180 research outputs found
On the discrepancy of random low degree set systems
Motivated by the celebrated Beck-Fiala conjecture, we consider the random
setting where there are elements and sets and each element lies in
randomly chosen sets. In this setting, Ezra and Lovett showed an discrepancy bound in the regime when and an bound
when .
In this paper, we give a tight bound for the entire range of
and , under a mild assumption that . The
result is based on two steps. First, applying the partial coloring method to
the case when and using the properties of the random set
system we show that the overall discrepancy incurred is at most .
Second, we reduce the general case to that of using LP
duality and a careful counting argument
An Algorithm for Koml\'os Conjecture Matching Banaszczyk's bound
We consider the problem of finding a low discrepancy coloring for sparse set
systems where each element lies in at most t sets. We give an efficient
algorithm that finds a coloring with discrepancy O((t log n)^{1/2}), matching
the best known non-constructive bound for the problem due to Banaszczyk. The
previous algorithms only achieved an O(t^{1/2} log n) bound. The result also
extends to the more general Koml\'{o}s setting and gives an algorithmic
O(log^{1/2} n) bound
Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma
We consider integer programming problems in standard form where , and . We show that such an integer program can be solved in time , where is an upper bound on each
absolute value of an entry in . This improves upon the longstanding best
bound of Papadimitriou (1981) of , where in addition,
the absolute values of the entries of also need to be bounded by .
Our result relies on a lemma of Steinitz that states that a set of vectors in
that is contained in the unit ball of a norm and that sum up to zero can
be ordered such that all partial sums are of norm bounded by . We also use
the Steinitz lemma to show that the -distance of an optimal integer and
fractional solution, also under the presence of upper bounds on the variables,
is bounded by . Here is again an
upper bound on the absolute values of the entries of . The novel strength of
our bound is that it is independent of . We provide evidence for the
significance of our bound by applying it to general knapsack problems where we
obtain structural and algorithmic results that improve upon the recent
literature.Comment: We achieve much milder dependence of the running time on the largest
entry in $b
Discrepancy and Signed Domination in Graphs and Hypergraphs
For a graph G, a signed domination function of G is a two-colouring of the
vertices of G with colours +1 and -1 such that the closed neighbourhood of
every vertex contains more +1's than -1's. This concept is closely related to
combinatorial discrepancy theory as shown by Fueredi and Mubayi [J. Combin.
Theory, Ser. B 76 (1999) 223-239]. The signed domination number of G is the
minimum of the sum of colours for all vertices, taken over all signed
domination functions of G. In this paper, we present new upper and lower bounds
for the signed domination number. These new bounds improve a number of known
results.Comment: 12 page
Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem
An important theorem of Banaszczyk (Random Structures & Algorithms `98)
states that for any sequence of vectors of norm at most and any
convex body of Gaussian measure in , there exists a
signed combination of these vectors which lands inside . A major open
problem is to devise a constructive version of Banaszczyk's vector balancing
theorem, i.e. to find an efficient algorithm which constructs the signed
combination.
We make progress towards this goal along several fronts. As our first
contribution, we show an equivalence between Banaszczyk's theorem and the
existence of -subgaussian distributions over signed combinations. For the
case of symmetric convex bodies, our equivalence implies the existence of a
universal signing algorithm (i.e. independent of the body), which simply
samples from the subgaussian sign distribution and checks to see if the
associated combination lands inside the body. For asymmetric convex bodies, we
provide a novel recentering procedure, which allows us to reduce to the case
where the body is symmetric.
As our second main contribution, we show that the above framework can be
efficiently implemented when the vectors have length ,
recovering Banaszczyk's results under this stronger assumption. More precisely,
we use random walk techniques to produce the required -subgaussian
signing distributions when the vectors have length , and
use a stochastic gradient ascent method to implement the recentering procedure
for asymmetric bodies
Improved Algorithmic Bounds for Discrepancy of Sparse Set Systems
We consider the problem of finding a low discrepancy coloring for sparse set
systems where each element lies in at most sets. We give an algorithm that
finds a coloring with discrepancy where is the
maximum cardinality of a set. This improves upon the previous constructive
bound of based on algorithmic variants of the partial
coloring method, and for small (e.g.) comes close to
the non-constructive bound due to Banaszczyk. Previously,
no algorithmic results better than were known even for . Our method is quite robust and we give several refinements and
extensions. For example, the coloring we obtain satisfies the stronger
size-sensitive property that each set in the set system incurs an discrepancy. Another variant can be used to
essentially match Banaszczyk's bound for a wide class of instances even where
is arbitrarily large. Finally, these results also extend directly to the
more general Koml\'{o}s setting
- …