Tiểu luận Thuật toán và phương pháp giải quyết vấn đề Bài toán người du lịch
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ĐẠI HỌC CÔNG NGHỆ THÔNG TIN
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### EFA
.ŠH|•$R&~š|eC˜~1
Chu trình (có hướng) Hamilton GDE|•$R&~S%-')FX.”.ŠH1
Đường đi (có hướng) Hamilton G.$B&.|•$R&~S%-')FX.”.ŠH1
$,!*XDEFG•.‚POz&S].”C,X.$B&.FG
•.‚POz&S].”D•01
ŠHŽFDE|•$R&~FG&XGđồ thị Hamilton.
Ví dụ1›V.ŠH
@#
ŠHDwJ&•.$B&.˜G}D,E".”&CC}CŒ•O!G…C$&•
DEFG
F→O→→P→}→Œ→&→F
##G HIJK
Định lí (S’.ŠH•DEFG1_.•
|~ŠHGw&1
|~X.”MF•O!GR%yOz&2C,•.W&FGw‚
‚DE1
|~+€•F.k.”O-Jœ^&"Gw‚W&CE.ŠH•G
S–•].Fk'LGw&1
Chứng minh
K./|~,|~G†w1K./|~S*DF•v+GJ€•F.J
.”O-Jœ^&"Gw‚W&EDEOHFDF/-
J'L1
Hệ quả
(S’.ŠHn.”•.$B&.FG1_.•
|~ŠHGw&1
|~•„-n−2.”O!≥2C,™.”•.W&FGw‚‚
.$B&.1
|~+€•F.k.”O-Jœ^&"Gw‚W&CE.ŠH•G
S–•].Fk•0'LGw&1
A
Ví dụ 11›V.ŠH
@G
ŠHJ&•DEFG1
!,!*C+ŠDEFGE•'(•Y1eEO!
P}&|,
2
~šP}&|,
A
~š7w'(‚R,
2
,‚R,
A
J&‚
DE1{]•GGAwJ&†•Y.$ƒCZx1
F•&•†"'P\&Dv+'HGUDw1+Ou.2.”,
2
,,
A
^&
"Gw‚W&E.ŠH•GG7.”.‚G!'C•7'LGw
&1$,!*}K./|~MFHGUDwE.ŠHJ&•DE
FG1
Ví dụ 21Ž&Dz&.ŠHSFJ&•.$B&.FG1
@$
(S’G.$B&.FG1ŠHDw•0[.”Cw.$B&.•0Y
|ž~1
‘Cách 11eE.ŠH•2NwS]J&zDw.$B&'(G02
|žž~1
›V".”‚!'šŸ0C2C7CACYC[CN 1X˜
Cš0C111CNCG!'ƒ'"
Gw‚.”∈J&zDw.$B&.1eEXy'.”D&J&J/
Fw"!'˜
•&DBF•&y'‚1$,!*S]Gw‚".”
MFJ&zDwG
Jš
( )
∑
=
N
0i
i
Ecard
eES]Gw‚0.”zDw.$B&./-G2w™!'
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0
C˜
2
C˜
7
C˜
N
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A
C˜
Y
C˜
[
•„-71eE,!*
Y
J≥Až0•7ž7š071
Zx,R|žž~1
‘Cách 21›V“.”•G1Z*G".”J&J/F•&y'‚C
D&.••„-N.”•2Gw‚∈,2.”•G•„-0
Gw‚∈1$,!*S]∈„-GNž2•2š0[CZx,R|ž~1
‘Cách 31›"N.”0C2C7CACYC[CN,"Gw‚E.ŠH•G
•“.”G!'CŽ“'LGw&1}K)(21A12|~C.ŠHJ&•
.$B&.FG1
Ví dụ 31|Bài toán xếp chỗ ngồi~“&$BO^&&Š•D&OD•AGL1™
GLX.$ƒ€+'&Š}‚Žv1—*F*.‹™&Š™GLSFJ&•
2&$B&Š&LF%0GL1
FG!'.ŠH“.”0C2C111C“C.””&$B1F.y.”0Z,"
.”•GDw.$B&D•$E,–1™"€+'G‚DEFGMF
.ŠH1
DEŽ-$E,–G
0→2→7→A→Y→[→N→•→“→0
@L
›F*GLG$ƒDE"&•π¡A}/J.Š&ŠF!.$ƒ"
DEC•&G""€+'CSF
@&
0→7→Y→2→N→A→“→[→•→0
[
@M
0→Y→N→7→“→2→•→A→[→0
@N
0→N→“→Y→•→7→[→2→A→0
##$ HIJ<O
Định lý 1ŠH.M_
,RnG…|n≥7~•|n−0~¡2DEFG•&.
