Minimum Cost Flow
Algorithms and Networks
This lecture
• The minimum cost flow problem: statement
and applications
• The cycle cancelling algorithm
• A polynomial time variant of cycle
cancelling
• The successive shortest paths algorithm
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Algorithms and Networks: Minimum Cost Flow
Minimum Cost Flow I
• Edges have
– Capacity: c(u,v): bound on amount of flow that
can go through the edge
– Cost: cost(u,v): cost that must be paid per unit
of flow that goes through the edge.
• Cost of flow f:
– Sum over all (u,v) of: f (u,v) * cost(u,v).
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Algorithms and Networks: Minimum Cost Flow
Minimum cost flow problem
• Given: Network G, c, cost, s, t, and a target
flow value r.
• Question: Find a flow from s to t with value
r, with minimum cost.
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Algorithms and Networks: Minimum Cost Flow
Unbounded capacities
• Some edges may have unbounded capacities
– If there is a cycle of negative cost with only
edges with unbounded capacity:
• Arbitrary small cost (degenerate case)
• Finding such a cycle: shortest paths algorithms (see
lectures by Johan)
– Otherwise: simple transformation to bounded
capacities
• E.g., set each unbounded capacity to sum of all
bounded capacities
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Algorithms and Networks: Minimum Cost Flow
Separate demands
• Similar to
max-flow with
multiple
sources and
multiple sinks
• Or: work with
“demand”,
which can be
positive or
negative
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s1
t1
t
G
s
sk
tr
Algorithms and Networks: Minimum Cost Flow
Nonnegative arc costs
• It is possible to transform the network to case
where all costs are nonnegative.
• In case of negative costs: assume bounded
capacity. Modify network to equivalent one with
nonnegative costs:
Or work
with separate
demands
Cost -r
b
s
Cost 0
Capacity c
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b
Capacity c
Cost r
Capacity c
a
a
Cost 0
Capacity c
t
Algorithms and Networks: Minimum Cost Flow
Applications
• Transport problems
• Minimum cost matchings
• Reconstruction of Left Ventricle from X-ray
projections
– Image: 2d bit array; known are sums of columns, rows;
probabilities for each bit
– Look for image with correct row and column sums of
maximum probability
– Can be modelled as minimum cost flow problem
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Algorithms and Networks: Minimum Cost Flow
Application: Optimal loading
of hopping airplane
1
2
n
bij weight units (or passengers) can be transported from i to j
Each gives a profit of fij
Plane can never carry more than p units
How much units do we transport of each type for maximum profit?
14
13
12
1
9
Capacity i  j: p
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Node ij has supply bij
Cost from ij to i: - fij
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34
Node i has demand
sum over all bji
2
3
4
All other arcs
infinite cap.
All other costs 0
Algorithms and Networks: Minimum Cost Flow
Cycles and circulations;
paths and flows
• We can view:
– Cycles as a special type of circulations
– Paths as a special type of flows
– If we have a directed cycle c, then we can see it as
a circulation:
• c(e) = 1 when e is on c
• c(e) = 0 otherwise
– If we have a path p from s to t, we can see it as a
flow with value 1:
• p(e) = 1 when e is on p
• p(e) = 0 otherwise
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Algorithms and Networks: Minimum Cost Flow
First: useful theorems on flows
and circulations
Theorem Let f be a circulation. Then f is a nonnegative
linear combination of cycles in G.
Proof.
• Ignore lower bounds.
• If f(v,w) = 0 everywhere, then it clearly holds.
• Otherwise, there is a directed cycle c in G with each edge e on c:
f(e)>0
• Let r be the minimum flow of an edge on c, i.e., r = min {f(e) | e
on c}
• Use induction: the theorem also holds for f – cr:
f-cr(e) = f(e) when e is not on c
f-cr(e) = f(e) – r when e is on c
• …
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A&N: Maximum flow
Corollary for flows
• Corollary: Let f be a flow from s to t. f is a
nonnegative linear combination of cycles in
G and paths from s to t.
• Proof: Add edge from t to s with flow equal
to value(f). Now, we have a circulation.
Apply previous theorem. Cycles that use
(t,s) correspond to a path from s to t. …
QED
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Algorithms and Networks: Minimum Cost Flow
More on these linear
combinations
• If circulation f is integer, then f can be written as
an`integer scalared’ linear combination of cycles
in G
• A flow f with each f(e) an integer is the integer
scalared linear combination of cycles and paths
from s to t
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Algorithms and Networks: Minimum Cost Flow
Algorithms for finding minimum
cost flow
• Here: two simple algorithms, based on the
residual network
• Better algorithms exist, see also the exercise
sets
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Algorithms and Networks: Minimum Cost Flow
Residual network
• Define the residual network Gf of a flow f
– Capacities as for maximum flow algorithms
– If f (u,v)>0, then
costf (u,v) = cost(u,v), and
costf (v,u) = – cost(u,v).
