MAX CUT in Weighted Random Intersection Graphs and Discrepancy of Sparse Random Set Systems
Let V be a set of n vertices, M a set of m labels, and let R be an m × n matrix ofs independent Bernoulli random variables with probability of success p ; columns of R are incidence vectors of label sets assigned to vertices. A random instance G ( V , E , R T R ) of the weighted random intersection...
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| Vydáno v: | Algorithmica Ročník 85; číslo 9; s. 2817 - 2842 |
|---|---|
| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.09.2023
Springer Nature B.V |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Let
V
be a set of
n
vertices,
M
a set of
m
labels, and let
R
be an
m
×
n
matrix ofs independent Bernoulli random variables with probability of success
p
; columns of
R
are incidence vectors of label sets assigned to vertices. A random instance
G
(
V
,
E
,
R
T
R
)
of the weighted random intersection graph model is constructed by drawing an edge with weight equal to the number of common labels (namely
[
R
T
R
]
v
,
u
) between any two vertices
u
,
v
for which this weight is strictly larger than 0. In this paper we study the average case analysis of
Weighted Max Cut
, assuming the input is a weighted random intersection graph, i.e. given
G
(
V
,
E
,
R
T
R
)
we wish to find a partition of
V
into two sets so that the total weight of the edges having exactly one endpoint in each set is maximized. In particular, we initially prove that the weight of a maximum cut of
G
(
V
,
E
,
R
T
R
)
is concentrated around its expected value, and then show that, when the number of labels is much smaller than the number of vertices (in particular,
m
=
n
α
,
α
<
1
), a random partition of the vertices achieves asymptotically optimal cut weight with high probability. Furthermore, in the case
n
=
m
and constant average degree (i.e.
p
=
Θ
(
1
)
n
), we show that with high probability, a majority type randomized algorithm outputs a cut with weight that is larger than the weight of a random cut by a multiplicative constant strictly larger than 1. Then, we formally prove a connection between the computational problem of finding a (weighted) maximum cut in
G
(
V
,
E
,
R
T
R
)
and the problem of finding a 2-coloring that achieves minimum discrepancy for a set system
Σ
with incidence matrix
R
(i.e. minimum imbalance over all sets in
Σ
). We exploit this connection by proposing a (weak) bipartization algorithm for the case
m
=
n
,
p
=
Θ
(
1
)
n
that, when it terminates, its output can be used to find a 2-coloring with minimum discrepancy in a set system with incidence matrix
R
. In fact, with high probability, the latter 2-coloring corresponds to a bipartition with maximum cut-weight in
G
(
V
,
E
,
R
T
R
)
. Finally, we prove that our (weak) bipartization algorithm terminates in polynomial time, with high probability, at least when
p
=
c
n
,
c
<
1
. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-023-01121-3 |