Moran Feldman

Loay Mualem, Moran Feldman

In recent years, maximization of DR-submodular continuous functions became an important research field, with many real-worlds applications in the domains of machine learning, communication systems, operation research and economics. Most of the works in this field study maximization subject to down-closed convex set constraints due to an inapproximability result by Vondrák (2013). However, Durr et al. (2021) showed that one can bypass this inapproximability by proving approximation ratios that are functions of $m$, the minimum $\ell_{\infty}$-norm of any feasible vector. Given this observation, it is possible to get results for maximizing a DR-submodular function subject to general convex set constraints, which has led to multiple works on this problem. The most recent of which is a polynomial time $\tfrac{1}{4}(1 - m)$-approximation offline algorithm due to Du (2022). However, only a sub-exponential time $\tfrac{1}{3\sqrt{3}}(1 - m)$-approximation algorithm is known for the corresponding online problem. In this work, we present a polynomial time online algorithm matching the $\tfrac{1}{4}(1 - m)$-approximation of the state-of-the-art offline algorithm. We also present an inapproximability result showing that our online algorithm and Du's (2022) offline algorithm are both optimal in a strong sense. Finally, we study the empirical performance of our algorithm and the algorithm of Du (which was only theoretically studied previously), and show that they consistently outperform previously suggested algorithms on revenue maximization, location summarization and quadratic programming applications.

Moran Feldman, Justin Ward

We study the problem of maximizing a non-negative monotone submodular objective $f$ subject to the intersection of $k$ arbitrary matroid constraints. The natural greedy algorithm guarantees $(k+1)$-approximation for this problem, and the state-of-the-art algorithm only improves this approximation ratio to $k$. We give a $\frac{2k\ln2}{1+\ln2}+O(\sqrt{k})<0.819k+O(\sqrt{k})$ approximation for this problem. Our result is the first multiplicative improvement over the approximation ratio of the greedy algorithm for general $k$. We further show that our algorithm can be used to obtain roughly the same approximation ratio also for the more general problem in which the objective is not guaranteed to be monotone (the sublinear term in the approximation ratio becomes $O(k^{2/3})$ rather than $O(\sqrt{k})$ in this case). All of our results hold also when the $k$-matroid intersection constraint is replaced with a more general matroid $k$-parity constraint. Furthermore, unlike the case in many of the previous works, our algorithms run in time that is independent of $k$ and polynomial in the size of the ground set. Our algorithms are based on a hybrid greedy local search approach recently introduced by Singer and Thiery (STOC 2025) for the weighted matroid $k$-intersection problem, which is a special case of the problem we consider. Leveraging their approach in the submodular setting requires several non-trivial insights and algorithmic modifications since the marginals of a submodular function $f$, which correspond to the weights in the weighted case, are not independent of the algorithm's internal randomness. In the special weighted case studied by Singer and Thiery, our algorithms reduce to a variant of their algorithm with an improved approximation ratio of $(k+1)\ln2<0.694k+0.694$, compared to an approximation ratio of $\frac{k+1}{2\ln2}\approx0.722k+0.722$ guaranteed by Singer and Thiery.

Niv Buchbinder, Moran Feldman, Siyue Liu et al.

We study random order semi-streaming algorithms for submodular maximization under a wide range of combinatorial constraint classes, including matroids, matroid $p$-parity, $p$-exchange systems and $p$-systems. For most of these classes of constraints, our results are the first improvement over what is known to be achievable for adversarial order. For matroids, matching and $p$-matchoids, previous random order results were known, and we improve over some of these as well. In the case of matroids, our improved results show a separation between adversarial and random order semi-streaming algorithms, and exponentially improve the number of passes necessary for getting $1 - 1/e - \varepsilon$ approximation for maximizing a monotone submodular function subject to a matroid constraint. We also prove a new hardness result showing a similar separation for $p$-systems. Our results are based on two new technical tools. One tool provides a general way to translate offline algorithms for many classes of constraints into random order semi-streaming algorithms. The other tool is a semi-streaming variant of a recently proposed offline algorithm for matroid constraints.

