Max Flow, Min Cut, and Matchings
NB: Some problems below were taken from, or inspired by, exercises in the
book Algorithm Design by Jon Kleinberg and ยด
Eva Tardos (Pearson Interna-
tional Edition, 2006).
1. The figure below shows a flow network on which an s-tflow is shown.
The capacity of each edge appears as a label next to the edge, and the
numbers in boxes give the amount of flow sent on each edge. (Edges
without boxed numbers have no flow being sent on them.)
(a) What is the value of this flow?
(b) Is this a maximum s-tflow in this graph? If not, find a maximum
s-tflow.
(c) Find a minimum s-tcut. (Specify which vertices belong to the
sets of the cut.)
llll
l
l
- - -
๎๎๎๎๎๎๎๎๎๎๎๎๎
๎1
PPPPPPPPPPPPP
Pq
PPPPPPPPPPPPP
Pq
๎๎๎๎๎๎๎๎๎๎๎๎๎
๎1
๎๎๎๎๎
@@@@R
@@@@R
๎๎๎๎๎
s tx y
w
z
44
77
10 6
2
2
2
2
10 7
10 4
77
66
2. Find all minimum s-tcuts in the following graph. The capacity of
each edge appears as a label next to the edge.
s
u
w
t
2
2
2
2
2
3. Consider the flow network Hbelow with source sand sink t. The edge
capacities are the numbers given near each edge.
1
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