The Steiner Forest problem, also known as the Generalized Steiner Tree problem, is a fundamental optimization problem on edge-weighted graphs where, given a set of vertex pairs, the goal is to select a minimum-cost subgraph such that each pair is connected. This problem generalizes the Steiner Tree problem, first introduced in 1811, for which the best known approximation factor is 1.39 by [Byrka, Grandoni, Rothvoβ, and Sanità, 2010]. The celebrated work of [Agrawal, Klein, and Ravi, 1989], along with refinements by [Goemans and Williamson, 1992], established a 2-approximation for Steiner Forest over 35 years ago. Pioneering iterative rounding techniques by [Jain, 1998] later extended these results to higher connectivity settings. Despite the long-standing importance of this problem, breaking the approximation factor of 2 has remained a major challenge, raising suspicions that achieving a better factor might indeed be hard. In this paper, we break the approximation barrier of 2 by designing a novel deterministic algorithm that achieves a $\mathbf{2} \mathbf{- 1 0}^{\mathbf{- 1 1}}$ approximation for this fundamental problem. As a key component of our approach, we also introduce a novel dual-based local search algorithm for the Steiner Tree problem with an approximation guarantee of 1.943, which is of independent interest.

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.

In this paper, we study the fundamental problems of maintaining the diameter and a $k$-center clustering of a dynamic point set $P \subset \mathbb{R}^d$, where points may be inserted or deleted over time and the ambient dimension $d$ is not constant and may be high. Our focus is on designing algorithms that remain effective even in the presence of an \emph{adaptive adversary}—an adversary that, at any time $t$, knows the entire history of the algorithm’s outputs as well as all the random bits used by the algorithm up to that point. We present a fully dynamic algorithm that maintains a $2$-approximate diameter with a \emph{worst-case} update time of $poly(d, \log n)$, where $n$ is the length of the stream. Our result is achieved by identifying a robust representative of the dataset that requires infrequent updates, combined with a careful deamortization. To the best of our knowledge, this is the first efficient fully-dynamic algorithm for diameter in high dimensions that \emph{simultaneously} achieves a $2$-approximation guarantee and robustness against an adaptive adversary. We also give an improved dynamic $(4+\epsilon)$-approximation algorithm for the $k$-center problem, also resilient to an adaptive adversary. Our clustering algorithm achieves an amortized update time of $k^{2.5} d \cdot poly(\epsilon^{-1}, \log n)$, improving upon the amortized update time of $k^6 d \cdot poly( \epsilon^{-1}, \log n)$ by Biabani et al. [NeurIPS'24].

Submodular maximization subject to a $p$-matchoid constraint has various applications in machine learning, particularly in tasks such as feature selection, video and text summarization, movie recommendation, graph-based learning, and constraint-based optimization. We study this problem in the dynamic setting, where a sequence of insertions and deletions of elements to a $p$-matchoid $\mathcal{M}(\mathcal{V},\mathcal{I})$ occurs over time and the goal is to efficiently maintain an approximate solution. We propose a dynamic algorithm for non-monotone submodular maximization under a $p$-matchoid constraint. For a $p$-matchoid $\mathcal{M}(\mathcal{V},\mathcal{I})$ of rank $k$, defined by a collection of $m$ matroids, our algorithm guarantees a $(2p + 2\sqrt{p(p+1)} + 1 + \epsilon)$-approximate solution at any time $t$ in the update sequence, with an expected amortized query complexity of $O(\epsilon^{-3} pk^4 \log^2(k))$ per update.

Approximation algorithms for the prize-collecting Steiner forest problem (PCSF) have been a subject of research for over three decades, starting with the seminal works of Agrawal, Klein, and Ravi [1, 2] and Goemans and Williamson [15, 16] on Steiner forest and prize-collecting problems. In this paper, we propose and analyze a natural deterministic algorithm for PCSF that achieves a 2-approximate solution in polynomial time. This represents a significant improvement compared to the previously best known algorithm with a 2.54-approximation factor developed by Hajiaghayi and Jain [20] in 2006. Furthermore, Könemann, Olver, Pashkovich, Ravi, Swamy, and Vygen [25] have established an integrality gap of at least 9/4 for the natural LP relaxation for PCSF. However, we surpass this gap through the utilization of an iterative algorithm and a novel analysis technique. Since 2 is the best known approximation guarantee for Steiner forest problem [2] (see also [16]), which is a special case of PCSF, our result matches this factor and closes the gap between the Steiner forest problem and its generalized version, PCSF.

We initiate the study of the submodular cover problem in a dynamic setting where the elements of the ground set are inserted and deleted. In the classical submodular cover problem, we are given a monotone submodular function $f : 2^{V} \to \mathbb{R}^{\ge 0}$ and the goal is to obtain a set $S \subseteq V$ that minimizes the cost subject to the constraint $f(S) = f(V)$. This is a classical problem in computer science and generalizes the Set Cover problem, 2-Set Cover, and dominating set problem among others. We consider this problem in a dynamic setting where there are updates to our set $V$, in the form of insertions and deletions of elements from a ground set $\mathcal{V}$, and the goal is to maintain an approximately optimal solution with low query complexity per update. For this problem, we propose a randomized algorithm that, in expectation, obtains a $(1-O(\epsilon), O(\epsilon^{-1}))$-bicriteria approximation using polylogarithmic query complexity per update.

Approximation algorithms for the prize-collecting Steiner forest problem (PCSF) have been a subject of research for over three decades, starting with the seminal works of Agrawal, Klein, and Ravi [1, 2] and Goemans and Williamson [14, 15] on Steiner forest and prize-collecting problems. In this paper, we propose and analyze a natural deterministic algorithm for PCSF that achieves a 2-approximate solution in polynomial time. This represents a significant improvement compared to the previously best known algorithm with a 2.54-approximation factor developed by Hajiaghayi and Jain [19] in 2006. Furthermore, Könemann, Olver, Pashkovich, Ravi, Swamy, and Vygen [24] have established an integrality gap of at least 9/4 for the natural LP relaxation for PCSF. However, we surpass this gap through the utilization of a combinatorial algorithm and a novel analysis technique. Since 2 is the best known approximation guarantee for Steiner forest problem [2] (see also [15]), which is a special case of PCSF, our result matches this factor and closes the gap between the Steiner forest problem and its generalized version, PCSF.

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.

Prize-Collecting Steiner Tree (PCST) is a generalization of the Steiner Tree problem, a fundamental problem in computer science. In the classic Steiner Tree problem, we aim to connect a set of vertices known as terminals using the minimum-weight tree in a given weighted graph. In this generalized version, each vertex has a penalty, and there is flexibility to decide whether to connect each vertex or pay its associated penalty, making the problem more realistic and practical.

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.

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.