MEIO/UPC-UB : NETWORK FLOWS
Network Flows
UPCOPENCOURSEWARE number 34414
Topic 6:
Minimum Cost Network Flow
Problems
F.-Javier Heredia
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(Chap. 9,10 & 11 Ahuja,Magnanti,Orlin)
• Pseudo-polynomial algorithms (AMO, Chap. 9)
• Polynomial algorithms (AMO, Chap. 10)
• Simplex algorithm (AMO, Chap. 11)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 2
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
6.- Minimum Cost Network Flow Problems
• Definitions and assumptions.
• Applications.
• Optimality Conditions:
– Negative Cycle Optimality Conditions.
– Reduced Cost Optimality Conditions.
– Complementary Slackness Optimality Conditions.
• Minimum Cost Flow Duality.
• Algorithms: (assignments)
– Cycle-canceling and negative cycle O.C.
– Successive shortest-path and Reduced cost O.C.
– Out-of-kilter algorithm and Complementary Slackness O.C.
• The dual-primal network flow algorithm (assignment).
• Relaxation methods.
• Summary.
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 3
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
MCNF: Basic algorithms (Chap. 9 AMO)
• Given the network G=(N,A) with costs c, capacities u and
supplies/demands b, the Minimum Cost Network Flow Problem
(MCNFP) is defined as:
 min

s.a.:






z=
∑cx
(i, j)∈A
∑
{ j:( i , j )∈A}
xij −
∑
ij ij
=
x ji b(i ) ∀i ∈ N
(1a)
(1b)
{ j:( j ,i )∈A}
0 ≤ xij ≤ uij
∀(i, j ) ∈ A
F.-Javier Heredia http://gnom.upc.edu/heredia
(1c)
MCNFP- 4
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Minimum Cost Network Flow Problem (MCNFP)
1. All data (cost, supply, demand and capacity) are integral (needed for
some proofs, and some running time analysis).
2. The network is directed.
3. The supplies/demands at the nodes satisfy the condition
Σi∈N b(i) = 0
and the minimum cost flow problem has a feasible solution.
4.
The network G contains an uncapacitated directed path between every
pair of nodes
•
If necessary artificial uncapacitated arcs (1,j) and (j,1) ∀j∈N will be added
with infinity costs.
5. All arc costs cij are nonnegative.
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 5
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
MCNFP: Assumptions
• To create a feasible
solution, add a
dummy node d.
• Add an arc from d
3
2
-4
4
3
5
2
-5
1
1
to each demand
node, each with a
large cost M, and
large capacity.
3
6
d
• Add an arc to d from each supply node, each with a large
cost M, and a large capacity.
• In an optimal solution, arcs with large cost will have a
flow of 0.
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 6
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Artificial Solutions
• Residual capacity rij :
cij , uij
i
xij
j
cij , rij = uij - xij
xij
j
i
xji
-cij , rij = xji
• Residual network G(x) : nodes N and arcs (i,j)
replaced with arcs (i,j) and (j,i) with positive
residual capacity rij
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 7
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Residual network
• Def: Augmenting Cycle W in G w.r.t. x : any cycle W (not
necessarily oriented) s.t. “x+f(W)” feasible.
– Prop.: W augmenting cycle in G w.r.t. x ⇔ W directed cycle in G(x)
• Th. (Augmenting Cycle Theorem.): Let x and x0 be any two feasible
Proof: AMO, page. 83
Interpretation:
-4
i
xij
bi
( cij , uij )
j
bj
2
2
5
1
4
1
3
2
x0 x
3
i
2
3 1
4
3
3
4
3
5
c(x0) + c( f(W) ) = 35 – 6 = c(x)
-6
1
4
3
3
rij
j
G(x0)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 8
Minimum Cost Network Flow Problems
solutions of a network flow problem. Then x equals x0 plus the flow of at
most m directed cycles in G(x0). Furthermore, the cost of x equals the cost
of x0 plus the cost of flow on these augmenting cycles.
(2,3)
MEIO/UPC-UB : NETWORK FLOWS
Augmenting Cycle Theorem
• Ta (Negative Cycle Optimality Cond.): A feasible solution x* is
an optimal solution of the minimum cost flow problem if and only if
satisfies the negative cycle optimality conditions: namely, the
residual network G(x*) contains no negative cost (directed) cycle.
Proof:
⇒ : a pos. flow along any neg.cycle of G(x) always improve the o.f.: if x*
optimal, G(x*) cannot contain a neg. cycle.
⇐ : Assume x* feasible and G(x*) without neg. cycle. Then, by the
Augmenting Cycle Th., for any feasible flow x≠x* holds
c(x) =c( x* ) + c( f(W) ) ≥ c( x* )
and thus, x* is optimal
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 9
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Negative cycle optimality conditions
• Potentials nodes:
• Reduced costs:
π ( i ) ∀i ∈ N
cijπ =cij − π (i ) + π ( j ) ∀(i, j ) ∈ A
– Red. cost in the red. Net. G(x):
•
cπji =
−cijπ
∀(i, j ) ∈ G ( x )
Properties:
a) For any directed path P from node k to node l,
π
=
c
∑ (i , j )∈P ij
∑
b) For any directed cycle W,
•
c − π ( k ) + π (l )
( i , j )∈P ij
∑
π
c
= ∑ ( i , j )∈W cij
ij
( i , j )∈W
Consequences of properties (a) and (b):
i.
(a) ⇒ P is the shortest path w.r.t. c ⇔ P is the shortest path w.r.t. cπ
ii.
(b) ⇒ W is a neg. cycle w.r.t. c ⇔ W is a neg. cycle w.r.t. cπ
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 10
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Reduced cost optimality conditions (1/2)
• Th. (Reduced Cost Optimality Cond.): A feasible solution x* is an optimal
solution of the minimum cost flow problem if and only if some set of node
potencials π satisfy the following reduced cost optimality conditions:
Proof:
cijπ ≥ 0 ∀(i, j ) ∈ G ( x*)
(1)
–
We will see the equivalence with the Neg. Cycle. Opt. Cond (NCOC).
–
RCOC ⇒ NCOC:
–

