12,367 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 $n$ elements and $m$ sets and each element lies in $t$
randomly chosen sets. In this setting, Ezra and Lovett showed an $O((t \log
t)^{1/2})$ discrepancy bound in the regime when $n \leq m$ and an $O(1)$ bound
when $n \gg m^t$.
In this paper, we give a tight $O(\sqrt{t})$ bound for the entire range of
$n$ and $m$, under a mild assumption that $t = \Omega (\log \log m)^2$. The
result is based on two steps. First, applying the partial coloring method to
the case when $n = m \log^{O(1)} m$ and using the properties of the random set
system we show that the overall discrepancy incurred is at most $O(\sqrt{t})$.
Second, we reduce the general case to that of $n \leq m \log^{O(1)}m$ using LP
duality and a careful counting argument

### k-Trails: Recognition, Complexity, and Approximations

The notion of degree-constrained spanning hierarchies, also called k-trails,
was recently introduced in the context of network routing problems. They
describe graphs that are homomorphic images of connected graphs of degree at
most k. First results highlight several interesting advantages of k-trails
compared to previous routing approaches. However, so far, only little is known
regarding computational aspects of k-trails.
In this work we aim to fill this gap by presenting how k-trails can be
analyzed using techniques from algorithmic matroid theory. Exploiting this
connection, we resolve several open questions about k-trails. In particular, we
show that one can recognize efficiently whether a graph is a k-trail.
Furthermore, we show that deciding whether a graph contains a k-trail is
NP-complete; however, every graph that contains a k-trail is a (k+1)-trail.
Moreover, further leveraging the connection to matroids, we consider the
problem of finding a minimum weight k-trail contained in a graph G. We show
that one can efficiently find a (2k-1)-trail contained in G whose weight is no
more than the cheapest k-trail contained in G, even when allowing negative
weights.
The above results settle several open questions raised by Molnar, Newman, and
Sebo

### Improved Approximation Algorithms for Stochastic Matching

In this paper we consider the Stochastic Matching problem, which is motivated
by applications in kidney exchange and online dating. We are given an
undirected graph in which every edge is assigned a probability of existence and
a positive profit, and each node is assigned a positive integer called timeout.
We know whether an edge exists or not only after probing it. On this random
graph we are executing a process, which one-by-one probes the edges and
gradually constructs a matching. The process is constrained in two ways: once
an edge is taken it cannot be removed from the matching, and the timeout of
node $v$ upper-bounds the number of edges incident to $v$ that can be probed.
The goal is to maximize the expected profit of the constructed matching.
For this problem Bansal et al. (Algorithmica 2012) provided a
$3$-approximation algorithm for bipartite graphs, and a $4$-approximation for
general graphs. In this work we improve the approximation factors to $2.845$
and $3.709$, respectively.
We also consider an online version of the bipartite case, where one side of
the partition arrives node by node, and each time a node $b$ arrives we have to
decide which edges incident to $b$ we want to probe, and in which order. Here
we present a $4.07$-approximation, improving on the $7.92$-approximation of
Bansal et al.
The main technical ingredient in our result is a novel way of probing edges
according to a random but non-uniform permutation. Patching this method with an
algorithm that works best for large probability edges (plus some additional
ideas) leads to our improved approximation factors

### The Complexity of Scheduling for p-norms of Flow and Stretch

We consider computing optimal k-norm preemptive schedules of jobs that arrive
over time. In particular, we show that computing the optimal k-norm of flow
schedule, is strongly NP-hard for k in (0, 1) and integers k in (1, infinity).
Further we show that computing the optimal k-norm of stretch schedule, is
strongly NP-hard for k in (0, 1) and integers k in (1, infinity).Comment: Conference version accepted to IPCO 201

### Matroid Online Bipartite Matching and Vertex Cover

The Adwords and Online Bipartite Matching problems have enjoyed a renewed
attention over the past decade due to their connection to Internet advertising.
Our community has contributed, among other things, new models (notably
stochastic) and extensions to the classical formulations to address the issues
that arise from practical needs. In this paper, we propose a new generalization
based on matroids and show that many of the previous results extend to this
more general setting. Because of the rich structures and expressive power of
matroids, our new setting is potentially of interest both in theory and in
practice.
In the classical version of the problem, the offline side of a bipartite
graph is known initially while vertices from the online side arrive one at a
time along with their incident edges. The objective is to maintain a decent
approximate matching from which no edge can be removed. Our generalization,
called Matroid Online Bipartite Matching, additionally requires that the set of
matched offline vertices be independent in a given matroid. In particular, the
case of partition matroids corresponds to the natural scenario where each
advertiser manages multiple ads with a fixed total budget.
Our algorithms attain the same performance as the classical version of the
problems considered, which are often provably the best possible. We present
$1-1/e$-competitive algorithms for Matroid Online Bipartite Matching under the
small bid assumption, as well as a $1-1/e$-competitive algorithm for Matroid
Online Bipartite Matching in the random arrival model. A key technical
ingredient of our results is a carefully designed primal-dual waterfilling
procedure that accommodates for matroid constraints. This is inspired by the
extension of our recent charging scheme for Online Bipartite Vertex Cover.Comment: 19 pages, to appear in EC'1

