E = E â Edgestar(NodeStar(r)). The reduced problem has the same shape precisely that the original, while replacing the graph G = (V,E) by the graph G = (V ,E ).
!"!
# #
!
"
# " $
!" #$ % "$!$ &'( % % &' ()
* % * % + , -. * % /01 /21 3 4 4 + 4 5 * % % 5 6 * 4 & ) % % & ) % 7 * 8 /!1 * 6 * & ) * * %
2!00
+
/1 /$1 /91 : % ; 3
/ ⎧ ⎪ ⎨
max ct .x (IP ) ⎪ A.x ≤ b ⎩ x ∈ {0, 1}n
6 x = (x1 , ...., xn)t c = (c1 , ...., cn)t b = (b1 , ...., bm )t A = ai,j 1 ≤ i ≤ m 1 ≤ j ≤ n + % > ⎧ ⎪ ⎨
max ct .x (SC) w t .A.x ≤ w t b ⎪ ⎩ x ∈ {0, 1}n
6 w = (w1, ...., wm)t ≥ 0
G = (V, E) S ⊆ V * 7 + % +
? + 3 /!!1 ( % 6 G = (V, E) * V = {1, ..., n} @ % i ∈ V * > Nodestar(i) = {j : {i, j} ∈ E} di = card(N odestar(i)) d0 = |E| =
2!02
+ % xi % i ∈ V * xi = 1 % 4 + % * > ⎧ max x0 = (xi : i ∈ V ) (IP ) ⎪ xi + xj ≤ 1, {i, j} ∈ E ⎩ xi binary, i ∈ V ⎪ ⎨
+ 54 & ) * & ) * 4 4 *>
(di xi : i ∈ V ) ≤ d0
+ & ) > ⎧ ⎪ x0 = (xi : i ∈ ⎨ max (di xi : i ∈ V ) ≤ d0 (SC) ⎪ ⎩
V)
xi binary, i ∈ V
6 5 (≤) & ) % h ∈ H & xi * * dhi ≥ 0) * >
(dhi xi : i ∈ V ) ≤ dh0
+ * wh % >
( (dhi : i ∈ V ) − dh0 ) wh = dh0
& ) * 5 ! 3
: &) xr 5 & %3 ) 7 A + xr 4 !
2!0
A 7 5 !
xr * > dr = min(di )
4
r ∈ V 4 Nodestar(r) = min(Nodestar(i) : i ∈ V ) xr = 1
!
+ xr = 1 & ) 5 xj = 0 xj xr + xj ≤ 1 + & ) * * * V E % 4 *> V = V − Nodestar(r) E = E − Edgestar(NodeStar(r))
+ * G = (V, E) G = (V , E ) * * > di = di − |N odestar(i) ∩ Nodestar(r)| d0 = d0 − Sizestar(Nodestar(r))
) *+ , - %
! + di A 0 ; i = 1 V = V 2 : & ) j > i {i, j} ∈ E j ∈ V &) % 7 % & ) % 7 % % 7 V ; i = min{q : q > i et q ∈ V } i % & %) % V > V * : B * i := i 0
! " #
w
- G = (V, E) * w = (w1 , ...., wm)t * m = d0 & ) 6 % : * >
2!0$
" #
; &) % & ) *
xr = 1 % j ∈ Nodestar(r) 1 ≤ k ≤ m ak,j = 1 wk = 0 1 ≤ j ≤ n 1 ≤ k ≤ m "
xr = 1 j ∈ &) * mk=1 a = dj & m k,j t % 7) * w .A.x = l=1 wl .al,1.x1 +...+ml=1 wl.al,n .xn&C) + xr = 1 xj = 0 + 7 * * k ∈ {1, ..., m} ak,j = 1 * 3 & ) &C) * wk = 0 "
# $
+ &) > di =
wk =0 (ak,i )
d0 =
wk =0
wk
* * % %&'
! 0 2
; w = (1, ...., 1)t V = ∅ wA = w =0(ak,.) ( i = index(min(wA) : (wA)i = 0) xi = 1 V = V ∪ {i} j ∈ , &) ak,j = 1 wk = 0 mk=1 wk = 0 B * 0
"" ( #
k
6 &' () * ( /!!1 4 +4 /!01 + > 6 ? B;B "0 ' &
2!09
/!1) + + ! * d0 * dm % + XS XF XDBG % 4 ( &' () %
d0
dm
XS
XF
XDBG
&'(
')*+
*,'
(-
(-
(-
(+
'.)*
*-/
'/
'*
'/
)&
*,/
&&-
&&/.*
'-+
'-+
'-+
))
*.&
)/&--
(/,&
)/&
).(
)/&
)'
*).
&&/.*
(/((
)'(
)'(
)'(
#
+*
/'(
'-&
)+
)+
)+
&
(/*
&',/,
&(+-+
.//
./&
.//
)
/*.
',).
&',+,
'/.
'.&
'/.
'
/)&
',+'
&',/,
'/-
'/-
'/-
0
(-
&*/
).*
'(
'(
'(
1 &
)&&
)-
*&--
&)-
&)-
&)-
1 )
)&&
)'/
'.*&
&)+
&))
&)+
1 '
)-/
//./
'.*-
&)'
&)'
&)'
&23 .
)()
')*+
,.
2
&),
&'(
'23 *
*-.
'.)*
(*
2
&.)
&,'
)2 *
&*,
.*&
'+
2
/(
+*
'2 '
./
&&-
&&
2
)'
)+
*2 '
+,
)/&--
&'
2
'+
',
&)-
&)-
/'(
&'
2
&+
))
&
&,+
',).
&)&
2
&--
&--
.
&(/
',+'
&.,
2
(+
((
& 2 4 5
$ % & ; G = (V, E) & ) , -. & ) : v∈V c(v) (G) = minc v∈V c(v) + ( & ) ( &() + ( X1 , X2, ..., Xk 4- Xi 1 ≤ i ≤ k (G) = i∈(1,...,k) i.card(Xi ) * card(X1 ) ≥ card(X2 ) ≥ .... ≥ card(Xk ) )
+ 3
+ * 3
2!0
- ! 0 2 $
6 V = ∅ &' () % V
4 V = V \V V 7 * 0 @ *
) #
B * * % 0""0 ! 9 (. &0 D) !( 6 ? ' /!1 + + 0 * χ(G) * LBth UBkok s(G)kok /!21 (G) & ) s(G) (
(G)
%
χ(G)
LBth
U Bkok
s(G)kok
#
+*
/-)
&&
&),
)*'
&&
)*'
6 //
'/
.(-
+
.+
&'(
(
&'(
(
).-
&)(
'(+
(
&./
'*+
(
'*'
&-
% &&
.--
&)(
&&+-
)-
'&(
+/)
)-
+..
))
&)-
&)-
&)+/
,
&./
*/-
,
**/
,
$ '
&&
)-
*
&+
)&
*
)&
*
$ *
)'
+&
.
''
*.
.
*.
.
$ .
*+
)'/
/
/)
,'
/
,'
/
$ /
,.
+..
+
&&/
&(,
+
&(,
+
$ +
&,&
)'/-
(
)&,
'()
(
'(&
(
) 2 7848 -+9
' * &' () % % + % & ) B * 8
5
2!0=
( /!1 3 -, . . + +> B ' !"&0)> !=2-0"0 &!EE=) /01 + ( . &!EE) 3 % B F 0> $ =9"-=
2!0E
/!21 G * L 4 4 L3 3 L* B * ,,(J"
