72 research outputs found
Constrained Submodular Maximization via New Bounds for DR-Submodular Functions
Submodular maximization under various constraints is a fundamental problem
studied continuously, in both computer science and operations research, since
the late 's. A central technique in this field is to approximately
optimize the multilinear extension of the submodular objective, and then round
the solution. The use of this technique requires a solver able to approximately
maximize multilinear extensions. Following a long line of work, Buchbinder and
Feldman (2019) described such a solver guaranteeing -approximation for
down-closed constraints, while Oveis Gharan and Vondr\'ak (2011) showed that no
solver can guarantee better than -approximation. In this paper, we
present a solver guaranteeing -approximation, which significantly
reduces the gap between the best known solver and the inapproximability result.
The design and analysis of our solver are based on a novel bound that we prove
for DR-submodular functions. This bound improves over a previous bound due to
Feldman et al. (2011) that is used by essentially all state-of-the-art results
for constrained maximization of general submodular/DR-submodular functions.
Hence, we believe that our new bound is likely to find many additional
applications in related problems, and to be a key component for further
improvement.Comment: 48 page
A Randomness Threshold for Online Bipartite Matching, via Lossless Online Rounding
Over three decades ago, Karp, Vazirani and Vazirani (STOC'90) introduced the
online bipartite matching problem. They observed that deterministic algorithms'
competitive ratio for this problem is no greater than , and proved that
randomized algorithms can do better. A natural question thus arises: \emph{how
random is random}? i.e., how much randomness is needed to outperform
deterministic algorithms? The \textsc{ranking} algorithm of Karp et
al.~requires random bits, which, ignoring polylog terms,
remained unimproved. On the other hand, Pena and Borodin (TCS'19) established a
lower bound of random bits for any
competitive ratio.
We close this doubly-exponential gap, proving that, surprisingly, the lower
bound is tight. In fact, we prove a \emph{sharp threshold} of random bits for the randomness necessary and sufficient to
outperform deterministic algorithms for this problem, as well as its
vertex-weighted generalization. This implies the same threshold for the advice
complexity (nondeterminism) of these problems.
Similar to recent breakthroughs in the online matching literature, for
edge-weighted matching (Fahrbach et al.~FOCS'20) and adwords (Huang et
al.~FOCS'20), our algorithms break the barrier of by randomizing matching
choices over two neighbors. Unlike these works, our approach does not rely on
the recently-introduced OCS machinery, nor the more established randomized
primal-dual method. Instead, our work revisits a highly-successful online
design technique, which was nonetheless under-utilized in the area of online
matching, namely (lossless) online rounding of fractional algorithms. While
this technique is known to be hopeless for online matching in general, we show
that it is nonetheless applicable to carefully designed fractional algorithms
with additional (non-convex) constraints
Online Algorithms for Maximum Cardinality Matching with Edge Arrivals
In the adversarial edge arrival model for maximum cardinality matching, edges of an unknown graph are revealed one-by-one in arbitrary order, and should be irrevocably accepted or rejected. Here, the goal of an online algorithm is to maximize the number of accepted edges while maintaining a feasible matching at any point in time. For this model, the standard greedy heuristic is 1/2-competitive, and on the other hand, no algorithm that outperforms this ratio is currently known, even for very simple graphs.
We present a clean Min-Index framework for devising a family of randomized algorithms, and provide a number of positive and negative results in this context. Among these results, we present a 5/9-competitive algorithm when the underlying graph is a forest, and prove that this ratio is best possible within the Min-Index framework. In addition, we prove a new general upper bound of 2/(3+1/phi^2) ~ 0.5914 on the competitiveness of any algorithm in the edge arrival model. Interestingly, this bound holds even for an easier model in which vertices (along with their adjacent edges) arrive online, and when the underlying graph is a tree of maximum degree at most 3
Comparing Apples and Oranges: Query Tradeoff in Submodular Maximization
Fast algorithms for submodular maximization problems have a vast potential
use in applicative settings, such as machine learning, social networks, and
economics. Though fast algorithms were known for some special cases, only
recently Badanidiyuru and Vondr\'{a}k (2014) were the first to explicitly look
for such algorithms in the general case of maximizing a monotone submodular
function subject to a matroid independence constraint. The algorithm of
Badanidiyuru and Vondr\'{a}k matches the best possible approximation guarantee,
while trying to reduce the number of value oracle queries the algorithm
performs.
Our main result is a new algorithm for this general case which establishes a
surprising tradeoff between two seemingly unrelated quantities: the number of
value oracle queries and the number of matroid independence queries performed
by the algorithm. Specifically, one can decrease the former by increasing the
latter and vice versa, while maintaining the best possible approximation
guarantee. Such a tradeoff is very useful since various applications might
incur significantly different costs in querying the value and matroid
independence oracles. Furthermore, in case the rank of the matroid is ,
where is the size of the ground set and is an absolute constant smaller
than , the total number of oracle queries our algorithm uses can be made to
have a smaller magnitude compared to that needed by Badanidiyuru and
Vondr\'{a}k. We also provide even faster algorithms for the well studied
special cases of a cardinality constraint and a partition matroid independence
constraint, both of which capture many real-world applications and have been
widely studied both theorically and in practice.Comment: 29 pages, accepted to SODA 201
Metrical Service Systems with Transformations
We consider a generalization of the fundamental online metrical service systems (MSS) problem where the feasible region can be transformed between requests. In this problem, which we call T-MSS, an algorithm maintains a point in a metric space and has to serve a sequence of requests. Each request is a map (transformation) : → between subsets and of the metric space. To serve it, the algorithm has to go to a point ∈ , paying the distance from its previous position. Then, the transformation is applied, modifying the algorithm’s state to ( ). Such transformations can model, e.g., changes to the environment that are outside of an algorithm’s control, and we therefore do not charge any additional cost to the algorithm when the transformation is applied. The transformations also allow to model requests occurring in the -taxi problem.
We show that for -Lipschitz transformations, the competitive ratio is Θ()−2 on -point metrics. Here, the upper bound is achieved by a deterministic algorithm and the lower bound holds even for randomized algorithms. For the -taxi problem, we prove a competitive ratio of Õ(( log )2). For chasing convex bodies, we show that even with contracting transformations no competitive algorithm exists.
The problem T-MSS has a striking connection to the following deep mathematical question: Given a finite metric space M, what is the required cardinality of an extension M̂ ⊇ M where each partial isometry on M extends to an automorphism? We give partial answers for special cases
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