Moran Feldman, Christopher Harshaw, Amin Karbasi

We present SimultaneousGreedys, a deterministic algorithm for constrained submodular maximization. At a high level, the algorithm maintains $\ell$ solutions and greedily updates them in a simultaneous fashion. SimultaneousGreedys achieves the tightest known approximation guarantees for both $k$-extendible systems and the more general $k$-systems, which are $(k+1)^2/k = k + \mathcal{O}(1)$ and $(1 + \sqrt{k+2})^2 = k + \mathcal{O}(\sqrt{k})$, respectively. This is in contrast to previous algorithms, which are designed to provide tight approximation guarantees in one setting, but not both. We also improve the analysis of RepeatedGreedy, showing that it achieves an approximation ratio of $k + \mathcal{O}(\sqrt{k})$ for $k$-systems when allowed to run for $\mathcal{O}(\sqrt{k})$ iterations, an improvement in both the runtime and approximation over previous analyses. We demonstrate that both algorithms may be modified to run in nearly linear time with an arbitrarily small loss in the approximation. Both SimultaneousGreedys and RepeatedGreedy are flexible enough to incorporate the intersection of $m$ additional knapsack constraints, while retaining similar approximation guarantees: both algorithms yield an approximation guarantee of roughly $k + 2m + \mathcal{O}(\sqrt{k+m})$ for $k$-systems and SimultaneousGreedys enjoys an improved approximation guarantee of $k+2m + \mathcal{O}(\sqrt{m})$ for $k$-extendible systems. To complement our algorithmic contributions, we provide a hardness result which states that no algorithm making polynomially many oracle queries can achieve an approximation better than $k + 1/2 + \varepsilon$. We also present SubmodularGreedy.jl, a Julia package which implements these algorithms and may be downloaded at https://github.com/crharshaw/SubmodularGreedy.jl . Finally, we test the effectiveness of these algorithms on real datasets.

Murad Tukan, Loay Mualem, Moran Feldman

Non-monotone constrained submodular maximization plays a crucial role in various machine learning applications. However, existing algorithms often struggle with a trade-off between approximation guarantees and practical efficiency. The current state-of-the-art is a recent $0.401$-approximation algorithm, but its computational complexity makes it highly impractical. The best practical algorithms for the problem only guarantee $1/e$-approximation. In this work, we present a novel algorithm for submodular maximization subject to a cardinality constraint that combines a guarantee of $0.385$-approximation with a low and practical query complexity of $O(n+k^2)$. Furthermore, we evaluate the empirical performance of our algorithm in experiments based on various machine learning applications, including Movie Recommendation, Image Summarization, and more. These experiments demonstrate the efficacy of our approach.

Moran Feldman, Amin Karbasi

Recent announcements from frontier AI model labs have highlighted strong results on high-school and undergraduate math competitions. Yet it remains unclear whether large language models can solve new, simple conjectures in more advanced areas of mathematics. We propose the Gödel Test: evaluating whether a model can produce correct proofs for very simple, previously unsolved conjectures. To this end, we study the performance of GPT-5 on five conjectures in combinatorial optimization. For each problem, we provided one or two source papers from which the conjecture arose, withheld our own conjecture, and then assessed the model's reasoning in detail. On the three easier problems, GPT-5 produced nearly correct solutions; for Problem 2 it even derived a different approximation guarantee that, upon checking, refuted our conjecture while providing a valid solution. The model failed on Problem 4, which required combining results from two papers. On Problem 5, a harder case without a validated conjecture, GPT-5 proposed the same algorithm we had in mind but failed in the analysis, suggesting the proof is more challenging than expected. Although our sample is small, the results point to meaningful progress on routine reasoning, occasional flashes of originality, and clear limitations when cross-paper synthesis is required. GPT-5 may represent an early step toward frontier models eventually passing the Gödel Test.

Loay Mualem, Ethan R. Elenberg, Moran Feldman et al.

Despite the rich existing literature about minimax optimization in continuous settings, only very partial results of this kind have been obtained for combinatorial settings. In this paper, we fill this gap by providing a characterization of submodular minimax optimization, the problem of finding a set (for either the min or the max player) that is effective against every possible response. We show when and under what conditions we can find such sets. We also demonstrate how minimax submodular optimization provides robust solutions for downstream machine learning applications such as (i) efficient prompt engineering for question answering, (ii) prompt engineering for dialog state tracking, (iii) identifying robust waiting locations for ride-sharing, (iv) ride-share difficulty kernelization, and (v) finding adversarial images. Our experiments demonstrate that our proposed algorithms consistently outperform other baselines.

Loay Mualem, Moran Feldman

Over the last two decades, submodular function maximization has been the workhorse of many discrete optimization problems in machine learning applications. Traditionally, the study of submodular functions was based on binary function properties. However, such properties have an inherit weakness, namely, if an algorithm assumes functions that have a particular property, then it provides no guarantee for functions that violate this property, even when the violation is very slight. Therefore, recent works began to consider continuous versions of function properties. Probably the most significant among these (so far) are the submodularity ratio and the curvature, which were studied extensively together and separately. The monotonicity property of set functions plays a central role in submodular maximization. Nevertheless, and despite all the above works, no continuous version of this property has been suggested to date (as far as we know). This is unfortunate since submoduar functions that are almost monotone often arise in machine learning applications. In this work we fill this gap by defining the monotonicity ratio, which is a continues version of the monotonicity property. We then show that for many standard submodular maximization algorithms one can prove new approximation guarantees that depend on the monotonicity ratio; leading to improved approximation ratios for the common machine learning applications of movie recommendation, quadratic programming and image summarization.

