[cs.DS]
9 Aug 2010
Constructive Algorithms for Discrepancy Minimization
Nikhil Bansal
*
Abstract
Given a set system
(
V,
S
)
,
V
=
{
1
,...,n
}
and
S
=
{
S
1
,...,S
m
}
, the minimum discrepancy
problem is to find a 2coloring
X
:
V
→ {−
1
,
+1
}
, such that each set is colored as evenly as
possible, i.e. find
X
to minimize
max
j
∈
[
m
]
v
v
v
∑
i
∈
S
j
X
(
i
)
v
v
v
.
In this paper we give the first polynomial time algorithms for discrepancy minimization that
achieve bounds similar to those known existentially using the socalled Entropy Method. We also
give a first approximationlike result for discrepancy. Specifically we give efficient randomized
algorithms to:
1. Construct an
O
(
n
1
/
2
)
discrepancy coloring for general sets systems when
m
=
O
(
n
)
, match
ing the celebrated result of Spencer [17] up to constant factors. Previously, no algorithmic
guarantee better than the random coloring bound, i.e.
O
((
n
log
n
)
1
/
2
)
, was known. More
generally, for
m
≥
n
, we obtain a discrepancy bound of
O
(
n
1
/
2
log(2
m/n
))
.
2. Construct a coloring with discrepancy
O
(
t
1
/
2
log
n
)
, if each element lies in at most
t
sets. This
matches the (nonconstructive) result of Srinivasan [19].
3. Construct a coloring with discrepancy
O
(
λ
log(
nm
))
, where
λ
is the hereditary discrepancy
of the set system.
The main idea in our algorithms is to produce a coloring over time by letting the color of the elements
perform a random walk (with tiny increments) starting from 0 until they reach
−
1
or
+1
. At each
time step the random hops for various elements are correlated using the solution to a semidefinite
program, where this program is determined by the current state and the entropy method.
1
Introduction
Let
(
V,
S
)
be a setsystem, where
V
=
{
1
,... ,n
}
are the elements and
S
=
{
S
1
,... ,S
m
}
is a collec
tion of subsets of
V
. Given a
{−
1
,
+1
}
coloring
X
of elements in
V
, let
X
(
S
j
) =
∑
i
∈
S
j
X
(
i
)
denote
the discrepancy of
X
for set
S
. The discrepancy of the collection
S
is defined as
disc
(
S
) = min
X
max
j
∈
[
m
]
X
(
S
j
)

.
Understanding the discrepancy of various setsystems has been a major area of research both in math
ematics and computer science, and this study has revealed fascinating connections to various areas of
mathematics. Discrepancy also has a range of applications to several topics in computer science such as
probabilistic and approximation algorithms, computational geometry, numerical integration, derandom
ization, communication complexity, machine learning, optimization and so on. We shall not attempt to
describe these connections and applications here, but refer the reader to [6, 9, 12].
∗
IBM T. J. Watson Research Center, Yorktown Heights, NY 10598. Email: