289 research outputs found
Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms
Constrained submodular maximization problems have long been studied, with
near-optimal results known under a variety of constraints when the submodular
function is monotone. The case of non-monotone submodular maximization is less
understood: the first approximation algorithms even for the unconstrainted
setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC
'09, APPROX '09) show how to approximately maximize non-monotone submodular
functions when the constraints are given by the intersection of p matroid
constraints; their algorithm is based on local-search procedures that consider
p-swaps, and hence the running time may be n^Omega(p), implying their algorithm
is polynomial-time only for constantly many matroids. In this paper, we give
algorithms that work for p-independence systems (which generalize constraints
given by the intersection of p matroids), where the running time is poly(n,p).
Our algorithm essentially reduces the non-monotone maximization problem to
multiple runs of the greedy algorithm previously used in the monotone case.
Our idea of using existing algorithms for monotone functions to solve the
non-monotone case also works for maximizing a submodular function with respect
to a knapsack constraint: we get a simple greedy-based constant-factor
approximation for this problem.
With these simpler algorithms, we are able to adapt our approach to
constrained non-monotone submodular maximization to the (online) secretary
setting, where elements arrive one at a time in random order, and the algorithm
must make irrevocable decisions about whether or not to select each element as
it arrives. We give constant approximations in this secretary setting when the
algorithm is constrained subject to a uniform matroid or a partition matroid,
and give an O(log k) approximation when it is constrained by a general matroid
of rank k.Comment: In the Proceedings of WINE 201
Online Contention Resolution Schemes
We introduce a new rounding technique designed for online optimization
problems, which is related to contention resolution schemes, a technique
initially introduced in the context of submodular function maximization. Our
rounding technique, which we call online contention resolution schemes (OCRSs),
is applicable to many online selection problems, including Bayesian online
selection, oblivious posted pricing mechanisms, and stochastic probing models.
It allows for handling a wide set of constraints, and shares many strong
properties of offline contention resolution schemes. In particular, OCRSs for
different constraint families can be combined to obtain an OCRS for their
intersection. Moreover, we can approximately maximize submodular functions in
the online settings we consider.
We, thus, get a broadly applicable framework for several online selection
problems, which improves on previous approaches in terms of the types of
constraints that can be handled, the objective functions that can be dealt
with, and the assumptions on the strength of the adversary. Furthermore, we
resolve two open problems from the literature; namely, we present the first
constant-factor constrained oblivious posted price mechanism for matroid
constraints, and the first constant-factor algorithm for weighted stochastic
probing with deadlines.Comment: 33 pages. To appear in SODA 201
Faster Submodular Maximization for Several Classes of Matroids
The maximization of submodular functions have found widespread application in areas such as machine learning, combinatorial optimization, and economics, where practitioners often wish to enforce various constraints; the matroid constraint has been investigated extensively due to its algorithmic properties and expressive power. Though tight approximation algorithms for general matroid constraints exist in theory, the running times of such algorithms typically scale quadratically, and are not practical for truly large scale settings. Recent progress has focused on fast algorithms for important classes of matroids given in explicit form. Currently, nearly-linear time algorithms only exist for graphic and partition matroids [Alina Ene and Huy L. Nguyen, 2019]. In this work, we develop algorithms for monotone submodular maximization constrained by graphic, transversal matroids, or laminar matroids in time near-linear in the size of their representation. Our algorithms achieve an optimal approximation of 1-1/e-ε and both generalize and accelerate the results of Ene and Nguyen [Alina Ene and Huy L. Nguyen, 2019]. In fact, the running time of our algorithm cannot be improved within the fast continuous greedy framework of Badanidiyuru and Vondrák [Ashwinkumar Badanidiyuru and Jan Vondrák, 2014].