x* optimal w.r.t. The RCOC ⇒

Prop. (b) ⇒
cπ ∑
∑=
( i , j )∈W
ij
∑
( i , j )∈W
( i , j )∈W
cijπ ≥ 0 ∀ directed cycle W ∈ G ( x*)
cij ≥ 0 ⇒ G(x*) contains no negative cycle.
RCOC ⇐ NCOC:


x* NCOC ⇒ G(x*) contains no negative cycles.
Let d(·) be the shortest path labels of the label correcting algorithm from node 1 over
G(x*) and let π = -d. Then, by the convergence theorem of the label-correcting alg. (1)
holds true.
Remark: relation between the RCOC over G(x*) and the simplex optimality
conditions for saturated arcs xij = uij
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 11
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Reduced cost optimality conditions (2/2)
• Th. (Complementary Slackness Optimality Conditions): A
feasible solution x* is an optimal solution of the minimum cost flow
problems if and only if for some set of node potentials π, the
reduced costs and flow values satisfy the following complementary
slackness optimality conditions for every arc ( i , j ) ∈ A:
If cijπ > 0
If 0 < xij < uij
If cijπ < 0
0
, then xij* =
0
, then cijπ =
, then xij* =
uij
Proof: AMO, pag. 310 (quite technical!!)
•
Relation with the optimality condition of the simplex algorithm
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 12
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Complementary Slackness Opt. Cond.
Complementary Slackness
optimality conditions (CSOC)
If cijπ > 0
If 0 < xij < uij
If cijπ < 0
, then xij* =
0
, then cijπ =
0
, then xij* =
uij
Kilter Diagram for arc (i, j)
Francesc López Ramos y Fernándo Saráchaga Gutierrez
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 13
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
CSOC & Kilter Diagram
Complementary Slackness
optimality conditions (CSOC)
(i, j )
(i, j ) B − D − E
(i, j ) A−C
≡ ( xij , cijπ )
→ in − kilter
→ out − of − kilter
In-Kilter and Out-kilter arc
Diagram
Francesc López Ramos y Fernándo Saráchaga Gutierrez
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 14
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
In-Kilter / Out-kilter
• Definition,
– The k(i, j) is a magnitude of the change in x(i, j) required to
make the arc (i, j) an in-kilter arc while keeping C(i, j, π)
fixed.
• Consequently,
– If (i, j) is in-kilter then k(i, j) = 0.
– We can measure how far we are from the optimal solution
as follows:
K=
∑ k (i, j )
( i , j )∈ A
– The smaller the value of K, the closer we are from the
optimal solution.
Francesc López Ramos y Fernándo Saráchaga Gutierrez
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 15
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Kilter Number (k(i, j))

k(i, j) has a close relation with CSOC as follows:
1)
if
cijπ > 0
then kij = xij
2)
if
cijπ < 0
then kij = uij − xij
cijπ = 0 & xij > uij
then kij = xij − uij
cijπ = 0 & xij < 0
then kij = − xij
3) 3.1) if
3.2) if

