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Leyla Biabani

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5 papers

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SODA Conference 2024 Conference Paper

Dynamic Algorithms for Matroid Submodular Maximization

Kiarash Banihashem
Leyla Biabani
Samira Goudarzi
MohammadTaghi Hajiaghayi
Peyman Jabbarzade
Morteza Monemizadeh

Submodular maximization under matroid and cardinality constraints are classical problems with a wide range of applications in machine learning, auction theory, and combinatorial optimization. In this paper, we consider these problems in the dynamic setting where (1) we have oracle access to a monotone submodular function f: 2 V → ℝ + and (2) we are given a sequence S of insertions and deletions of elements of an underlying ground set V. We develop the first fully dynamic algorithm for the submodular maximization problem under the matroid constraint that maintains a (4 + ɛ )-approximation solution (0 < ɛ ≤ 1) using an expected query complexity of O(k log( k ) log 3 (k/ɛ)), which is indeed parameterized by the rank k of the matroid M(V, I) as well. Chen and Peng [52] at STOC’22 studied the complexity of this problem in the insertion-only dynamic model (a restricted version of the fully dynamic model where deletion is not allowed), and they raised the following important open question: “for fully dynamic streams [sequences of insertions and deletions of elements], there is no known constant-factor approximation algorithm with poly(k) amortized queries for matroid constraints. ” Our dynamic algorithm answers this question as well as an open problem of Lattanzi et al. [109] (NeurIPS’20) affirmatively. As a byproduct, for the submodular maximization under the cardinality constraint k, we propose a parameterized (by the cardinality constraint k) dynamic algorithm that maintains a (2 + ɛ )-approximate solution of the sequence S at any time t using an expected query complexity of O(kɛ -1 log 2 ( k )), which is an improvement upon the dynamic algorithm that Monemizadeh [125] (NeurIPS’20) developed for this problem using an expected query complexity O (k 2 ɛ -3 log 5 ( n )). In particular, this dynamic algorithm is the first one for this problem whose query complexity is independent of the size of ground set V (i. e. , n = | V |). We develop our dynamic algorithm for the submodular maximization problem under the matroid or cardinality constraint by designing a randomized leveled data structure that supports insertion and deletion operations, maintaining an approximate solution for the given problem. In addition, we develop a fast construction algorithm for our data structure that uses a one-pass over a random permutation of the elements and utilizes monotonicity property of our problems which has a subtle proof in the matroid case. We believe these techniques could also be useful for other optimization problems in the area of dynamic algorithms.

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NeurIPS Conference 2024 Conference Paper

Improved Guarantees for Fully Dynamic $k$-Center Clustering with Outliers in General Metric Spaces

Leyla Biabani
Annika Hennes
Denise La Gordt Dillie
Morteza Monemizadeh
Melanie Schmidt

The metric $k$-center clustering problem with $z$ outliers, also known as $(k, z)$-center clustering, involves clustering a given point set $P$ in a metric space $(M, d)$ using at most $k$ balls, minimizing the maximum ball radius while excluding up to $z$ points from the clustering. This problem holds fundamental significance in various domains such as machine learning, data mining, and database systems. This paper addresses the fully dynamic version of the problem, where the point set undergoes continuous updates (insertions and deletions) over time. The objective is to maintain an approximate $(k, z)$-center clustering with efficient update times. We propose a novel fully dynamic algorithm that maintains a $(4+\epsilon)$-approximate solution to the $(k, z)$-center clustering problem that covers all but at most $(1+\epsilon)z$ points at any time in the sequence with probability $1-k/e^{\Omega(\log k)}$. The algorithm achieves an expected amortized update time of $\mathcal{O}(\epsilon^{-2} k^6\log(k) \log(\Delta))$, and is applicable to general metric spaces. Our dynamic algorithm presents a significant improvement over the recent dynamic $(14+\epsilon)$-approximation algorithm by Chan, Lattanzi, Sozio, and Wang for this problem.