Siddharth Mitra, Moran Feldman, Amin Karbasi

It has been well established that first order optimization methods can converge to the maximal objective value of concave functions and provide constant factor approximation guarantees for (non-convex/non-concave) continuous submodular functions. In this work, we initiate the study of the maximization of functions of the form $F(x) = G(x) +C(x)$ over a solvable convex body $P$, where $G$ is a smooth DR-submodular function and $C$ is a smooth concave function. This class of functions is a strict extension of both concave and continuous DR-submodular functions for which no theoretical guarantee is known. We provide a suite of Frank-Wolfe style algorithms, which, depending on the nature of the objective function (i.e., if $G$ and $C$ are monotone or not, and non-negative or not) and on the nature of the set $P$ (i.e., whether it is downward closed or not), provide $1-1/e$, $1/e$, or $1/2$ approximation guarantees. We then use our algorithms to get a framework to smoothly interpolate between choosing a diverse set of elements from a given ground set (corresponding to the mode of a determinantal point process) and choosing a clustered set of elements (corresponding to the maxima of a suitable concave function). Additionally, we apply our algorithms to various functions in the above class (DR-submodular + concave) in both constrained and unconstrained settings, and show that our algorithms consistently outperform natural baselines.

Christopher Harshaw, Ehsan Kazemi, Moran Feldman et al.

We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set, and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual offline setting, we present SampleGreedy, which obtains a $(p + 2 + o(1))$-approximation for maximizing a submodular function subject to a $p$-extendible system using $O(n + nk/p)$ evaluation and feasibility queries, where $k$ is the size of the largest feasible set. The approximation ratio improves to $p+1$ and $p$ for monotone submodular and linear objectives, respectively. In the streaming setting, we present SampleStreaming, which obtains a $(4p +2 - o(1))$-approximation for maximizing a submodular function subject to a $p$-matchoid using $O(k)$ memory and $O(km/p)$ evaluation and feasibility queries per element, where $m$ is the number of matroids defining the $p$-matchoid. The approximation ratio improves to $4p$ for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks.

Moran Feldman, Amin Karbasi

In this paper, we propose the first continuous optimization algorithms that achieve a constant factor approximation guarantee for the problem of monotone continuous submodular maximization subject to a linear constraint. We first prove that a simple variant of the vanilla coordinate ascent, called Coordinate-Ascent+, achieves a $(\frac{e-1}{2e-1}-\varepsilon)$-approximation guarantee while performing $O(n/\varepsilon)$ iterations, where the computational complexity of each iteration is roughly $O(n/\sqrt{\varepsilon}+n\log n)$ (here, $n$ denotes the dimension of the optimization problem). We then propose Coordinate-Ascent++, that achieves the tight $(1-1/e-\varepsilon)$-approximation guarantee while performing the same number of iterations, but at a higher computational complexity of roughly $O(n^3/\varepsilon^{2.5} + n^3 \log n / \varepsilon^2)$ per iteration. However, the computation of each round of Coordinate-Ascent++ can be easily parallelized so that the computational cost per machine scales as $O(n/\sqrt{\varepsilon}+n\log n)$.

Wenxin Li, Moran Feldman, Ehsan Kazemi et al.

In this paper, we provide the first deterministic algorithm that achieves the tight $1-1/e$ approximation guarantee for submodular maximization under a cardinality (size) constraint while making a number of queries that scales only linearly with the size of the ground set $n$. To complement our result, we also show strong information-theoretic lower bounds. More specifically, we show that when the maximum cardinality allowed for a solution is constant, no algorithm making a sub-linear number of function evaluations can guarantee any constant approximation ratio. Furthermore, when the constraint allows the selection of a constant fraction of the ground set, we show that any algorithm making fewer than $Ω(n/\log(n))$ function evaluations cannot perform better than an algorithm that simply outputs a uniformly random subset of the ground set of the right size. We then provide a variant of our deterministic algorithm for the more general knapsack constraint, which is the first linear-time algorithm that achieves $1/2$-approximation guarantee for this constraint. Finally, we extend our results to the general case of maximizing a monotone submodular function subject to the intersection of a $p$-set system and multiple knapsack constraints. We extensively evaluate the performance of our algorithms on multiple real-life machine learning applications, including movie recommendation, location summarization, twitter text summarization and video summarization.

Ehsan Kazemi, Shervin Minaee, Moran Feldman et al.