To achieve near-linear running time, we make use of dynamic data structures that maintain bases with approximate maximum cardinality and weight under certain element updates. These data structures need to support a weight decrease operation and a novel Freeze operation that allows the algorithm to freeze elements (i.e. force to be contained) in its basis regardless of future data structure operations. For the laminar matroid, we present a new dynamic data structure using the top tree interface of Alstrup, Holm, de Lichtenberg, and Thorup [Stephen Alstrup et al., 2005] that maintains the maximum weight basis under insertions and deletions of elements in O(log n) time. This data structure needs to support certain subtree query and path update operations that are performed every insertion and deletion that are non-trivial to handle in conjunction. For the transversal matroid the Freeze operation corresponds to requiring the data structure to keep a certain set S of vertices matched, a property that we call S-stability. While there is a large body of work on dynamic matching algorithms, none are S-stable and maintain an approximate maximum weight matching under vertex updates. We give the first such algorithm for bipartite graphs with total running time linear (up to log factors) in the number of edges
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Diversity Maximization Under Matroid Constraints
Aggregator websites typically present documents in the form of representative clusters. In order for users to get a broader perspective,it is important to deliver a diversified set of representative documents in those clusters. One approach to diversification is to maximize the average dissimilarity among documents. Another way to capture diversity is to avoid showing several documents from the same category (e.g. from the same news channel). We model the latter approach as a (partition) matroid constraint, and study diversity maximization problems under matroid constraints. We present the first constant-factor approximation algorithm for this problem,using a new technique. Our local search 0:5-approximation algorithm is also the first constant-factor approximation for the maxdispersion problem under matroid constraints. Our combinatorial proof technique for maximizing diversity under matroid constraints uses the existence of a family of Latin squares which may also be of independent interest. In order to apply these diversity maximization algorithms in the context of aggregator websites and as a preprocessing step for our diversity maximization tool, we develop greedy clustering algorithms that maximize weighted coverage of a predefined set of topics. Our algorithms are based on computing a set of cluster centers, where clusters are formed around them. We show the better performance of our algorithms for diversity and coverage maximization by running experiments on real (Twitter) and synthetic data in the context of real-time search over micro-posts. Finally we perform a user study validating our algorithms and diversity metrics
Online Matroid Intersection: Submodular Water-Filling and Matroidal Welfare Maximization
We study two problems in online matroid intersection. First, we consider the
problem of maximizing the size of a common independent set between a general
matroid and a partition matroid whose parts arrive online. This captures the
classic online bipartite matching problem when both matroids are partition
matroids. Our main result is a -competitive algorithm for
the fractional version of this problem. This applies even for the poly-matroid
setting, where the rank function of the offline matroid is replaced with a
general monotone submodular function. The key new ingredient for this result is
the construction of a ''water level'' vector for poly-matroids, which allows us
to generalize the classic water-filling algorithm for online bipartite
matching. This construction reveals connections to submodular utility
allocation markets and principal partition sequences of matroids.
Our second result concerns the Online Submodular Welfare Maximization (OSWM)
problem, in which items arriving online are allocated among a set of agents
with the goal of maximizing their overall utility. If the utility function of
each agent is a monotone, submodular function over the set of available items,
then a simple greedy algorithm achieves a competitive ratio of .
Kapralov, Post, and Vondr\'ak showed that in this case, no polynomial time
algorithm achieves a competitive ratio of for any
unless NP = RP (SODA, 2013). We extend the RANKING algorithm
of Karp, Vazirani, and Vazirani (STOC, 1990) to achieve an optimal
-competitive algorithm for OSWM in the case that the utility
function of each agent is the rank function of a matroid
The Submodular Secretary Problem Goes Linear
During the last decade, the matroid secretary problem (MSP) became one of the
most prominent classes of online selection problems. Partially linked to its
numerous applications in mechanism design, substantial interest arose also in
the study of nonlinear versions of MSP, with a focus on the submodular matroid
secretary problem (SMSP). So far, O(1)-competitive algorithms have been
obtained for SMSP over some basic matroid classes. This created some hope that,
analogously to the matroid secretary conjecture, one may even obtain
O(1)-competitive algorithms for SMSP over any matroid. However, up to now, most
questions related to SMSP remained open, including whether SMSP may be
substantially more difficult than MSP; and more generally, to what extend MSP
and SMSP are related.
Our goal is to address these points by presenting general black-box
reductions from SMSP to MSP. In particular, we show that any O(1)-competitive
algorithm for MSP, even restricted to a particular matroid class, can be
transformed in a black-box way to an O(1)-competitive algorithm for SMSP over
the same matroid class. This implies that the matroid secretary conjecture is
equivalent to the same conjecture for SMSP. Hence, in this sense SMSP is not
harder than MSP. Also, to find O(1)-competitive algorithms for SMSP over a
particular matroid class, it suffices to consider MSP over the same matroid
class. Using our reductions we obtain many first and improved O(1)-competitive
algorithms for SMSP over various matroid classes by leveraging known algorithms
for MSP. Moreover, our reductions imply an O(loglog(rank))-competitive
algorithm for SMSP, thus, matching the currently best asymptotic algorithm for
MSP, and substantially improving on the previously best
O(log(rank))-competitive algorithm for SMSP
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