In order to use k(i, j) in the OFKA and thus, in the G(x), it is
redefined as follows:
 0
kij = 
rij
if
if
cijπ ≥ 0
∀(i, j ) ∈ G ( x)
π
cij < 0
Francesc López Ramos y Fernándo Saráchaga Gutierrez
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 16
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Computation of k(i, j)’s
x(i, j) = -2 , c(i, j, π) > 0 →
k(i , j) = |-2| = 2
x(i, j) = 3 , c(i, j, π) > 0 →
cij
k(i , j) = |3| = 3
1
1
x(i, j) = 6 , c(i, j, π) = 0 →
x(i, j) = -2 , c(i, j, π) = 0 →
k(i , j) = -(-2) = 2
k(i , j) = 6 - 4 = 2
3.2
3.1
xij
uij = 4
2
2
x(i, j) = -2 , c(i, j, π) < 0 →
x(i, j) = 2 , c(i, j, π) < 0 →
k(i , j) = |4 - (-2)| = 6
k(i , j) = |4 - 2| = 2
Francesc López Ramos y Fernándo Saráchaga Gutierrez
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 17
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Cases
MEIO/UPC-UB : NETWORK FLOWS
Relation with G(x)
G
G(x)
i
xij = 0
rij = uij
G
xij = 0, cij(π) ≥ 0
j
cij(π) ≥ 0
rij
rij = uij - xij
j
cij(π) ≥ 0
0 < xij < uij
i
(x factible)
j
0 < xij < uij,
cij(π)= - cji(π) ≥0 ⇒ cij(π) = 0
rji = xij
xij = uij
i
rji = uji
j
xij = uij, cij(π) = - cji(π) ≤ 0
cji(π) ≥ 0
Francesc López Ramos y Fernándo Saráchaga Gutierrez
F.-Javier Heredia http://gnom.upc.edu/heredia
18
MCNFP- 18
Minimum Cost Network Flow Problems
i
algorithm out-of-kilter;
begin
π: = 0;
establish a feasible flow x in the network;
define the residual network G(x) and compute the kilter numbers of arcs;
while the network contains an out-of-kilter arc do
begin
select an out-of-kilter arc (p, q) in G(x);
define the length of each arc (i, j) in G(x) as max {0, C(i, j, π)};
let d(·) denote the shortest path distances from node q to all other nodes in
G(x) – {(q, p)} and let P denote a shortest path from node q to node p;
update π´(i) : = π(i) – d(i) for all i є N;
if C(p, q, π´) < 0 then
begin
W : = P U {(p, q)};
delta : = min { r(i, j) : (i, j) є W };
augment delta units of flow along W;
update x, G(x) and the reduced costs;
end;
end;
end;
Francesc López Ramos y Fernándo Saráchaga Gutierrez
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 19
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Out-of-kilter algorithm
• Def. (Dual Minimum Cost Flow Problem) :
 max =
w(π , α ) ∑ b(i )π (i ) − ∑ uijαij

i∈N
( i , j )∈A
 s.t.:


π (i ) − π ( j ) − αij ≤ cij ∀(i, j ) ∈ A

αij ≥ 0 ∀(i, j ) ∈ A


π ( j ) unrestricted ∀j ∈ N

(1a)
(1b)
(1c)
(1d)
– It is easy to verify that this is the dual of the standard form of
the MCNFP (exercise).
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 20
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Minimum Cost Flow Duality
• Th. (Weak Duality Theorem): Let z(x) denote the objective
function of some feasible solution x of the minimum cost flow
problem and let w(π,α) denote the objective function value of some
feasible solution (π,α) of its dual. Then w(π,α) ≤ z(x).
• Th. (Strong Duality Theorem): Any minimum cost flow problem
with optimal solution x* has a dual minimum cost flow problem with
optimal solution π satisfying the property that z(x*)=w(π).
• Th. (Rel. Duality - Comp. Slack. O.C.): If x* is an optimal solution
of the minimum cost flow problem, and π is an optimal solution of
the dual minimum cost flow problem, the pair (x*, π) satisfies the
complementary slackness optimality conditions.
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 21
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Minimum Cost Flow Duality
• Def. Lagrangian relaxed problem LR(π):




=
w(π ) min  ∑ cij xij + ∑ π (i ) b(i ) − ∑ xij + ∑ x ji  0 ≤ xij ≤ uij ∀(i, j ) ∈ A
i∈N
{ j:( i , j )∈A}
{ j:( j ,i )∈A}
x  (i , j )∈A



• Alternative expressions of LR(π):


=
w(π ) min  ∑ cij xij + ∑ π (i )e(i ) 0 ≤ xij ≤ uij ∀(i, j ) ∈ A
i∈N
x  (i , j )∈A



π
=
w(π ) min  ∑ cij xij + ∑ π (i )b(i ) 0 ≤ xij ≤ uij ∀(i, j ) ∈ A
i∈N
x  (i , j )∈A