### New developments in iterated rounding

Iterated rounding is a relatively recent technique in algorithm design, that despite its simplicity has led to several remarkable new results and also simpler proofs of many previous results. We will briefly survey some applications of the method, including some recent developments and giving a high level overview of the ideas

### Local Guarantees in Graph Cuts and Clustering

Correlation Clustering is an elegant model that captures fundamental graph
cut problems such as Min $s-t$ Cut, Multiway Cut, and Multicut, extensively
studied in combinatorial optimization. Here, we are given a graph with edges
labeled $+$ or $-$ and the goal is to produce a clustering that agrees with the
labels as much as possible: $+$ edges within clusters and $-$ edges across
clusters. The classical approach towards Correlation Clustering (and other
graph cut problems) is to optimize a global objective. We depart from this and
study local objectives: minimizing the maximum number of disagreements for
edges incident on a single node, and the analogous max min agreements
objective. This naturally gives rise to a family of basic min-max graph cut
problems. A prototypical representative is Min Max $s-t$ Cut: find an $s-t$ cut
minimizing the largest number of cut edges incident on any node. We present the
following results: $(1)$ an $O(\sqrt{n})$-approximation for the problem of
minimizing the maximum total weight of disagreement edges incident on any node
(thus providing the first known approximation for the above family of min-max
graph cut problems), $(2)$ a remarkably simple $7$-approximation for minimizing
local disagreements in complete graphs (improving upon the previous best known
approximation of $48$), and $(3)$ a $1/(2+\varepsilon)$-approximation for
maximizing the minimum total weight of agreement edges incident on any node,
hence improving upon the $1/(4+\varepsilon)$-approximation that follows from
the study of approximate pure Nash equilibria in cut and party affiliation
games

### On the number of matroids

We consider the problem of determining $m_n$, the number of matroids on $n$
elements. The best known lower bound on $m_n$ is due to Knuth (1974) who showed
that $\log \log m_n$ is at least $n-3/2\log n-1$. On the other hand, Piff
(1973) showed that $\log\log m_n\leq n-\log n+\log\log n +O(1)$, and it has
been conjectured since that the right answer is perhaps closer to Knuth's
bound.
We show that this is indeed the case, and prove an upper bound on $\log\log
m_n$ that is within an additive $1+o(1)$ term of Knuth's lower bound. Our proof
is based on using some structural properties of non-bases in a matroid together
with some properties of independent sets in the Johnson graph to give a
compressed representation of matroids.Comment: Final version, 17 page

### An entropy argument for counting matroids

We show how a direct application of Shearers' Lemma gives an almost optimum
bound on the number of matroids on $n$ elements.Comment: Short note, 4 page

### Sequential item pricing for unlimited supply

We investigate the extent to which price updates can increase the revenue of
a seller with little prior information on demand. We study prior-free revenue
maximization for a seller with unlimited supply of n item types facing m myopic
buyers present for k < log n days. For the static (k = 1) case, Balcan et al.
[2] show that one random item price (the same on each item) yields revenue
within a \Theta(log m + log n) factor of optimum and this factor is tight. We
define the hereditary maximizers property of buyer valuations (satisfied by any
multi-unit or gross substitutes valuation) that is sufficient for a significant
improvement of the approximation factor in the dynamic (k > 1) setting. Our
main result is a non-increasing, randomized, schedule of k equal item prices
with expected revenue within a O((log m + log n) / k) factor of optimum for
private valuations with hereditary maximizers. This factor is almost tight: we
show that any pricing scheme over k days has a revenue approximation factor of
at least (log m + log n) / (3k). We obtain analogous matching lower and upper
bounds of \Theta((log n) / k) if all valuations have the same maximum. We
expect our upper bound technique to be of broader interest; for example, it can
significantly improve the result of Akhlaghpour et al. [1]. We also initiate
the study of revenue maximization given allocative externalities (i.e.
influences) between buyers with combinatorial valuations. We provide a rather
general model of positive influence of others' ownership of items on a buyer's
valuation. For affine, submodular externalities and valuations with hereditary
maximizers we present an influence-and-exploit (Hartline et al. [13]) marketing
strategy based on our algorithm for private valuations. This strategy preserves
our approximation factor, despite an affine increase (due to externalities) in
the optimum revenue.Comment: 18 pages, 1 figur

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