Grph width-prmeters nd lgorithms Jisu Jeong (KAIST) joint work with Sigve Hortemo Sæther nd Jn Arne Telle (University of Bergen) 2015 KMS Annul Meeting 2015.10.24. YONSEI UNIVERSITY
Grph width-prmeters tree-width (Hlin 1976, Roertson nd Seymour 1984) rnh-width (Roertson nd Seymour 1991) rving-width (Seymour nd Thoms 1994) lique-width (Courelle nd Olriu 2000) rnk-width (Oum nd Seymour 2006) oolen-width (Bui-Xun, Telle, Vtshelle 2011) mximum mthing-width (Vtshelle 2012)
Tree-width tree-width (Hlin 1976, Roertson nd Seymour 1984) A mesure of how tree-like the grph is. tree tree-like Figures from http://fptshool.mimuw.edu.pl/slides/le6.pdf
Tree-width tree-width (Hlin 1976, Roertson nd Seymour 1984) A mesure of how tree-like the grph is. tree tree-like Figures from http://fptshool.mimuw.edu.pl/slides/le6.pdf
Tree-width tree-width (Hlin 1976, Roertson nd Seymour 1984) A mesure of how tree-like the grph is. Figures from http://fptshool.mimuw.edu.pl/slides/le6.pdf
A tree-deomposition of grph GG is pir (TT, XX tt tt VV TT ) onsisting of tree TT nd fmily XX tt tt VV TT of susets XX tt of VV GG, lled gs, stisfying the following three onditions: 1. eh vertex of GG is in t lest one g, 2. for eh edge uuuu of GG, there exists g tht ontins oth uu nd vv, 3. if XX ii nd XX jj oth ontin vertex vv, then ll gs XX kk in the pth etween XX ii nd XX jj ontin vv s well. d i j h g e f d j j j h h e d i h h e g f
A tree-deomposition of grph GG is pir (TT, XX tt tt VV TT ) onsisting of tree TT nd fmily XX tt tt VV TT of susets XX tt of VV GG, lled gs, stisfying the following three onditions: 1. eh vertex of GG is in t lest one g, 2. for eh edge uuuu of GG, there exists g tht ontins oth uu nd vv, 3. if XX ii nd XX jj oth ontin vertex vv, then ll gs XX kk in the pth etween XX ii nd XX jj ontin vv s well. The width of tree-deomposition (TT, XX tt tt VV TT ) is mx XX tt 1. The tree-width of grph GG, denoted y tttt(gg), is the minimum width over ll possile tree-deompositions of G.
Exmples tree-width 1 forest no yle tree-width 2 series-prllel grph no KK 4 minor Figures from wikipedi
Exmples tree-width 1 forest no yle tree-width 2 series-prllel grph no KK 4 minor The tree-width of kk kk grid is kk. The tree-width of KK nn is nn 1. 4 4 grid KK 5
Algorithm using tree-deomposition Exerise Given tree-deomposition of width tt of grph GG, 3-COLORABILITY n e solved in time OO(tt3 tt nn).
Algorithm using tree-deomposition Exerise Given tree-deomposition of width tt of grph GG, 3-COLORABILITY n e solved in time OO(tt3 tt nn). d
d d
d 1 2 1 3 2 1 2 3 3 1 3 2
d 1 2 1 3 2 1 2 3 3 1 3 2 1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1
d 1 2 3 1 3 2 2 1 3 2 3 1 3 1 2 3 2 1
d 1 2 3 2 3 1 3 2 3 2 2 1 3 1 3 2 3 1 3 1 3 1 2 1 2 3 2 1 2 1
d 2 3 3 2 1 3 3 1 1 2 2 1 d 2 3 1 3 2 1 1 3 2 3 1 2 1 2 3 2 1 3
d d 1 2 1 2 3 2 3 2 3 1 1 3 1 3 2 3 2 3 2 1 2 1 2 1 3 1 3 1 3 2 2 3 2 3 1 3 1 3 1 2 3 1 3 1 2 1 2 1 2 3 3 2 3 2 1 2 1 2 1 3
The numer of olumns of tle is t most tt + 11. d 1 2 The numer of rows of 1 2 3 2 3 d 2 3 1 1 3 1 tle 3 2 is t 3most 2 33 tt+11. 3 2 1 2 1 2 1 3 1 3 1 3 2 2 3 The 2 3 numer 1 3of 1 tles 3 1 2 3 1 3 2 3 1 2 3 2 1 1 2 is t most OO(nn). 2 1 1 2 3 2 1 3
Algorithm using tree-deomposition Esy exerise Given tree-deomposition of width tt of grph GG, 3-COLORABILITY n e solved in time OO(tt3 tt nn). Diffiult exerise Given tree-deomposition of width tt of grph GG, Minimum Dominting Set Prolem n e solved in time OO(4 tt nn).
