Maximum Matching Width: new characterizations and a fast algorithm for dominating set
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1 Maxmum Matcn Wdt: nw caractrzatons and a ast alortm or domnatn st 정지수 (KAIST) jont work wt Sv Hortmo Sætr and Jan Arn Tll (Unvrsty o Brn) 1st Koran Worksop on Grap Tory KAIST
2 Grap wdt paramtrs tr-wdt (Haln 1976, Robrtson and Symour 1984) branc-wdt (Robrtson and Symour 1991) carvn-wdt (Symour and Tomas 1994) clqu-wdt (Courcll and Olaru 2000) rank-wdt (Oum and Symour 2006) boolan-wdt (Bu-Xuan, Tll, Vatsll 2011) maxmum matcn-wdt (Vatsll 2012)
3 Tr-wdt tr-wdt (Haln 1976, Robrtson and Symour 1984) A masur o ow tr-lk t rap s. tr tr-lk Furs rom ttp://ptscool.mmuw.du.pl/slds/lc6.pd
4 Tr-wdt tr-wdt (Haln 1976, Robrtson and Symour 1984) A masur o ow tr-lk t rap s. tr tr-lk Furs rom ttp://ptscool.mmuw.du.pl/slds/lc6.pd
5 A tr-dcomposton o a rap GG s a par (TT, XX tt tt VV TT ) consstn o a tr TT and a amly XX tt tt VV TT o substs XX tt o VV GG, calld bas, satsyn t ollown tr condtons: 1. ac vrtx o GG s n at last on ba, 2. or ac d uuuu o GG, tr xsts a ba tat contans bot uu and vv, 3. XX and XX jj bot contan a vrtx vv, tn all bas XX kk n t pat btwn XX and XX jj contan vv as wll. a b c j d a j b j b b d b c d j
6 T wdt o a tr-dcomposton (TT, XX tt tt VV TT ) s max XX tt 1. T tr-wdt o a rap GG, dnotd by tttt(gg), s t mnmum wdt ovr all possbl tr-dcompostons o G. a b c d j a b j j b j b c d d b
7 Exampls tr-wdt 1 a orst no cycl tr-wdt 2 a srs-paralll rap no KK 4 mnor Furs rom wkpda
8 Exampls tr-wdt 1 a orst no cycl tr-wdt 2 a srs-paralll rap no KK 4 mnor T tr-wdt o a kk kk rd s kk. T tr-wdt o KK nn s nn rd KK 5
9 by Fdor V. Fomn (ttp://ptscool.mmuw.du.pl/s lds/lc6.pd)
10 Courcll s torm Torm (Courcll 1990) Lt PP b a proprty tat can b xprssd n MMMMMM 2 loc. I GG s a rap o boundd tr-wdt, tn tr xsts a lnar-tm alortm tat tsts wtr GG as proprty PP... 3-COLORING, HAMILTONICITY, SUBGRAPH ISOMORPHISM, MINOR TEST, VERTEX COVER, DOMINATING SET, INDEPENDENT SET, STEINER TREE, FEEDBACK VERTEX SET
11 Excludd Grd Torm T tr-wdt o a kk kk rd s kk. I a rap contans a lar rd as a mnor, tn ts tr-wdt s also lar. Torm (Robrtson, Symour, Tomas 1994) I t tr-wdt o a rap GG s at last 20 2kk5, tn GG as a kk kk rd as a mnor. Torm (Robrtson, Symour, Tomas 1994) I t tr-wdt o a planar rap GG s at last 6kk 4, tn GG as a kk kk rd as a mnor.
12 Grap wdt paramtrs tr-wdt (Haln 1976, Robrtson and Symour 1984) branc-wdt (Robrtson and Symour 1991) carvn-wdt (Symour and Tomas 1994) clqu-wdt (Courcll and Olaru 2000) rank-wdt (Oum and Symour 2006) boolan-wdt (Bu-Xuan, Tll, Vatsll 2011) maxmum matcn-wdt (Vatsll 2012)
13 A branc-dcomposton TT, LL ovr t vrtcs o a rap GG conssts o a tr TT wr all ntrnal vrtcs av dr 3 and a bjctv uncton LL rom t lavs o TT to t vrtcs o GG. T valu o an d ㅅ o TT s t sz o t maxmum matcn o GG[ aa, bb,, cc, dd,,,, ]. a b c d a b c d a b c d ㅅ GG TT, LL GG[ aa, bb,, cc, dd,,,, ]
14 A branc-dcomposton TT, LL ovr t vrtcs o a rap GG conssts o a tr TT wr all ntrnal vrtcs av dr 3 and a bjctv uncton LL rom t lavs o TT to t vrtcs o GG. T valu o an d ㅅ o TT s t sz o t maxmum matcn o GG[ aa, bb,, cc, dd,,,, ]. a b c d a b c d a b c d 2 GG TT, LL GG[ aa, bb,, cc, dd,,,, ]
15 A branc-dcomposton TT, LL ovr t vrtcs o a rap GG conssts o a tr TT wr all ntrnal vrtcs av dr 3 and a bjctv uncton LL rom t lavs o TT to t vrtcs o GG. T wdt o a branc-dcomposton (TT, LL) s t maxmum valu amon all ds. T maxmum matcn-wdt (mm-wdt, mmmmmm(gg)) o a rap GG s t mnmum wdt ovr all possbl branc-dcompostons ovr VV(GG). a b c d GG a b c d T wdt o (TT, LL) s TT, LL
16 Wy mm-wdt? Torm (Vatsll 2012) For vry rap GG, mmmmmm GG tttt GG mmmmmm GG. A rap GG as boundd tr-wdt and only GG as boundd mm-wdt.
