True or False: Flow network with $n$ internal nodes has $2^n$ different cuts with all min capacity
Prove whether true or false: For every $n >0$, there exists a flow network with $n$ internal nodes such that there are $2^n$ different cuts that all have minimum capacity.
I have no idea how to do this. Please help!
discrete-mathematics graph-theory algorithms network-flow network
edited Nov 23 at 22:24
Key Flex
7,46941232
asked Nov 23 at 22:15
Tom Cruise
6
add a comment |
Prove whether true or false: For every $n >0$, there exists a flow network with $n$ internal nodes such that there are $2^n$ different cuts that all have minimum capacity.
I have no idea how to do this. Please help!
discrete-mathematics graph-theory algorithms network-flow network
edited Nov 23 at 22:24
Key Flex
7,46941232
asked Nov 23 at 22:15
Tom Cruise
6
Why the down comment on the question?
– Tom Cruise
Nov 23 at 23:46
add a comment |
Prove whether true or false: For every $n >0$, there exists a flow network with $n$ internal nodes such that there are $2^n$ different cuts that all have minimum capacity.
I have no idea how to do this. Please help!
discrete-mathematics graph-theory algorithms network-flow network
edited Nov 23 at 22:24
Key Flex
7,46941232
asked Nov 23 at 22:15
Tom Cruise
6
Prove whether true or false: For every $n >0$, there exists a flow network with $n$ internal nodes such that there are $2^n$ different cuts that all have minimum capacity.
I have no idea how to do this. Please help!
discrete-mathematics graph-theory algorithms network-flow network
discrete-mathematics graph-theory algorithms network-flow network
edited Nov 23 at 22:24
Key Flex
7,46941232
asked Nov 23 at 22:15
Tom Cruise
6
edited Nov 23 at 22:24
Key Flex
7,46941232
asked Nov 23 at 22:15
Tom Cruise
6
edited Nov 23 at 22:24
Key Flex
7,46941232
edited Nov 23 at 22:24
Key Flex
7,46941232
edited Nov 23 at 22:24
Key Flex
7,46941232
7,46941232
asked Nov 23 at 22:15
Tom Cruise
6
asked Nov 23 at 22:15
Tom Cruise
6
asked Nov 23 at 22:15
Tom Cruise
6
6
Why the down comment on the question?
– Tom Cruise
Nov 23 at 23:46
add a comment |
Why the down comment on the question?
– Tom Cruise
Nov 23 at 23:46
Why the down comment on the question?
– Tom Cruise
Nov 23 at 23:46
Why the down comment on the question?
– Tom Cruise
Nov 23 at 23:46
add a comment |
1 Answer
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oldest
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Yes, there is indeed such a network. So the answer to your true/false question is True.
Hint: Consider a network $cal{N}$ with a source $s$, sink $t$, and in addition, internal nodes $v_1,ldots, v_n$ (so $n+2$ nodes total), where the arcs in $cal{N}$ are ${(s,v_i)$; $i=1,ldots, n}$ $+$ ${(v_i,t)$; $i=1,ldots, n}$, and all arcs in $cal{N}$ have capacity 1.
LEFT FOR YOU TO CHECK: That there are indeed $2^n$ different min-capacity cuts.
answered Nov 24 at 12:01
Mike
2,804211
Yes Mike, this makes sense, thanks man!
– Tom Cruise
Nov 24 at 19:44
add a comment |
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1 Answer
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active
oldest
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1 Answer
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active
oldest
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active
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active
oldest
votes
Yes, there is indeed such a network. So the answer to your true/false question is True.
Hint: Consider a network $cal{N}$ with a source $s$, sink $t$, and in addition, internal nodes $v_1,ldots, v_n$ (so $n+2$ nodes total), where the arcs in $cal{N}$ are ${(s,v_i)$; $i=1,ldots, n}$ $+$ ${(v_i,t)$; $i=1,ldots, n}$, and all arcs in $cal{N}$ have capacity 1.
LEFT FOR YOU TO CHECK: That there are indeed $2^n$ different min-capacity cuts.
answered Nov 24 at 12:01
Mike
2,804211
Yes Mike, this makes sense, thanks man!
– Tom Cruise
Nov 24 at 19:44
add a comment |
Yes, there is indeed such a network. So the answer to your true/false question is True.
Hint: Consider a network $cal{N}$ with a source $s$, sink $t$, and in addition, internal nodes $v_1,ldots, v_n$ (so $n+2$ nodes total), where the arcs in $cal{N}$ are ${(s,v_i)$; $i=1,ldots, n}$ $+$ ${(v_i,t)$; $i=1,ldots, n}$, and all arcs in $cal{N}$ have capacity 1.
LEFT FOR YOU TO CHECK: That there are indeed $2^n$ different min-capacity cuts.
answered Nov 24 at 12:01
Mike
2,804211
Yes Mike, this makes sense, thanks man!
– Tom Cruise
Nov 24 at 19:44
add a comment |
Yes, there is indeed such a network. So the answer to your true/false question is True.
Hint: Consider a network $cal{N}$ with a source $s$, sink $t$, and in addition, internal nodes $v_1,ldots, v_n$ (so $n+2$ nodes total), where the arcs in $cal{N}$ are ${(s,v_i)$; $i=1,ldots, n}$ $+$ ${(v_i,t)$; $i=1,ldots, n}$, and all arcs in $cal{N}$ have capacity 1.
LEFT FOR YOU TO CHECK: That there are indeed $2^n$ different min-capacity cuts.
answered Nov 24 at 12:01
Mike
2,804211
Yes, there is indeed such a network. So the answer to your true/false question is True.
Hint: Consider a network $cal{N}$ with a source $s$, sink $t$, and in addition, internal nodes $v_1,ldots, v_n$ (so $n+2$ nodes total), where the arcs in $cal{N}$ are ${(s,v_i)$; $i=1,ldots, n}$ $+$ ${(v_i,t)$; $i=1,ldots, n}$, and all arcs in $cal{N}$ have capacity 1.
LEFT FOR YOU TO CHECK: That there are indeed $2^n$ different min-capacity cuts.
answered Nov 24 at 12:01
Mike
2,804211
edited Nov 24 at 13:28
answered Nov 24 at 12:01
Mike
2,804211
answered Nov 24 at 12:01
Mike
2,804211
answered Nov 24 at 12:01
Mike
2,804211
2,804211
Yes Mike, this makes sense, thanks man!
– Tom Cruise
Nov 24 at 19:44
add a comment |
Yes Mike, this makes sense, thanks man!
– Tom Cruise
Nov 24 at 19:44
Yes Mike, this makes sense, thanks man!
– Tom Cruise
Nov 24 at 19:44
Yes Mike, this makes sense, thanks man!
– Tom Cruise
Nov 24 at 19:44
add a comment |
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Why the down comment on the question?
– Tom Cruise
Nov 23 at 23:46