Algorithm using tree-deomposition Diffiult exerise Given tree-deomposition of width tt of grph GG, Minimum Dominting Set Prolem n e solved in time OO(4 tt nn). In dominting set D Dominted y D Not in D, ut do not hve to e dominted y D (will e dominted lter) TRUE / FALSE the size of D (if D is not dominting set, then )
Algorithm using tree-deomposition Theorem (vn Rooij, Bodlender, Rossmnith 2009) Minimum Dominting Set Prolem n e solved in time OO(3 tt nn) when grph nd its tree-deomposition of width tt is given. Theorem (Lokshtnov, Mrx, Surh 2011) Minimum Dominting Set Prolem nnot e solved in time OO((3 εε) tt nn) where tt is the tree-width of the given grph.
New width-prmeter Mximum mthing width (mmw) Theorem (Vtshelle 2012) For every grph GG, mmmmmm GG tttt GG + 1 3 mmmmmm GG. A grph GG hs ounded tree-width if nd only if GG hs ounded mm-width.
Algorithm using mmw Theorem (J., Sæther, Telle 2015) Minimum Dominting Set Prolem n e solved in time OO(8 mm nn) when grph nd its mm-deomposition of mm-width mm is given.
Using tree-width: OO(3 tt nn) Using mm-width: OO(8 mm nn) Our lgorithm is fster when 8 mm < 3 tt, tht is, 1.893 mmmmmm GG < tttt GG. Note tht for every grph GG, mmmmmm GG tttt GG + 1 3 mmmmmm GG.
Wht if only grph is n input? Theorem (Oum, Seymour 2006) Given grph GG, rnh-deomposition over VV(GG) of mmwidth t most 3mmmmmm GG + 1 n e found in time OO 2 3mmmmmm(GG). Runtime : OO 8 mm = OO (8 3mmmmmm(GG) ) Theorem (Amir 2010) Given grph GG, tree-deomposition over VV(GG) of width t most 3.67tttt GG n e found in time OO 2 3.7tttt(GG). Runtime : OO 3 tt = OO (3 3.67tttt(GG) )
Wht if only grph is n input? Theorem (Oum, Seymour 2006) Given grph GG, rnh-deomposition over VV(GG) of mmwidth t most 3mmmmmm GG + 1 n e found in time OO 2 3mmmmmm(GG). Runtime : OO 8 mm = OO (8 3mmmmmm(GG) ) Theorem (Amir 2010) Given grph GG, tree-deomposition over VV(GG) of width t most 3.67tttt GG n e found in time OO 2 3.7tttt(GG). Runtime : OO 3 tt = OO (3 3.67tttt(GG) ) Our lgorithm is fster if n input grph GG stisfies 1.55 mmmmmm GG < tttt GG.
Open questions Theorem (J., Sæther, Telle 2015) Minimum Dominting Set Prolem n e solved in time OO(8 mm nn) when grph nd its mm-deomposition of mm-width mm is given. Improve OO(8 mm nn) or show tht it is tight Other prolems Other width-prmeters
Theorem (vn Rooij, Bodlender, Rossmnith 2009) Minimum Dominting Set Prolem n e solved in time Using OO(3 tt nn) tree-width: when grph OO (3 nd tt ) its tree-deomposition of width tt is given. Theorem (J., Sæther, Telle 2015) Minimum Dominting Set Prolem n e solved in time OO(8 mm nn) when grph nd its mm-deomposition of mm-width mm is given. Our lgorithm is fster when 8 mm < 3 tt, tht is, 1.893 mmmmmm GG < tttt GG. Thnk you
New hrteriztion For ny kk 2, grph GG on verties vv 1, vv 2,, vv nn hs tree-width (mm-width, rnh-width) t most kk if nd only if there re sutrees TT 1, TT 2,, TT nn of tree TT where ll internl verties hve degree 3 suh tht 1) if vv ii vv jj EE(GG), then TT ii nd TT jj hve t lest one vertex (vertex, edge) of TT in ommon, 2) for eh vertex (edge, edge) of TT, there re t most kk 1 (t most kk, t most kk) sutrees ontining it. Pul, Telle, Edge-mximl grphs of rnhwidth k: the k-rnhes. 2009