(1)
(2)
• Solution of problem LR(π): trivial, from formulation (2).
xij*

0
if cijπ > 0 Property 1: If a pseudoflow x of the minimum cost

flow problem satisfies the reduced cost
if cijπ < 0
uij

optimality conditions for some π, then x is an

π
0
any x ∈ 0, uij  if cij =
optimal solution of LR(π)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 22
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Relaxation Algorithm (1)
•
Lemma:
a) For any node potentials π, w(π) ≤ z*
b) For some choice of node potentials π*, w(π*) = z*
Proof:
a) If x* opt. sol. of the MCNFP ⇒ ∀π, x* feasible for LR(π) and w(π) ≤ z*.
b) Let π*, x* satisfy the RCOC ⇒ By Prop. 1 : x* is the opt. solution of
LR(π*) ⇒ w(π*) = z*.
•
Rationale of the Relaxation algorithm:
–
Maintains a pair (π, x), x a pseudoflow, s.t. satisfy the RCOC
–
It modifies π to π’ and x to x’ so that x’ is an optimal solution of LR(π’)
and w(π’) > w(π).
–
Keeping π unchanged, it modifies x to x’ so that x’ is also an optimal
solution of LR(π) and the excess of at least one node decreases
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 23
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Relaxation Algorithm (2)
At some stage of the algorithm:
• S: nodes of a tree T rooted at node s s.t.
– Every node s∈S : e(s) ≥0.
– Every arc (i,j)∈T : cπij =0
• S=N-S ; (S,S) onward arcs of cut [S, S] in G(x) ; (S,S) backward arcs
of cut [S, S] in G(x).
=
e( S )
=
e(i ) ; r (π , S )
∑
i∈S
3
S
8
s 1
∑
( i , j )∈( S , S )
cijπ =0
(0,2 )
-1
2
0
rij
5
(0,3 )
6 4
3
(2,4 )
5
7
4
(cπij , rij )
-2
8
(3,2 )
e(i)
i
S
0
e(j)
j
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 24
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Relaxation Algorithm (3)
3
S
8
s 1
(0,2 )
-1
2
5
(0,3 )
0
6 4
3
(2,4 )
5
7
-2
4
8
(3,2 )
e(i)
i
(cπij , rij )
S
0
e(j)
j
• Case 1: if e(S) > r(π,S) then increase w(π)
– Modify x saturating all (i,j) ∈ (S,S) s.t. cπij =0 (w(π) invariant, due to (2))
– The change in x reduces e(S) : e(S) ← e(S)- r(π,S) ≥ 0.
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 25
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Relaxation Algorithm (4)
MEIO/UPC-UB : NETWORK FLOWS
Relaxation Algorithm (5)
S
8
s 1
(0,2 )
1
2
-3
5
(0,3 )
6 7
3
(2,4 )
5
7
-2
4
8
(3,2 )
e(i)
i
(cπij , rij )
S
0
e(j)
j
• Case 1: if e(S) > r(π,S) then increase w(π)
–
–
–
–
Modify x saturating all (i,j) ∈ (S,S) s.t. cπij =0 (w(π) invariant, due to (2))
The change in x reduces e(S) : e(S) ← e(S)- r(π,S) ≥ 0.
At this point cπij > 0 ∀(i,j)∈(S,S): α:=min{ cπij | (i,j) ∈ (S,S) } > 0 (*)
Increase π(i) : π’(i) := π(i) + α ∀i∈S
 w(π) exp. (1) : w(π’) := w(π) + [e(S)-r(π,S)] α > w(π)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 26
Minimum Cost Network Flow Problems
1
MEIO/UPC-UB : NETWORK FLOWS
Relaxation Algorithm (6)
1
S
8
s 1
(2,2 )
1
2
-3
5
(2,3 )
6 7
3
(0,4 )
5
7
-2
4
8
(1,2 )
e(i)
i
(cπij , rij )
S
0
e(j)
j
Modify x saturating all (i,j) ∈ (S,S) s.t. cπij =0 (w(π) invariant, due to (2))
The change in x reduces e(S) : e(S) ← e(S)- r(π,S) ≥ 0.
At this point cπij > 0 ∀(i,j)∈(S,S): α:=min{ cπij | (i,j) ∈ (S,S) } > 0 (*)
Increase π(i) : π’(i) := π(i) + α ∀i∈S
 w(π) exp. (1) : w(π’) := w(π) + [e(S)-r(π,S)] α > w(π)
– The RCOC is conserved: cπij := cπij - α ≥ 0 ∀(i,j) ∈(S,S) due to (*)
(and the rest of cπij cannot be < 0)
–
–
–
–
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 27
Minimum Cost Network Flow Problems
• Case 1: if e(S) > r(π,S) then increase w(π)
• Case 2: if e(S) ≤ r(π,S):
– r(π,S) ≥ e(S) > 0 ⇒ ∃(i,j)∈(S,S) s.t. cπij = 0
– If e(j) ≥ 0 then: S:=S+{ j }
– If e(j) < 0 then : augmenting flow δ trough P(s→j)