‚J&&FF|ŽGJ&•&~1
Chứng minh
$%&v$GB&(O"€+'“&$BDwOD•CF€Z*Pv&"€+'
}DEFGDw.ŠHSF|š2J•0~
0→2→7→111→2J→2J•0→0
@P
N
›F*DEGLG$ƒ‚&•π¡J}/J.Š&ŠF!.$ƒJ
"€+'1
Định lý 2 (Dirac):G.%.ŠHn.”|n≥7~1+O!P}&|,~≥n¡2,RX
.”vMFCE•DEFG1
Định lý 3:G.ŠH.%n.”|n≥7~1+O!P}&|,~≥|−0~¡2,RX.”
,MFE•.$B&.FG1
Chứng minh
+š0E•.$B&FGL$B&G0.”1
(S’¢01FG!'.ŠHOz&"w,.”,,-("],
,RX.”MF1ŠH••0.”,P}&|,~š1eRX.”∈F•
P}&
|~šP}&
|~•0≥|−0~¡2•0š|•0~¡2
}.HGU?DFE•DEFG1u.,,"R,F
.$ƒ.$B&.FGD&1
Định lý 4:G.ŠH.%n.”|≥7~1(S’,,GF.”J&J/
FMFSF
P}&|~•P}&|,~≥1
_.••DEFGJ,”J.ŠH•|C,~|.ŠHw
|C,~~•DEFG1
Định lý 5:G.ŠH.%&(.”1(S’£,¤GF.ŠH.$ƒ
•Oz&)*']-(y'.”J&J/F•‹&"O!„-Oz&1
_.•F•£š¤1
•.HGUDwF•†.H&IFJ"Kbao đóngMF.ŠH1
Định nghĩa 6: Bao đóng|~MF.ŠHn.”G.ŠH.$ƒ•Oz&"C
})*'C]-("y'.”J&J/F‹&S]O!„-Oz&n
.+JJ&•y'.”$,!*hF1
Ví dụ 2.4.51"€Z*Pv&OF.•&1
@Q
Định lý 7:ŠH•DEFGJ,”JOF.•&MF•DE
FG1
Định lý 8: +OF.•&|~š_
|≥7~E.ŠH•DEFG1
Định lý 9RECS6?TUG.%.ŠH.”|≥7~1+P}&|~•P}&|,~≥
,RXy'.”J&J/FE.ŠH•DEFG1
Định lý 10:G.%.ŠH.”|≥7~,1+≥|−0C2~•2E
.ŠH•DEFG1
•
Định lý 11:.ŠH.%G.ŠHG$¥&'Z,RF!'.”e
0
,e
2
SF
FDP|e
0
~šFDP|e
2
~š≥2
+O!P}&|,~¢¡2,RX.”,MFCE•DEFG1
Đồ thị có hướng.
.ŠH•$R&,Rn.”1F•"J+)('"O†D&".H
GUSF
Định lý 12:|điều kiện đủ tồn tại chu trình có hướng Hamilton~
F1|}*}G~+.ŠHGw&,
P}&|~•P}&|,~≥2n−0∀C,∈J&J/F
E•DE•$R&FG1
O1|GF‘D~+.ŠHGw&,
P}&|,~≥n∀,∈
E•DE•$R&FG1
1|ˆPFGG~+
P}&
<
|~•P}&
|,~≥n∀C,∈J&Š&•.+,
E•DE•$R&FG1
P1+
P}&
|,~≥¡2#P}&
<
|,~≥¡2∀,∈
E•DE•$R&FG1
•Định lý 13:|điều kiện đủ tồn tại đường đi có hướng Hamilton~
a. Nếu
P}&|~•P}&|,~≥2−7∀C,∈J&J/F
thì G có đường đi có hướng Hamilton.
b. Nếu
P}&|,~≥−0∀,∈
thì G có đường đi có hướng Hamilton.
c. Nếu
P}&
<
|~•P}&
|,~≥−0∀C,∈J&Š&•.+,
thì G có đường đi có hướng Hamilton.
d. Nếu
P}&
|,~≥¡2#P}&
<
|,~≥¡2∀,∈
thì G có đường đi có hướng Hamilton.
