Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTowards Nearly-linear Time Algorithms for Submodular Maximization with a Matroid Constraint
85
0
0.0
(
0
)
Added by
Alina Ene
Publication date
2018
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a emph{general matroid constraint} with a $1 - 1/e - epsilon$ approximation that achieves a fast running time provided we have a fast data structure for maintaining a maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the emph{first algorithms} for these classes of matroids that achieve a nearly-optimal, $1 - 1/e - epsilon$ approximation, using a nearly-linear number of function evaluations and arithmetic operations.
rate research
Read More
We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, $1 - 1/e - epsilon$ approximation, using $(1/epsilon)^{O(1/epsilon^4)} n log^2{n}$ function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to $Omega(n^2)$ running times, since evaluating the multilinear extension is expensive. Our algorithm maintains a fractional solution with only a constant number of entries that are strictly fractional, which allows us to overcome this obstacle.
In this paper, we study the tradeoff between the approximation guarantee and adaptivity for the problem of maximizing a monotone submodular function subject to a cardinality constraint. The adaptivity of an algorithm is the number of sequential rounds of queries it makes to the evaluation oracle of the function, where in every round the algorithm is allowed to make polynomially-many parallel queries. Adaptivity is an important consideration in settings where the objective function is estimated using samples and in applications where adaptivity is the main running time bottleneck. Previous algorithms achieving a nearly-optimal $1 - 1/e - epsilon$ approximation require $Omega(n)$ rounds of adaptivity. In this work, we give the first algorithm that achieves a $1 - 1/e - epsilon$ approximation using $O(ln{n} / epsilon^2)$ rounds of adaptivity. The number of function evaluations and additional running time of the algorithm are $O(n mathrm{poly}(log{n}, 1/epsilon))$.
We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $1-1/e-epsilon$ approximation for monotone functions and a $1/e-epsilon$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(log^2{n}/epsilon^3)$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e-epsilon$ approximation using $O(log(n/epsilon) log(1/epsilon) log(n+m)/ epsilon^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $1-1/e-epsilon$ approximation in $O(log(n/epsilon)log(m)/epsilon^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.
The growing need to deal with massive instances motivates the design of algorithms balancing the quality of the solution with applicability. For the latter, an important measure is the emph{adaptive complexity}, capturing the number of sequential rounds of parallel computation needed. In this work we obtain the first emph{constant factor} approximation algorithm for non-monotone submodular maximization subject to a knapsack constraint with emph{near-optimal} $O(log n)$ adaptive complexity. Low adaptivity by itself, however, is not enough: one needs to account for the total number of function evaluations (or value queries) as well. Our algorithm asks $tilde{O}(n^2)$ value queries, but can be modified to run with only $tilde{O}(n)$ instead, while retaining a low adaptive complexity of $O(log^2n)$. Besides the above improvement in adaptivity, this is also the first emph{combinatorial} approach with sublinear adaptive complexity for the problem and yields algorithms comparable to the state-of-the-art even for the special cases of cardinality constraints or monotone objectives. Finally, we showcase our algorithms applicability on real-world datasets.
Which ads should we display in sponsored search in order to maximize our revenue? How should we dynamically rank information sources to maximize the value of the ranking? These applications exhibit strong diminishing returns: Redundancy decreases the marginal utility of each ad or information source. We show that these and other problems can be formalized as repeatedly selecting an assignment of items to positions to maximize a sequence of monotone submodular functions that arrive one by one. We present an efficient algorithm for this general problem and analyze it in the no-regret model. Our algorithm possesses strong theoretical guarantees, such as a performance ratio that converges to the optimal constant of 1 - 1/e. We empirically evaluate our algorithm on two real-world online optimization problems on the web: ad allocation with submodular utilities, and dynamically ranking blogs to detect information cascades. Finally, we present a second algorithm that handles the more general case in which the feasible sets are given by a matroid constraint, while still maintaining a 1 - 1/e asymptotic performance ratio.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