e(S) decreases.
w(π) invariant (cπij =0 ∀(i,j) ∈P(s →j)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 28
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Relaxation Algorithm (7)
Relaxation algorithm: given a MCNFP in standard form.
begin
procedure adjust-potential;
x:= 0 and π:= 0;
begin
while the network contains a node s
for every arc (i,j)∈(S,S) with cπij =0 do send rij units
with e(s)>0 do
of flow on the arc (i,j);
begin
compute α := min{cπij : (i,j)∈(S,S) and rij > 0};
S := {s};
for every node j∈S do π(j) := π(j) + α;
if e(S) > r(π,S) then adjust-potencial
end;
else
repeat
select an arc (i,j)∈(S,S) in the residual network with cπij =0;
if e(j) ≥ 0 then set pred(j):= i and add node j to S;
procedure adjust-flow;
until e(j) < 0 or e(S) > r(π,S);
if e(S) > r(π,S) then adjust-potential begin
identify directed P(s→j) from pred();
else adjust-flow;
δ:=min[ e(s) , -e(j) , min{rij : (i,j) ∈P}];
endif
augment δ along P;
end;
update imbalances and residual capac.;
end;
where: e( S )
end;
=
e(i ) ; r (π , S )
∑
i∈S
∑
( i , j )∈( S , S )
π
=0
cij
rij
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 29
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Relaxation algorithm (8)
•
Convergence:
The relaxation algorithm terminates with an optimal flow x*.
Proof:
i.
The rel. Alg. terminates with e(s)=0 ∀s∈N ⇒ x* is a feasible flow.
ii.
The pair (π*,x*) satisfies the RCOC.
•
Finite convergence: the relaxation algorithm terminates
in a finite number of iterations (for problems with integral
data).
•
Worst case analysis: O(m2nCU2) (worst than cyclecancelling, succ. shortest path and primal-dual)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 30
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Relaxation algorithm - convergence
25
1
(4,10)
(5,25)
2
3
b(i) i
4
0
(2,25)
(cij ,uij )
5
-25
j b(j)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 31
Minimum Cost Network Flow Problems
0
(1,20)
MEIO/UPC-UB : NETWORK FLOWS
Relaxation algorithm – example
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 32
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Relaxation algorithm – example (cont.)
The algorithms studied so far are pseudopolynomial-time algorithms, that
is, its worst-case complexity is bounded by a polynomial of n, m, C and U.
Algorithm
Worst-case complexity
Cycle cancelling O( nm2CU )
Succ. shortest path O(n3U ) ← O( nU∙S(n,m,nC))
Primal-dual O(mn3∙min{U,C}) ← O( min{nU, nC}∙{S(n,m,nC)+M(n,m,U)} )
Out-of-kilter O(mn2U ) ← O( mU∙S(n,m,nC))
Relaxation O(m2nCU2)
S(n,m,nC): running time of the shortest path alg. with cij ≥ 0 (O(n2) for Dijkstra).
M(n,m,U): running time of a max-flow alg. (O(mn2) for the SAPA)
Instead, polynomial-time algorithms have a worst-case complexity that
is bounded by a polynomial of n, m, logC and logU.
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 33
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Summary of pseudopolynomial-time MCNF algorithms
• Capacity Scaling algorithm.
• Cost Scaling Algorithm (assignment)
• Double Scaling Algorithm (assignment)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 34
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Polynomial MCNFP algorithms
• This algorithm is an enhancement of the successive shortest path
algorithm.
• It ensures that in each augmentation, sufficiently large flow (at
least Δ) is sent from an excess node to a deficit node.
• Δ-residual network G(x, Δ): subgraf of G(x) consisting on those
arcs whose residual capacity is at least Δ.
• The algorithm selects a node k with excess at least Δ, selects a
node l with deficit at least Δ, identifies a shortest path in G(x) from
node k to node l with residual capacity at least Δ, and augments
flow along this path.
• The algorithm terminates when there is no imbalanced node.
• Source: J. Orlin “Network Optimization” http://ocw.mit.edu/courses/sloan-schoolof-management/15-082j-network-optimization-fall-2010/
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 35
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
The Capacity Scaling Algorithm
algorithm capacity scaling;
begin
x: = 0 and π : = 0;
Δ: = 2 log U;
while Δ ≥ 1do
begin {Δ-scaling phases}
for every arc (i, j) in the residual network G(x) do;
if rij ≥ Δ and cijπ < 0 then
send rij units of flow along (i, j), update x and the imbalances e(i) and e(j);
end for;
S(Δ) : = {i ∈ N : e(i) ≥ Δ};
T(Δ) : = {i ∈ N : e(i) ≤ -Δ};
while S(Δ) ≠ Ф and T(Δ) ≠ Ф (and a path P from k to l over G(x,Δ)⇐ Ass.4) do
Select a node k ∈ S(Δ) and a node l ∈ T(Δ);
Determine a shortest path distance d(.) from node k to all other nodes in the
Δ-residual network G(x,Δ) with respect to the reduced costs cijπ;
Let P denote a shortest path from node k to node l in G(x, Δ);
Update π := π – d and cijπ := cijπ + d(i) - d(j);
Augment Δ units of flow along the path P;
Update x, S(Δ), T(Δ), and G(x, Δ);
end while;
Δ : = Δ/2;
end {Δ-scaling phases};
end while;
end;
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 36
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
The Capacity Scaling Algorithm (contd.)
0
0
4
2
4
7
5
2
0
1
1
6
3
0
2
5
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 37
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
The Original Costs and Node Potentials
5
10
2
-2
4
30
25
23
20
20
1
20
3
-7
25
5
-19
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 38
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
The Original Capacities and Supplies/Demands
5
10
2
-2
We send flow
from nodes
with excess ≥
∆ to nodes
with deficit ≥ ∆.
4
30
25
23
20
20
1
20
3
-7
25
5
We ignore
arcs with
capacity
≤ ∆.
-19
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 39
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Set ∆ = 16.
This begins the ∆-scaling phase.
7
shortest
path
distance
8
4
2
4
7
0
5
2
1
1
6
3
6
2
The shortest
path tree is
marked in bold
and blue.
5
8
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 40