Z*&BW&FS–&wŽ.$B&.,DE•$R&FGD&
.ŠH•$R&.M|•.ŠHG•.M~1D$R+G.HGUJ¦&.HSvŠ.$B&
.•$R&FGD&.ŠH•$R&.M1
Định lý 14 (Konig)X.ŠH•$R&.M./•.$B&.•$R&FG1
Định lý 15 (Camion)ŠH•$R&.M•DE•$R&FGJ,”
J•Gw&1
“
G V>>DWWVX>DW;CE
G## >JX>DW
‚&$B€-'"•‚'].•]R•
−
0
']J"C™'].W&‚GLCDŠ)F*,/']OF.L1uw.
}DEv.†.‚P‹&‚&" . .$B&.)FG&t-|J(&
"&hFF']•††GvG*&$B&yB&FL.y
'„MFDEC111,€}$D$R~1
›V .Š H .L* .M š|eC˜~C ,R ešŸ0C 2C 111C C • DX& S] ,R DX& S]
§
š|C§~•†J"
§
š|§C~1$,!*CF•†€}$G‚.ŠH•
$R&.L*.M¨¤}&IF,RXC§š0C2C111CC©§CG•|C§~C
|§C~
∈
˜1"DTEDEFG•.‚P&t-D&1
"‹+&*.—•GB&(Oz&"S’P\&'$%&'"'¨",
!¤1
G#G YWYWV
!""!G‚D&"'$%&'"'M*+&(O"]$
‹ƒ'1$$T&%O(MF•GD&)"DEEJ+F'Z"'$%&
"MFO"DFFF*/!'$G"WMFZ*EJ+,]
&t&."&"!"WCGOuh&"FO+ttGJ&
ŽF'$%&"]$1
›Vbài toán người du lịch1X
šŸc
ij
C§š0C2C111C
Gma trận chi phí1™DE
,š ,|0~→,|2~→111→,|−0~→,|~→,|0~
•†,+P$RP&
,š|,|0~C,|2~~C|,|2~C,|7~~C111C|,|−0~C,|~~C|,|~C,|0~~
D&.•™'L|,|−0~C,|~~&XG‚MFDE1
D&O"&$BPGHJ+EJ+GB&(FS–'Z!'
DEF!' ‚!'ŽF|C§~,!'J&ŽF*1F&X
,K.•G'Z"C™!'•Dw&XG"1eK'Z".$ƒ
OTZ*EJ+
!'-(DE
"DE)F|C§~ "DEJ&)F|C§~
eK'Z"S–.$ƒPvFDw)tƒ'GU.•'V'FDW&t
)"DEEJ+'$%&"]$1{FJ'Z"FS–„!P$RMF
\wD&™!'•Dw1eKEJ+S–EDw!'•!P$Ru
%1M\S–+'\.+J.$ƒDE.L*.MCŽG'$%&"MF
O"&$BPGH1{F.•F”L€Vh&!'•!P$Ru%&"DH
\wE.$ƒ1
0Q
G#$ " ZCSCVOAY[Y>DW
+ J& €" .H '] €- '" E•ª DEC ™ DE
Ž&,R‚",H.•MF!'Ÿ0C2C111C 1•+D$R']€-
'"E•-(G|
−
0~ªDE1
G#L A>?V?\>
a«D&‹&'„MF‚DES–ŽF.W&‚'L’Dw™P•&
,™‚MFFD!'„š|
§
~1?.•+FD•OR™'L’MF‚
P•&|F*‚‚~.^&‚&"DHE'„MF-(DES–&(.‚
G$ƒ&C,E+DE]$S–J&F*.‹1eE,!*++D•OR"
'L’MF™P•&,™‚.‚z&S]SF.$ƒFD!J&Z
,™‚•&$™P•&ŽF„-‚S]QCE‹&"z&S]D•..•S–
F!P$RMFXDE1M\D•OR*&XGM\DW&XC"z&
S]D•T™P•&|‚~&XGz&S]DW&XP•&|‚~CFD!.$ƒ&XGF
D!DW&X1
Thủ tục rút gọn:
◊Đầu vào FD!'„š|c
ij
~
◊Đầu ra FD!DW&X,‹&z&S]DW&XSum
◊Thuật toán
|~_T
Sum šQ¬|”"'P\&FD!'„OF.L~
|~aW&XP•&
eR™P•&D•0.+MFFD!vK
‘E'L’
D§
šαu-DwP•&1
‘D•-("'L’DwP•&.‚G$ƒ&α1
‘‚&PŠ Sum šSum•α
|~aW&X‚
eR™‚•0.+MFFD!vK
‘E'L’
šαu-Dw‚1
‘D•-("'L’Dw‚.‚G$ƒ&α1
‘‚&PŠ Sum šSum•α
G#& ]W
(S’FX'Z"|DCS~1$,!*"DES–.$ƒFG
F!'