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Algorithms and Networks: Minimum Cost Flow
Example
Capacity 5, cost 3
a
Suppose we send 1
flow from a to b
b
Capacity 4, cost 3
Capacity 2, cost 6
a
In Gf:
b
Capacity 2, cost 6
Capacity 1, cost -3
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Algorithms and Networks: Minimum Cost Flow
Cycle cancelling algorithm
• Make a feasible flow f in the network
• while Gf has a negative cycle do
– Find a negative cycle C in Gf
– Let D be the minimum residual capacity cf of an
edge on C
– Add D units of flow to each edge on C: this is a
new feasible flow of smaller cost
• Output f.
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Algorithms and Networks: Minimum Cost Flow
Cycle cancelling algorithm is
correct
• Theorem: a flow f has minimum costs, if and only
if Gf has no negative cycle.
• Proof:
– If Gf has negative cycle, then we can improve f to one
with smaller cost.
– Suppose f is a flow, and f’ is an optimal flow. f’ – f is a
circulation in Gf, hence a linear combination of cycles,
and if f is not optimal, then the total cost of these cycles
is negative, so there is a negative cycle in this set: it is a
cycle in Gf.
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Algorithms and Networks: Minimum Cost Flow
More on the cycle cancelling
algorithm
• No guarantee that it uses polynomial time.
• Corollary: if all capacities and target flow value
are integral, then there is an optimal integer
minimum cost flow.
– The cycle cancelling algorithm finds an integer flow in
this case.
• Variant: using always the minimum mean cost
cycle gives a polynomial time algorithm!
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Algorithms and Networks: Minimum Cost Flow
Minimum mean-cost circulation
algorithm
• Cycle cancelling algorithm but find always
the minimum mean cost cycle and use that.
– O(nm) time to find the cycle.
– A theorem shows that O(nm2 log2 n) iterations
are sufficient.
– O(n2m3 log2 n) algorithm.
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Algorithms and Networks: Minimum Cost Flow
Lower bounds
• Cycle cancelling algorithm also works with
lower bounds:
– First find a feasible flow, respecting lower
bounds (see lecture on max flow)
– Remaining algorithm the same, with the
residual network defined as for flows with
lower bounds
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Algorithms and Networks: Minimum Cost Flow
Successive shortest paths
• Start with flow f with f(u,v)=0 for all u,v.
• repeat until value(f ) = r
– Find the shortest path P in Gf from s to t
– Let q be the minimum residual capacity of an
edge on P.
– Send min(q,r – value(f)) additional units of
flow across P.
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Algorithms and Networks: Minimum Cost Flow
Time and correctness
• Time: possibly exponential time …
• Correctness: only if G has no cycle with
negative cost
• If G has a negative cycle, the algorithm may
give an incorrect answer, e.g., when we
have target flow r=0
• Proof of correctness: next
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Algorithms and Networks: Minimum Cost Flow
Correctness of successive
shortest paths algorithm
• Theorem: If G has no negative cycle, then
the successive shortest paths algorithm
gives the minimum cost flow.
• Proof follows from
• Claim: The ssp-algorithm maintains the
following invariant:
f has minimum cost among all flows with
value value(f).
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Algorithms and Networks: Minimum Cost Flow
Proof of claim
• Suppose we obtain f’ from f by sending flow over a path P.
• From invariant: f has minimum cost for flows with value equal
to value(f)
• Suppose f’’ is a minimum cost flow with same value as f’.
• f’’ – f is a flow in Gf. Hence, it can be written as a linear
combination of cycles in Gf and paths from s to t in Gf.
– All cycles in Gf have cost at least 0: otherwise f is not of minimum
cost
– All paths in this combination have cost at least the cost of P: this is
how the algorithm works (P is shortest path)
– It follows that cost(f’’ – f) >= cost (f’ – f)
– So, cost(f’’) >= cost(f’) which show minimality of cost of f’.
• QED
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Algorithms and Networks: Minimum Cost Flow
Finally
• More efficient algorithms exist
– Some use scaling techniques:
• Scaling on capacities
• Scaling on costs
– Scaling will be discussed in the chapter on
shortest paths
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Algorithms and Networks: Minimum Cost Flow