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research
Tight Bounds for Gomory-Hu-like Cut Counting
Authors
Rajesh Chitnis
Lior Kamma
Robert Krauthgamer
Publication date
1 January 2016
Publisher
Doi
Cite
View
on
arXiv
Abstract
By a classical result of Gomory and Hu (1961), in every edge-weighted graph
G
=
(
V
,
E
,
w
)
G=(V,E,w)
G
=
(
V
,
E
,
w
)
, the minimum
s
t
st
s
t
-cut values, when ranging over all
s
,
t
∈
V
s,t\in V
s
,
t
∈
V
, take at most
∣
V
∣
−
1
|V|-1
∣
V
∣
−
1
distinct values. That is, these
(
∣
V
∣
2
)
\binom{|V|}{2}
(
2
∣
V
∣
​
)
instances exhibit redundancy factor
Ω
(
∣
V
∣
)
\Omega(|V|)
Ω
(
∣
V
∣
)
. They further showed how to construct from
G
G
G
a tree
(
V
,
E
′
,
w
′
)
(V,E',w')
(
V
,
E
′
,
w
′
)
that stores all minimum
s
t
st
s
t
-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum
s
t
st
s
t
-cut problem. 1. Group-Cut: Consider the minimum
(
A
,
B
)
(A,B)
(
A
,
B
)
-cut, ranging over all subsets
A
,
B
⊆
V
A,B\subseteq V
A
,
B
⊆
V
of given sizes
∣
A
∣
=
α
|A|=\alpha
∣
A
∣
=
α
and
∣
B
∣
=
β
|B|=\beta
∣
B
∣
=
β
. The redundancy factor is
Ω
α
,
β
(
∣
V
∣
)
\Omega_{\alpha,\beta}(|V|)
Ω
α
,
β
​
(
∣
V
∣
)
. 2. Multiway-Cut: Consider the minimum cut separating every two vertices of
S
⊆
V
S\subseteq V
S
⊆
V
, ranging over all subsets of a given size
∣
S
∣
=
k
|S|=k
∣
S
∣
=
k
. The redundancy factor is
Ω
k
(
∣
V
∣
)
\Omega_{k}(|V|)
Ω
k
​
(
∣
V
∣
)
. 3. Multicut: Consider the minimum cut separating every demand-pair in
D
⊆
V
×
V
D\subseteq V\times V
D
⊆
V
×
V
, ranging over collections of
∣
D
∣
=
k
|D|=k
∣
D
∣
=
k
demand pairs. The redundancy factor is
Ω
k
(
∣
V
∣
k
)
\Omega_{k}(|V|^k)
Ω
k
​
(
∣
V
∣
k
)
. This result is a bit surprising, as the redundancy factor is much larger than in the first two problems. A natural application of these bounds is to construct small data structures that stores all relevant cut values, like the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.Comment: This version contains additional references to previous work (which have some overlap with our results), see Bibliographic Update 1.
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