The Maximum Submodular Matching (MSM) problem is a generalization of the classical Maximum Weight Matching (MWM) problem. In this problem, given a monotone submodular function f: 2^E → ℝ^{≥ 0} defined over subsets of edges of a graph G(V, E), we are asked to return a matching whose submodular value is maximum among all matchings in graph G(V, E). In this paper, we consider this problem in a fully dynamic setting against an oblivious adversary. In this setting, we are given a sequence 𝒮 of insertions and deletions of edges of the underlying graph G(V, E), along with an oracle access to the monotone submodular function f. The goal is to maintain a matching M such that, at any time t of sequence 𝒮, its submodular value is a good approximation of the value of the optimal submodular matching while keeping the number of operations minimal. We develop the first dynamic algorithm for the submodular matching problem, in which we maintain a matching whose submodular value is within expected (8 + ε)-approximation of the optimal submodular matching at any time t of sequence 𝒮 using expected amortized poly(log n, 1/(ε)) update time. Our approach incorporates a range of novel techniques, notably the concept of Uniform Hierarchical Caches (UHC) data structure along with its invariants, which lead to the first algorithm for fully dynamic submodular matching and may be of independent interest for designing dynamic algorithms for other problems.

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.

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 : 2V → ℝ+ and (2) we are given a sequence S of insertions and deletions of elements of an underlying ground set V.

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.

Given a point set P ⊆ X of size n in a metric space (X, dist) of doubling dimension d and two parameters k ∈ ℕ and z ∈ ℕ, the k-center problem with z outliers asks to return a set ${{\mathcal{C}}^ * } = \{ c_1^ * , \cdots ,c_k^ * \} \subseteq X$ of k centers such that the maximum distance of all but z points of P to their nearest center in C* is minimized. An (ε, k, z)-coreset for this problem is a weighted point set P* such that an optimal solution for the k-center problem with z outliers on P* gives a (1 ± ε)-approximation for the k-center problem with z outliers on P. We study the construction of such coresets in the Massively Parallel Computing (MPC) model, and in the insertion-only as well as the fully dynamic streaming model. We obtain the following results, for any given 0d+ z).• In the MPC model the data are distributed over m machines. One is the coordinator machine, which will contain the final answer, the others are worker machines.We present a deterministic 2-round algorithm using $O(\sqrt n )$ machines, where the worker machines have $O(\sqrt {nk/{\varepsilon ^d}} + \sqrt n \cdot \log (z + 1))$ local memory, and the coordinator has $O(\sqrt {nk/{\varepsilon ^d}} + \sqrt n \cdot \log (z + 1) + z)$ local memory. The algorithm can handle point sets P that are distributed arbitrarily (possibly adversarially) over the machines. We also present a randomized algorithm that uses only a single round, under the assumption that the input set P is initially distributed randomly over the machines. Then we present a deterministic algorithm that obtains a trade-off between the number of rounds, R, and the storage per machine.In the streaming model we have a single machine with limited storage, and P is revealed in a streaming fashion.○ We present the first lower bound for the insertion-only streaming model, where the points arrive one by one and no points are deleted. We show that any deterministic algorithm that maintains an (ε, k, z)-coreset must use Ω(k/εd+ z) space. We complement this by a deterministic streaming algorithm using O(k/εd+ z) space, which is thus optimal. ○ For the fully dynamic data streams, where points can be inserted as well as deleted we give a randomized algorithm for point sets from a d-dimensional discrete Euclidean space [Δ]d, where Δ ∈ ℕ indicates the size of the universe from which the coordinates are taken. Our algorithm uses only O((k/εd+ z)log4(kΔ/εδ)) space, and it is the first algorithm for this setting. We also present an Ω((k/εd)logΔ + z) lower bound for deterministic fully dynamic streaming algorithms. ○ For the sliding-window model, we show that any deterministic streaming algorithm that guarantees a (1 + ε)-approximation for the k-center problem with outliers in ℝdmust use Ω((kz/εd) logσ) space, where σ is the ratio of the largest and smallest distance between any two points in the stream. This (negatively) answers a question posed by De Berg, Monemizadeh, and Zhong [1].

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.

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.

In this paper we study the extraction of representative elements in the data stream model in the form of submodular maximization. Different from the previous work on streaming submodular maximization, we are interested only in the recent data, and study the maximization problem over sliding windows. We provide a general reduction from the sliding window model to the standard streaming model, and thus our approach works for general constraints as long as there is a corresponding streaming algorithm in the standard streaming model. As a consequence, we obtain the first algorithms in the sliding window model for maximizing a monotone/non-monotone submodular function under cardinality and matroid constraints. We also propose several heuristics and show their efficiency in real-world datasets.