11.3.1 STSP by Column Generation
 Recall the formulation of STSP (for 1-tree relaxation)
minimize
eE ce xe
subject to
e({i}) xe = 2 ,
i  N\{1}
e({1}) xe = 2 ,
eE(S) xe  |S| - 1 ,
S  N\{1}, S  , N\{1},
eE(N\{1}) xe = |N| - 2.
xe  { 0, 1 }.
(note that we use N to denote the set of nodes instead of V)
 min { eE cexe : e(i) xe = 2 for iN\{1}, xX1 }
where X1 = { xZ+m : e(1) xe = 2, eE(S) xe  |S|-1 for   S  N\{1},
eE(N\{1}) xe = |N| - 2 }
is the set of incidence vectors of one-trees.
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 STSP:
min { eE cexe : e(i) xe = 2 for iN\{1}, xX1 }
where X1 = { xZ+m : e(1) xe = 2, eE(S) xe  |S|-1 for   S  N\{1},
eE(N\{1}) xe = |N| - 2 }
is the set of incidence vectors of one-trees.
 Write xe = t : eE(t) t , t {0, 1}, where Gt = (N, Et) is the tth one-tree.
 e(i) xe = e(i) t : eE(t) t = t ditt = 2
(dit is degree of node i in the one-tree Gt.)
(In matrix notation, we have Ax = 2, where A is the node-edge incidence
matrix (for edges E(N\{1})).
x = T, where each column of T is incidence vector of a one-tree.
 AT = 2 (each column of AT is ATt ) )
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min t=1T1 (cxt)t
(DW)
t=1T1 ditt = 2,
t=1T1 t = 1
B|T1|
 LP relaxation is:
min t=1T1 (cxt)t
(LPM)
t=1T1 ditt = 2,
t=1T1 t = 1
R+T1

for i  N\{1}
for i  N\{1}
 Note that the convexity constraint t=1T1 t = 1 (t=1T1 2t = 2) may be
regarded as d1t = 2 for node 1.
 Subproblem: 1= min{eE(ce- ui - uj)xe - : xX1} (e=(i, j)E, i, jN\{1})
( cxt - iN\{1} ditui -  = cxt - iN\{1} ui  e(i) xet -  = eE (ce - ui - uj)xet )
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11.3.2 Strength of the LP Master
 Prop 11.1:
zLPM = max { k=1K ckxk : k=1K Akxk = b, xk  conv(Xk) for k = 1, … , K }
Pf) LPM is obtained from IP by substituting
xk = t=1Tk xk, tk, t, t=1Tk k, t = 1, k, t  0 for t = 1, … , Tk .
This is equivalent to substituting xk  conv(Xk). 
 Prop 11.1 implies that the previous column generation approach for STSP
provides the same bound as Lagrangean dual.
 Let wLD be the value of the Lagrangian dual when the joint constraint k=1K
Akxk = b are dualized.
Let zCUT be the optimal value obtained when cutting planes are added to the
LP relaxation of IP using exact separation algorithm for each of conv(Xk) for
k = 1, … , K.
Then Thm 11.2: zLPM = wLD = zCUT
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 Hence column generation approach (if it can be used for the problem) usually
provides strong bounds.
 It uses the dual variables obtained from LP relaxation as guides compared to
simple rule of subgradient method for Lagrangian dual. (Some more
discussions on this later.)
 Generated columns can be kept for overall optimization. (compare to
Lagrangian dual)
 Consider column generation algorithm for generalized assignment problem
and its merits compared to Lagrangian dual and cutting plane algorithms.
(See Savelsbergh, A Branch and Price Algorithm for the Generalized
Assignment Problem, Operations Research, 1997, Vol 45, No 6, p831-841)
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11.4 IP Column Generation for 0-1 IP
 Branch-and-price algorithm
How to branch after column generation?
 (IP) z = max { k=1K ckxk : k=1K Akxk = b,
Dkxk  dk for k = 1, … , K, xk  𝐵𝑛𝑘 for k = 1, … , K},
reformulation
z = max k=1K t=1Tk ( ckxk, t ) k, t
(IPM)
k=1K t=1Tk ( Akxk, t ) k, t = b
t=1Tk k, t = 1
for k = 1, … ,K
k, t  {0, 1} for t = 1, … , Tk and k = 1, … , K
𝑥𝑘 =
𝑇𝑘
𝑘,𝑡
𝑡=1 𝜆𝑘,𝑡 𝑥
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∈ {0,1}𝑛𝑘 if and only if 𝜆 is integer
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 If LP relaxation of IPM has fractional solution, two choices to branch : (1)
on x variable, (2) on  variable.
(Following are general schemes. Particular implementation may vary
and/or even very difficult.)
 Consider branch on x variables
If 𝜆 fractional in LP optimal, there is some  and j such that the
corresponding 0-1 variable xj has LP value 𝑥𝑗  that is fractional.
Split the set S of all feasible solutions into S0 = S  { x: xj = 0} and S1 =
S  { x: xj = 1}
Now as xjk = t=1Tk k, t xjk, t  {0, 1}, xjk =   {0, 1} implies that xjk, t =
 for all k, t with k, t > 0
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Hence the Master Problem at node Si = S  { x: xj = i} for i = 0, 1 is
z(Si) = max k   t ( ckxk, t ) k, t +
k   t ( Akxk, t ) k, t +
(IPM(Si))
t k, t = 1
,
𝜅
𝜅
,
𝑡:𝑥𝑗  𝑡 =𝑖(𝐴
𝑥 𝜅,𝑡 )𝜆𝜅,𝑡
𝑥 𝜅,𝑡 )𝜆𝜅,𝑡 = b
for k  
𝑡:𝑥𝑗  𝑡 =𝑖 𝜆𝜅,𝑡
,
𝑡:𝑥𝑗  𝑡 =𝑖(𝑐
=1
k, t  {0, 1} for t = 1, … , Tk , k = 1, … , K
The columns for k =  are affected.
Column generation for subproblem  and i = 0, 1
(Si) = max { (c - A) x -  : x  X , xj = i }
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 Branch on  variables
If k, t fractional, fix it to 0 and 1 respectively.
If fixed to 1, no problem in column generation.
( usually implemented by setting the lower and upper bound on k, t as 1)
If fixed to 0 ( set upper bound of k, t to 0), we need a scheme to prevent the
generation of the same column again in the column generation procedure.
 Also the branching may partition the set of feasible solutions unbalanced.
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