0
ŽF"DE)F|DCS~,
2
ŽF"DEJ&)F|DCS~1
•"!'
0
!P$Rβ,R&"DH€-'"••M\DW&X1
‘(-'FD!'„Oz&"GP•&D,‚S1
‘&•-DE
-|SCD~Oz&".y
SD
š∞1
+|DCS~G'Z"ŽFDT.E'(€V".—X]
D$R,SF|DCS~P—*]+'"$ESF
→ 111 → 111 →
|C§~ 111 |DCS~ 111 |JC~
,--("P&|C~Oz&".y
š∞1
aW&XFD!'„F•!P$Rβ šβ•|‹&z&S]DW&X~1
00
F•†+'\M\'Z"}"*,RFD!'„.—
.$ƒK”,&(0O!1eKX.†'Z"FS–OT\
+'}1
•"!'
2
‘-|DCS~Oz&".y
DS
š∞1
‘vKM\DW&X,RFD!'„$%&Ž&,„!P$R
β šβ•|‹&z&S]DW&X~"1
F•†+'\M\'Z"}"*,RFD!'„.—
.$ƒK”^&!P$R$%&Ž&1eKX.†'Z"FS–
OT\+'}1
G#M O>^_Y]W
‚"G&GFX'Z"|DCS~SF!P$RMF"
J&ŽF|DCS~S–•&/-1
Thủ tục chọn cạnh phân nhánh (r,s)
◊Đầu vào FD!DW&XO!J1
◊Đầu ra 'Z"|DCS~1
◊Thuật toán
|~_T α š−∞¬
|~eR™y'|C§~(c
ij
šQ|š0C111CJ¬§š0C111CJ~vK
‘›".H
DšŸ
≠§
SšŸ
§
≠
‘+α-D•SC.y
α šD•S¬ D š¬ S š§¬
G#N "G_!`
™DE•1{FJ.—X−2CF'(X]2•
G13W*FD!DW&X•O!2,G‚D&FP&SF
^&!P$Rβ1
D&D$B&ƒ'|~FXF|'C~,|)C,~C•D&D$B&ƒ'|~
FXF|'C,~,|)C~1‹&'„Gβ1
02
,G*0a0"b-cd"9e(*f-5"5*g-h67
# i>Iij>?\kCJ
†&()*+",-./MF.ŠHOz&"*„W&FLG$&h.ŠH
D&O‚RMF"*„1?.•W&FL.$FDF"'$%&'"'O†P®.Š
HOT"-DWPhGK1•/'$%&'"'O†P®.ŠH O†P®.ŠH
Oz&FD!J/CFD!Gw‚CPFS"|&~,PFS"J/1
D&./*.ŠH.$ƒG$DhOTFD!,&"6lmlmC,RGS]
.”MF.ŠH,<{¯°¯±°G'„.•.”.+.”§1
Ví dụ ŠHSF1
ŠHDw.$ƒG$,FD!$SF
0 2 7 A
0
∞
2Q Y 0Y
2 0Q
∞
0Y 2Q
7 • Y
∞
2Q
A Y 2 2Y
∞
G V>>DW
?vF,'LGU*+TDw!""!.$ƒ.H&IFOT"
$%&DESF
G## B?\>
Thủ tục rút gọn:
•Đầu vào FD!'„š|
§
~
•Đầu ra FD!DW&X,‹&z&S]DW&XSum
•Thuật toán
|~_T { šQ¬
|~aW&XP•&
eR™P•&D•0.+MFFD!vK
‘E'L’
D§
šαu-DwP•&1
‘D•-("'L’DwP•&.‚G$ƒ&α1
‘‚&PŠ { š{•α
|~aW&X‚
07
2
Y
A 7
0 2
2
Q
0
Y
Y0
Y
Y
2
Q
0
Q
2
Q
Y
•
2
eR™‚•0.+MFFD!vK
‘E'L’
šαu-Dw‚1
‘D•-("'L’Dw‚.‚G$ƒ&α1
‘‚&PŠ { š{•α
•Ví dụ1FD!'„MFO"&$BPGH,Rš[$SF
0 2 7 A Y [ α
0 ∞ 7 “7 07 77 “ 7
2 A ∞ NN A2 20 0[ A
7 AY 0N ∞ 7[ 0[ 2• 0[
A 7“ “Q •Q ∞ Y[ N N
Y 2• A[ •• 77 ∞ 2Y 2Y
[ 7 •• 0• A[ “2 ∞ 7
F+DW&XFD!*.†E!P$R-(DE
‘Khởi tạo
y { šQ¬
‘Rút gọn các dòng
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