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Select a supply node and find the shortest paths
MEIO/UPC-UB : NETWORK FLOWS
Update the Node Potentials and the Reduced Costs
-7
43
4
7
5
6
0
0
0
2
1
1
0
6
3
-6
2
0
To update a
node
potential,
substract the
shortest path
distance.
5
-8
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 41
Minimum Cost Network Flow Problems
2
-8
5
10
2
-2
Send flow
from node 1
to node 5.
4
30
25
23
20
1
20
3
-7
25
20
How much
flow should
be sent?
5
-19
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 42
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Send Flow Along a Shortest Path in G(x, 16)
5
10
2
-2
19 units of
flow were
sent from
node 1 to
node 5.
4
30
23
4
25
20
1
20
19
1
19
3
5
6
-7
-19
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 43
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Update the Residual Network
5
10
2
-2
4
30
25
4
20
1
e(j) ≤ -∆ for some j.
20
19
1
The ∆-scaling phase
continues when
e(i) ≥ ∆ for some i.
There is a path from
i to j.
19
3
5
6
-7
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 44
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
This ends the 16-scaling phase.
5
10
2
-2
4
30
25
4
20
1
e(j) ≤ -∆ for some j.
20
19
1
The ∆-scaling phase
continues when
e(i) ≥ ∆ for some i.
There is a path from
i to j.
19
3
5
6
-7
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 45
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
This begins and ends the 8-scaling phase.
5
10
2
-2
4
30
25
4
20
1
20
19
1
19
3
What would we
do if there were
arcs with
capacity at
least 4 and
negative
reduced cost?
5
6
-7
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 46
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
This begins 4-scaling phase.
-7
3
2
4
0
0
0
-8
6
1
1
0
0
0
3
5
The
shortest
path tree is
marked in
bold and
blue.
0
-6
-8
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 47
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Select a “large excess” node and find shortest paths.
MEIO/UPC-UB : NETWORK FLOWS
Update the Node Potentials and the Reduced Costs
0
2
1
0
0
4
2
0
1
To update a
node
potential,
subtract the
shortest path
distance.
4
-4
0
3
5
0
-6
-10
-8
-12
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 48
Minimum Cost Network Flow Problems
-7
-11
-8
5
10
2
-2
4
30
25
4
20
1
20
How much
flow should
be sent?
19
1
19
3
Send flow
from node
1 to node 7
5
6
-7
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 49
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Send Flow Along a Shortest Path in G(x, 4).
5
6
-2
2
4
4
26
0
4
4 units of
flow were
sent from
node 1 to
node 3
25
4
20
1
16
19
4
1
15
3
-7
-3
5
10
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 50
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Update the Residual Network
5
6
-2
2
4
4
26
0
There is no
node j with
e(j) ≤ -4.
25
4
20
1
16
19
4
1
15
3
5
10
-3
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 51
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
This ends the 4-scaling phase.
5
6
-2
2
There is no
node j with
e(j) ≤ -4.
4
4
26
0
25
4
20
1
16
19
4
1
15
3
5
10
-3
What would we
do if there were
arcs with
capacity at
least 4 and
negative
reduced cost?
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 52
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Begin the 2-scaling phase
5
6
-2
2
4
Send flow
from node
2 to node 4
4
26
0
25
4
20
1
16
19
4
1
15
3
5
How much
flow should
be sent?
10
-3
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 53
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Send flow along a shortest path
3
5
4
0
-2
2
4
2 units of flow
were sent from
node 2 to node 4
6
26
0
25
4
20
1
16
19
4
1
15
3
5
10
-3
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 54
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Update the Residual Network
3
0
4
2
4
Send flow
from node 2 to
node 3
6
26
0
25
4
20
1
16
19
4
1
15
3
5
How much
flow should
be sent?
10
-3
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 55
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Send Flow Along a Shortest Path
0
3
0
1
2
4
3 units of flow
were sent from
node 2 to node 3
9
26
0
25
4
20
1
13
19
7
1
12
3
5
13
-3
0
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 56
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Update the Residual Network
0
0
1
2
4
Are we
optimal?
9
26
0
25
4
20
1
13
19
7
1
12
3
5
13
0
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 57
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
This ends the 2-scaling phase.
0
0
1
2
4
9
26
0
25
4
20
1
13
19
7
1
Saturate any
arc whose
capacity is at
least 1 and
with negative
reduced cost.
12
3
reduced cost
0
is negative
5
13
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 58
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Begin the 1-scaling phase.
0
0
1
2
4
Send flow from
node 3 to node 1.
9
26
-1
25
4
20
1
13
20
7
12
3
Note: Node 1
is now a node
with deficit
5
13
1
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 59
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Update the Residual Network
0
0
2
2
4
1 unit of flow was
sent from node 3
to node 1.
8
27
0
25
3
20
1
∆:= ∆/2 = ½ →END
14
20
6
13
3
5
Is this flow
optimal?
12
0
0
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 60
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Update the Residual Network
MEIO/UPC-UB : NETWORK FLOWS
The Final Optimal Flow
5
10,8
2
-2
4
30,3
23
20
20,6
1
20,20
3
-7
25,13
5
-19
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 61
Minimum Cost Network Flow Problems
25
-7
0
2
4
1
0
0
-11
2
0
1
Flow is
at lower
bound.
-4
Flow is
at upper
bound
3
-10
0
5
-12
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 62
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
The Final Optimal Node Potentials and the
Reduced Costs
•
•
Convergence: as in the successive shortest-path algorithm.
Running time:
– A Δ-scaling phase performs at most 2(n+m)=O(m) augmentations:

