or more mtrois Theorem: For monotoni sumoulr funtions, greey lgorithm gives onstnt ftor pproximtion ( p ) F( A ) 2 F( greey A opt Greey gives over intersetion of p mtrois. ) MERL [Nemhuser, Fisher & Wolsey 78] 26

27 Exmple: Cmer network Groun set Configurtion: Sensing qulity moel Configurtion is fesile if no mer is pointe in two iretions t one MERL Slie ourtesy: Kruse 27

28 Exmple: Cmer network Groun set Configurtion: Sensing qulity moel Configurtion is fesile if no mer is pointe in two iretions t one This is prtition mtroi: Inepenene: MERL Slie ourtesy: Kruse 28

29 Greey lgorithm for mtrois: Given: finite set V Wnt: suh tht Greey lgorithm: Strt with While MERL Slie ourtesy: Kruse 29

30 Superpixel Segmenttion Pixel representtion Superpixel representtion Suset Seletion Prolem n optimiztion prolem on grph topology mx F( A ) sujet to n A results in K lusters Proues stte-of-the-rt results in superpixel segmenttion n lustering tsets. [Liu et l. 20, 203] MERL mm//yy 30

31 Referenes Oxley, Mtroi Theory, 20 (possily the est mteril, ut time-onsuming). The Coming of the Mtrois", Willim Cunninghm, 202. Welsh, Mtroi Theory, 975. Slies n Vieos: Jeff Bilmes, Sumoulr Funtions, Optimiztion, n Applitions to Mhine Lerning, 204. Feerio Aril, Mtroi Theory, MERL mm//yy 3