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ICML Conference 2023 Conference Paper

Dynamic Constrained Submodular Optimization with Polylogarithmic Update Time

Kiarash Banihashem
Leyla Biabani
Samira Goudarzi
MohammadTaghi Hajiaghayi
Peyman Jabbarzade
Morteza Monemizadeh

Maximizing a monotone submodular function under cardinality constraint $k$ is a core problem in machine learning and database with many basic applications, including video and data summarization, recommendation systems, feature extraction, exemplar clustering, and coverage problems. We study this classic problem in the fully dynamic model where a stream of insertions and deletions of elements of an underlying ground set is given and the goal is to maintain an approximate solution using a fast update time. A recent paper at NeurIPS’20 by Lattanzi, Mitrovic, Norouzi-Fard, Tarnawski, Zadimoghaddam claims to obtain a dynamic algorithm for this problem with a $(\frac{1}{2} -\epsilon)$ approximation ratio and a query complexity bounded by $\mathrm{poly}(\log(n), \log(k), \epsilon^{-1})$. However, as we explain in this paper, the analysis has some important gaps. Having a dynamic algorithm for the problem with polylogarithmic update time is even more important in light of a recent result by Chen and Peng at STOC’22 who show a matching lower bound for the problem – any randomized algorithm with a $\frac{1}{2}+\epsilon$ approximation ratio must have an amortized query complexity that is polynomial in $n$. In this paper, we develop a simpler algorithm for the problem that maintains a $(\frac{1}{2}-\epsilon)$-approximate solution for submodular maximization under cardinality constraint $k$ using a polylogarithmic amortized update time.

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NeurIPS Conference 2023 Conference Paper

Dynamic Non-monotone Submodular Maximization

Kiarash Banihashem
Leyla Biabani
Samira Goudarzi
MohammadTaghi Hajiaghayi
Peyman Jabbarzade
Morteza Monemizadeh

Maximizing submodular functions has been increasingly used in many applications of machine learning, such as data summarization, recommendation systems, and feature selection. Moreover, there has been a growing interest in both submodular maximization and dynamic algorithms. In 2020, Monemizadeh and Lattanzi, Mitrovic, Norouzi-Fard, Tarnawski, and Zadimoghaddam initiated developing dynamic algorithms for the monotone submodular maximization problem under the cardinality constraint $k$. In 2022, Chen and Peng studied the complexity of this problem and raised an important open question: "\emph{Can we extend [fully dynamic] results (algorithm or hardness) to non-monotone submodular maximization? }". We affirmatively answer their question by demonstrating a reduction from maximizing a non-monotone submodular function under the cardinality constraint $k$ to maximizing a monotone submodular function under the same constraint. Through this reduction, we obtain the first dynamic algorithms to solve the non-monotone submodular maximization problem under the cardinality constraint $k$. Our algorithms maintain an $(8+\epsilon)$-approximate of the solution and use expected amortized $O(\epsilon^{-3}k^3\log^3(n)\log(k))$ or $O(\epsilon^{-1}k^2\log^3(k))$ oracle queries per update, respectively. Furthermore, we showcase the benefits of our dynamic algorithm for video summarization and max-cut problems on several real-world data sets.

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NeurIPS Conference 2023 Conference Paper

Faster Query Times for Fully Dynamic $k$-Center Clustering with Outliers

Leyla Biabani
Annika Hennes
Morteza Monemizadeh
Melanie Schmidt

Given a point set $P\subseteq M$ from a metric space $(M, d)$ and numbers $k, z \in N$, the *metric $k$-center problem with $z$ outliers* is to find a set $C^\ast\subseteq P$ of $k$ points such that the maximum distance of all but at most $z$ outlier points of $P$ to their nearest center in ${C}^\ast$ is minimized. We consider this problem in the fully dynamic model, i. e. , under insertions and deletions of points, for the case that the metric space has a bounded doubling dimension $dim$. We utilize a hierarchical data structure to maintain the points and their neighborhoods, which enables us to efficiently find the clusters. In particular, our data structure can be queried at any time to generate a $(3+\varepsilon)$-approximate solution for input values of $k$ and $z$ in worst-case query time $\varepsilon^{-O(dim)}k \log{n} \log\log{\Delta}$, where $\Delta$ is the ratio between the maximum and minimum distance between two points in $P$. Moreover, it allows insertion/deletion of a point in worst-case update time $\varepsilon^{-O(dim)}\log{n}\log{\Delta}$. Our result achieves a significantly faster query time with respect to $k$ and $z$ than the current state-of-the-art by Pellizzoni, Pietracaprina, and Pucci, which uses $\varepsilon^{-O(dim)}(k+z)^2\log{\Delta}$ query time to obtain a $(3+\varepsilon)$-approximation.

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