In the Δ-scaling phase the sum of excesses is bounded by 2(n+m)Δ (see text).

Each augmentation reduces the excess by at least Δ units.
– Each augmentation solves a shortest path problem, S(n,m,nC)
– At most log U Δ-scaling phases.
– In the overall: O(m log U S(n,m,nC))
•
Theorem: the capacity scaling algorithm solves the minimum cost flow problems
in O( m log U S(n,m,nC) ).
– Successive shortest-path alg.: O( n U S(n,m,nC) ) (pseudo-polynomial) .
n
m
U
m*logU
n*U
n*U/m*logU
1000
2000
1,0E+01
6,64E+03
1,00E+04
1,51E+00
1000
2000
1,0E+02
1,33E+04
1,00E+05
7,53E+00
1000
2000
1,0E+03
1,99E+04
1,00E+06
5,02E+01
1000
2000
1,0E+04
2,66E+04
1,00E+07
3,76E+02
1000
2000
1,0E+05
3,32E+04
1,00E+08
3,01E+03
1000
2000
1,0E+06
3,99E+04
1,00E+09
2,51E+04
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 63
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
The Capacity Scaling Algorithm (contd.)
• Standard form.
• Spanning Tree Structure.
• Simplex specialization:
–
–
–
–
–
–
–
Computing the basic feasible solution.
Reduced cost calculation.
Leaving variable selection.
Base change .
Simplex Algorithm for Network Flows.
Example.
Exercises.
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 64
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
MCNF:
Network Simplex Algorithm. (Cap. 11 AMO)
• Mathematic formulation:
 min

s.a. :






z=
∑c x
(i,j)∈ A
∑ xij −
{ j:( i , j )∈A}
ij ij
(1.1a)
Balance
equations
∑ x ji = bi ∀i ∈ N (1.1b)
{ j:( j ,i )∈A}
xij ≥ 0 ∀(i, j ) ∈ A
(1.1c)
n
– Assuming that the network is balanced:
∑b = 0
i =1
• Matrix notation:
i
c′x
(1.2a)
 min

s.a. : N x = b (1.2b)

x ≥ 0 (1.2c)