17 Wy mm-wdt? W want to solv Grap Problms cntly. A Domnatn St o a rap GG s a st DD o vrtcs suc tat NN DD DD = VV(GG). Wat s t mnmum sz o a domnatn st o GG?
18 Wy mm-wdt? Usn tr-wdt Torm (van Rooj, Bodlandr, Rossmant 2009) Mnmum Domnatn St Problm can b solvd n tm OO (3 tt ) wn a rap and ts tr-dcomposton o wdt tt s vn. Torm (Lokstanov, Marx, Saurab 2011) Mnmum Domnatn St Problm cannot b solvd n tm OO ((3 εε) tt ) wr tt s t tr-wdt o t vn rap.
19 Wy mm-wdt? Usn mm-wdt Torm (J., Sætr, Tll 2015+) Mnmum Domnatn St Problm can b solvd n tm OO (8 mm ) wn a rap and ts branc-dcomposton o mmwdt mm s vn.
20 Wy mm-wdt? Usn tr-wdt: OO (3 tt ) Usn mm-wdt: OO (8 mm ) Our alortm s astr wn 8 mm < 3 tt, tat s, mmmmmm GG < tttt GG. Not tat or vry rap GG, mmmmmm GG tttt GG mmmmmm GG.
21 Wy mm-wdt? Usn mm-wdt Torm (J., Sætr, Tll 2015+) Mnmum Domnatn St Problm can b solvd n tm OO (8 mm ) wn a rap and ts branc-dcomposton o mmwdt mm s vn. Proo das 1. Nw caractrzaton o raps o mm-wdt at most kk 2. Fast Subst Convoluton
22 Nw caractrzaton (tr-wdt) a b c a b j b c d j d b j b d j rap tr-dcomposton
23 Nw caractrzaton (tr-wdt) For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as tr-wdt at most kk and only tr ar subtrs TT 1, TT 2,, TT nn o a tr TT wr all ntrnal vrtcs av dr 3 suc tat 1) vv vv jj EE(GG), tn TT and TT jj av at last on vrtx o TT n common, 2) or ac vrtx o TT, tr ar at most kk 1 subtrs contann t.
24 Nw caractrzaton (mm-wdt) a b c d a 1 1 b c d
25 Nw caractrzaton (mm-wdt) a a b c d 1 Torm (Kön 1931) 1 b c d For vry bpartt rap GG, t sz o a maxmum matcn s qual to t sz o a mnmum vrtx covr
26 Nw caractrzaton (mm-wdt) a b c d a a b c d b c d a b c d
27 Nw caractrzaton (mm-wdt) a b c d a b c d a b c d TT
28 Nw caractrzaton (mm-wdt) For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as mm-wdt at most kk and only tr ar subtrs TT 1, TT 2,, TT nn o a tr TT wr all ntrnal vrtcs av dr 3 suc tat 1) vv vv jj EE GG, tn TT and TT jj av at last on vrtx o TT n common, 2) or ac d o TT, tr ar at most kk subtrs contann t.
29 Torm (J., Sætr, Tll 2015+) For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as mm-wdt at most kk and only tr ar subtrs TT 1, TT 2,, TT nn o a tr TT wr all ntrnal vrtcs av dr 3 suc tat 1) vv vv jj EE GG, tn TT and TT jj av at last on vrtx o TT n common, 2) or ac d o TT, tr ar at most kk subtrs contann t.
30 Nw caractrzaton For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as tr-wdt (mm-wdt) at most kk and only tr ar subtrs TT 1, TT 2,, TT nn o a tr TT wr all ntrnal vrtcs av dr 3 suc tat 1) vv vv jj EE(GG), tn TT and TT jj av at last on vrtx o TT n common, 2) or ac vrtx (d) o TT, tr ar at most kk 1 (at most kk) subtrs contann t. Tank you
31 Nw caractrzaton (branc-wdt) For any kk 2, a rap GG on vrtcs vv 1, vv 2,, vv nn as branc-wdt at most kk and only tr ar a tr TT o max dr at most 3 and subtrs TT 1, TT 2,, TT nn suc tat 1) vv vv jj EE(GG) tn TT and TT jj av at last on d o TT n common, 2) or ac d o TT, tr ar at most kk subtrs usn t.
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