Combinatorial optimization
Papp László <lazsa@cs.bme.hu>
2022. spring 7th practice
1. Find a maximums−t ow in each of these networks and prove that they are maximum.
1
4
4
4 10
4
s t
8 3 6
2
2 3 6
3 2
12
2 2
s
t 2
2 8
3
2 2 2 3
2 3
2. In the left network given in the rst problem, replace the top-left edge's capacity (which was 8) by a parameter p. Determine the value of the maximum s−t ow as a function of p.
3. Let (G, s, t, c)~ be a network such that c(e) is divisible by 3 for each edge e. Which ones of these statements are true?
Each s−t cut's capacity is divisible by 3.
A minimum s−t cut's capacity is divisible by 3.
The value of a maximum ow is divisible by 3.
The value among each edge in a maximum ow is divisible by 3. 4. Is it possible that in the value of a maximum ow is17 in this network?
s
15 t
12 6
6 6
9 9 15
9 9
9
9
9 6
6 6
6 12
1
5. In the given network each number in a round bracket is the capacity of the corresponding edge. The numbers which are not contained in brackets are values of an S −T ow.
Determine the a, b and c values of the ow.
Determine whether it is a maximum S−T ow or not! If it is not a maximum S −T ow, then nd one! Hint: Use the Ford-Fulkerson algorithm!
A
5 (5)
12 (12)
3 (3) 7 (10)
2 (2)
4 (9)
2 (4) 0 (5)
14 (14)
2 (9)
8 (9) c (7)
a (16)
b (10)
S C D T
E F
B
6. Is it true, that in any network(G, s, t, c)~ , there is an edge e, such that if we decrease c(e) byε, then the maximums−t ow value decreases byε?
Is it true, that in any network (G, s, t, c)~ , there is an edge e, such that if we increase c(e) byε, then the maximums−t ow value increases byε?
7. Determine all possible values of p = c(ef)~ which implies that the maximum s−t ow value is 42.
s t
a b
c d
e
99
100 22 15
10 100 10
9 44 8 p
10
f
22 11
8. LetG~ be a directed graph and letcbe a capacity function over its edge set. Assume that there is an s−t ow of value m and at−w ow of value m. Prove that, then an s−w ow of value m exists! Hint: Use cuts.
9. Let s and t be opposite vertices of a cube. Orient the edges of the cube towards t. We have 4 pieces of 1, 4 pieces of 2 and 4 pieces of 3. How to assign these numbers to the edges as capacities to have the biggest possible maximum s−t ow?
10. In a (G, s, t, c)~ network, each edge is either red, white or green. If we consider only the red and white edges, then in the obtained network the maximum s−t ow's value is10. The same holds if we consider the network containing only the red and green edges, or the network containing the white and green edges. Prove that in the (G, s, t, c)~ network the maximum s−t ow's value is at least 15.
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