F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 65
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Standard form of the
Minimum Cost Flow Problem
• Ta : “Each spanning tree T of G define a base of the minimum cost
flow problem. Conversely, each base of the minimum cost flow
problem define a spanning tree of G.”
(1, 2) (3,1) (3,5) ( 2, 4)
1
3
2
5
4
1
2
B=
3
4
5
(3,1) (3,5) (1, 2) ( 2, 4)
3
1 −1 0 0
5
1
−1 0 0
0
1 1 0 B=2
4
0 0 0 −1
1
0 0 −1 0
+1 +1 0 0
0 −1 0 0
0 0 −1 +1
0 0 0 −1
−1 0 +1 0
• Ta: “The rows and columns of the B matrix linked to a spanning
tree T it can be rearranged so that B is triangular”
Procedure:
– Nodes are arranged in crossbar order : 3-5-2-4-1
– The arcs are arranged visiting the nodes according to the crossbar
order. Each node is associated with the only arc that relates it with its
predecessor: (3,1), (3,5), (1,2), (2,4)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 66
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Spanning tree and bases.
•
15
1
i
4
5
5
6
7
5
10
-15
15
1 5
10
-20 2
4
5
5
xij
j bj
3 0
-20 2
5
5
5
bi
3 0
15
10
6
7
10
-15
Procedure:
1. Select a leaf node.
2. Assign flow to its associated
arc.
3. Modify the predecessor
contribution.
4. Eliminate the node and the
arc treated.
xB T = [10,5,5,5,10,15]
(second crossbar order)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 67
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Feasible basic solution calculation.
• Reduced cost:
rij = cij - πT Nij = cij - πi + πj , ∀(i,j)∈A
– Any constant k it can be summed to the Lagrange
multipliers πi without change the value of rij :
rij = cij - πi + πj = cij – (πi+k) + (πj+k)
– This implies that we can fix arbitrarily the value of the
Lagrange multiplier in a node:
π1 = 0
– The rest of the Lagrange multipliers can be
calculated using the reduced cost expression of the
basic arcs:
rij = cij - πi + πj = 0 , ∀(i,j)∈T
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 68
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Reduce cost computation (I)
πi
i
1
5
2 -5
cij
j
πj
4
6
-7
2
5 -3
1
7
•
0
3
3
-2
1
-2
4
-1
Procedure:
1. Assign the value π1 = 0 to the
source node.
2. Visit the following node according
to the cross bar order.
3. Compute the Lagrange multiplier
based on:
cij - πi + πj = 0
4. Go to step 2.
πT = [ -5, -3, -7, -2, -2, -1]
(Cross bar order)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 69
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Langrange multipliers computation (II)
• No bàsic arc (k,l) introduction to base with flow θ :
xB(θ ) = B-1b - B-1 Nklθ = xB - yklθ = xB + ∆xB(θ )
1
i
(xBij , –yij )
2
j
(2,-1)
(4,0)
6
5
(3,-1)
• The vector ykl is defined by the
(3,0)
(1,+1)
3
(0,+1)
7
4
θ =+1
fundamental cycle associated to
(k,l):
– If (k,l) = (7,4) i θ =+1:
∆xB(+1 ) = - ykl = [0,-1,0,-1,+1,+1]T
• θ is increase until the flow of a
basic arc is cancelled.
– In the example, the arc (5,2) would
leave and (7,4) would enter to the
base with a value of 2
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 70
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Leaving variable selection
• Pivotation between the outgoing arc (5,2) and the entering arc (7,4):
1
1
2
2
5
6
7
3
3
4
4
7
• The values of xB are updated, π and r
from the new spanning tree
5
6
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 71
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Base change
Begin
A initial feasible tree T is defined;
While there are no basic arcs (i,j)∉ T with rij < 0 Do
The Lagrange multipliers π are defined and the flow x
associated to T;
An entering arc (k,l) is selected with rkl < 0;
The arc (k,l) is added to the tree and the outgoing arc (p,q);
The tree T is updated;
End while
End
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 72
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Simplex algorithm for network flows
MEIO/UPC-UB : NETWORK FLOWS
Example
i
0
2
9 1
cij
5
bj
j
-2
4
2
5
3
0
4
• Solve the problem of network
6 -5
flows associated to the
following network.
5
-2
(Solution)
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 73
Minimum Cost Network Flow Problems
bi
1. Formulate a network flow feasible problem of minimum cost with 6
nodes and 9 arcs.
2. Find a initial spanning tree feasible.
3. Solve the problem from the spanning tree found in the last section
with the simplex algorithm for network flows.
4. Find the optimal solution with the Giden support.
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 74
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Exercise
bi
i
0
2
9 1
j
-2
4
5
2
i πi
xij
j πj
• Solve the network flows
5
3
0
rij
bj
cij
6 -5
5
-2
4
• Begin:
1 0
9
-3
2 -3
2 7
-5 3 -5 5
4 -8
0
4
2
-2
problem associated to the
network.
6 -11
– Initial feasible tree:
T={(1,2),(2,3),(2,5),(2,4),(4,6)}
– xB=[9,0,2,7,5]T
– π = [0,-3,-5,-5,-8,-11]T
• 1st iteration:
– A\ T={(1,3),(3,5),(5,4),(5,6)}
r = [ -3, 4, 2, -2]T⇒ (1,3) entering
arc
– (2,3) outgoing arc
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 75
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Example (I)
1
0
-2 3
0
9
rij
1 -5 5
-2 3
4 -8
2
-2
0
j πj
2 -3
7
2
3
1
3
1 -5 5
5
-9 6
i πi
xij
• 2nd iteration:
6 -11
0
9
•
2 -3
2
7
2
4 -8
2
– Feasible tree:
T={(1,3),(1,2),(2,5),(2,4),(4,6)}
– xB=[0,9,2,7,5]T
(invariant for all arcs ∉ fundamental
cycles)
– π = [0,-2,-3,-5,-8,-11]T(invariant from
node 2)
– A\ T={(2,3),(3,5),(5,4),(5,6)}
r = [ 3, 1, 2, -2]T⇒ (5,6) entering arc
(4,6) outgoing arc.
3rd iteration:
– Feasible tree:
T={(1,3),(1,2),(2,5),(5,6),(2,4)}
– xB=[0,9,7,5,2]T
– π = [0,-2,-3,-5,-9,-8]T
– A\ T={(2,3),(3,5),(5,4),(4,6)}
r = [ 3, 1, 2, 2]T>0 ⇒ optimum
F.-Javier Heredia http://gnom.upc.edu/heredia
MCNFP- 76
Minimum Cost Network Flow Problems
MEIO/UPC-UB : NETWORK FLOWS
Example (II)