3-Colorability. CSE 589 Applied Algorithms Spring The Gadget. 3-CNF-Sat < P 3-Color. Reduction by Example. Properties of the Gadget
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1 3-Cololt CSE 589 Appled Alothms Sp 999 put: Gph G VE d ume. utput: Deteme f ll vetces c e coloed wth 3 colos such tht o two djcet vetces hve the sme colo. 3-Cololt Bch d Boud 3-colole ot 3-colole CSE Lectue 5 - Sp CF-St < P 3-Colo Gve 3-CF fomul F we hve to show how to costuct poloml tme ph G such tht: F s stsfle mples G s 3-colole G s 3-colole mples F s stsfle he Gdet hs s clssc educto tht uses det. Assume the oute vetces e coloed t most two colos. he det s 3-colole f d ol f the oute vetces e ot ll the sme colo. CSE Lectue 5 - Sp CSE Lectue 5 - Sp Popetes of the Gdet hee colole f d ol f oute vetces ot ll the sme colo. educto Emple F ot 3 colole s 3 colole CSE Lectue 5 - Sp CSE Lectue 5 - Sp 999 6
2 CSE Lectue 5 - Sp Stsfcto Emple F CSE Lectue 5 - Sp Stsfcto Emple F CSE Lectue 5 - Sp o-stsfcto Emple F CSE Lectue 5 - Sp 999 m the Gdet CSE Lectue 5 - Sp 999 Geel Costucto 3 F { j whee E V G whee : { { : { { : { { { { { { { { { { { : { { { 3 E V CSE Lectue 5 - Sp 999 eductos CF-St 3-CF-St Clque 3-Colo Ect Cove Suset Sum 3-Ptto B Pc
3 Ect Cove put: A set { u u u d susets S S S m utput: Deteme f thee s set of pwse dsjot set tht uo to tht s set X such tht: X { j X d X S m j mples S S φ CSE Lectue 5 - Sp j Emple of Ect Cove { c d e f h { c e{ f { d{ f h{ e h { f h { d Ect Cove { c e{ f h{ d CSE Lectue 5 - Sp Ptto put: A set of umes A { 3m d ume B wth the popetes tht B/4 < < B/ d 3m mb. utput: Deteme f A c e pttoed to S S S m such tht fo ll j S j B. ote: ech S must cot ectl 3 elemets. CSE Lectue 5 - Sp Emple of 3-Ptto A { B m 3 3-Ptto CSE Lectue 5 - Sp B Pc put: A set of umes A { 3m d umes B cpct d ume of s. utput: Deteme f A c e pttoed to S S S such tht fo ll j S j B. B Pc Emple A { B 4 B Pc Pefect ft! 4 4 CSE Lectue 5 - Sp CSE Lectue 5 - Sp
4 Cop wth P-Completeess Gve polem ppes to e hd wht do ou do? to fd ood lothm fo t. to show ts decso veso s P-complete o P-hd. Fl oth the polem pol s hd oe. Fo hd polem thee e m ths to t. Bch-d-oud lothm - fo ect soluto Appomte lothm - heustc Lod Blced Sp ee Cost Cte Gve ph G VE d sp tee. d m deee of vete of c sum of the sques of the deees d 3 c 4* + *4 + *9 6 Advte of c s tht t hs fe dtos. CSE Lectue 5 - Sp CSE Lectue 5 - Sp 999 Dev c Eve sp tee o vetces hs - edes. Hece the vee ume of edes pe vete s d -/ out. Let d e the deee of vete. he vce deee s d d / d d / Mm the vce s equvlet to mm d CSE Lectue 5 - Sp 999 Emples of c c 9* + *9 9 c * + 8* 34 CSE Lectue 5 - Sp 999 Aothe Emple Lod Blced Sp ee wth Mmum Vce put: dected ph G VE. uput: A sp tee tht mmes the sum of the sques of the deees of the vetces the tee. c 3* + 3*4 + *9 4 c * + 5*4 CSE Lectue 5 - Sp CSE Lectue 5 - Sp
5 Bch d Boud Stt wth tl tee wth cost c. Sstemtcll sech thouh ll foests ecusvel ch dd ew edes to the cuet foest. Dscotue sech f the foest cot e coted sp tee of smlle cost. hs s the oud step. hs s ette th ehustve sech ut t s stll ol vlule o ve smll polems. Emple of Bch d Boud tl cost CSE Lectue 5 - Sp CSE Lectue 5 - Sp Boud Codto Gphc of Boud Codto Let cf e the cost of the cuet foest of tees whee tee hd mmum deee vete d soted smllest to lest. Let B e the est cost of tee so f. he lowest possle cost of tee cot F s m F c F + d + d + d + d d d 3 d d 4 d 5 f mf > B the do ot cotue sech fom F. CSE Lectue 5 - Sp d < d < d 3 < d 4 < d 5 CSE Lectue 5 - Sp Emple of Boud F d cf * + 8* + *6 4 mf 4 + * + *4 - * + * + * +*4 - * + * 36 CSE Lectue 5 - Sp Bch d Boud Cotol he edes of G e E[..m] F s set of dces of edes tll empt hee s tl Best-ee wth Best-Cost LBS-SechF f F s tee the f cf < Best-Cost the Best-ee : F; Best-Cost : cf; else {F s ot tee fo lst-de-f + to m do f otcclef d mf < Best-Cost the F : uof; LBS-SechF; CSE Lectue 5 - Sp
6 otes o Bch d Boud Bch d oud s stll epoetl sech. o me t wo well m effceces should e mde. Elmte cop of the ptl soluto F o the ecusve cll. Mt cost of ptl soluto F d ts sequece of mmum deees to me computto of mf fst. se up tee fo ccle chec. educe use of epesve oud checs whe possle. Add moe oud checs CSE Lectue 5 - Sp
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