In this paper, we propose scalable methods for maximizing a regularized submodular function $f = g - \ell$ expressed as the difference between a monotone submodular function $g$ and a modular function $\ell$. Indeed, submodularity is inherently related to the notions of diversity, coverage, and representativeness. In particular, finding the mode of many popular probabilistic models of diversity, such as determinantal point processes, submodular probabilistic models, and strongly log-concave distributions, involves maximization of (regularized) submodular functions. Since a regularized function $f$ can potentially take on negative values, the classic theory of submodular maximization, which heavily relies on the non-negativity assumption of submodular functions, may not be applicable. To circumvent this challenge, we develop the first one-pass streaming algorithm for maximizing a regularized submodular function subject to a $k$-cardinality constraint. It returns a solution $S$ with the guarantee that $f(S)\geq(φ^{-2}-ε) \cdot g(OPT)-\ell (OPT)$, where $φ$ is the golden ratio. Furthermore, we develop the first distributed algorithm that returns a solution $S$ with the guarantee that $\mathbb{E}[f(S)] \geq (1-ε) [(1-e^{-1}) \cdot g(OPT)-\ell(OPT)]$ in $O(1/ ε)$ rounds of MapReduce computation, without keeping multiple copies of the entire dataset in each round (as it is usually done). We should highlight that our result, even for the unregularized case where the modular term $\ell$ is zero, improves the memory and communication complexity of the existing work by a factor of $O(1/ ε)$ while arguably provides a simpler distributed algorithm and a unifying analysis. We also empirically study the performance of our scalable methods on a set of real-life applications, including finding the mode of distributions, data summarization, and product recommendation.

Ran Haba, Ehsan Kazemi, Moran Feldman et al.

In this paper, we propose a novel framework that converts streaming algorithms for monotone submodular maximization into streaming algorithms for non-monotone submodular maximization. This reduction readily leads to the currently tightest deterministic approximation ratio for submodular maximization subject to a $k$-matchoid constraint. Moreover, we propose the first streaming algorithm for monotone submodular maximization subject to $k$-extendible and $k$-set system constraints. Together with our proposed reduction, we obtain $O(k\log k)$ and $O(k^2\log k)$ approximation ratio for submodular maximization subject to the above constraints, respectively. We extensively evaluate the empirical performance of our algorithm against the existing work in a series of experiments including finding the maximum independent set in randomly generated graphs, maximizing linear functions over social networks, movie recommendation, Yelp location summarization, and Twitter data summarization.

Christopher Harshaw, Moran Feldman, Justin Ward et al.

It is generally believed that submodular functions -- and the more general class of $γ$-weakly submodular functions -- may only be optimized under the non-negativity assumption $f(S) \geq 0$. In this paper, we show that once the function is expressed as the difference $f = g - c$, where $g$ is monotone, non-negative, and $γ$-weakly submodular and $c$ is non-negative modular, then strong approximation guarantees may be obtained. We present an algorithm for maximizing $g - c$ under a $k$-cardinality constraint which produces a random feasible set $S$ such that $\mathbb{E} \left[ g(S) - c(S) \right] \geq (1 - e^{-γ} - ε) g(OPT) - c(OPT)$, whose running time is $O (\frac{n}ε \log^2 \frac{1}ε)$, i.e., independent of $k$. We extend these results to the unconstrained setting by describing an algorithm with the same approximation guarantees and faster $O(\frac{n}ε \log\frac{1}ε)$ runtime. The main techniques underlying our algorithms are two-fold: the use of a surrogate objective which varies the relative importance between $g$ and $c$ throughout the algorithm, and a geometric sweep over possible $γ$ values. Our algorithmic guarantees are complemented by a hardness result showing that no polynomial-time algorithm which accesses $g$ through a value oracle can do better. We empirically demonstrate the success of our algorithms by applying them to experimental design on the Boston Housing dataset and directed vertex cover on the Email EU dataset.

Marko Mitrovic, Ehsan Kazemi, Moran Feldman et al.

In many machine learning applications, one needs to interactively select a sequence of items (e.g., recommending movies based on a user's feedback) or make sequential decisions in a certain order (e.g., guiding an agent through a series of states). Not only do sequences already pose a dauntingly large search space, but we must also take into account past observations, as well as the uncertainty of future outcomes. Without further structure, finding an optimal sequence is notoriously challenging, if not completely intractable. In this paper, we view the problem of adaptive and sequential decision making through the lens of submodularity and propose an adaptive greedy policy with strong theoretical guarantees. Additionally, to demonstrate the practical utility of our results, we run experiments on Amazon product recommendation and Wikipedia link prediction tasks.

Lin Chen, Moran Feldman, Amin Karbasi

In this paper, we consider the unconstrained submodular maximization problem. We propose the first algorithm for this problem that achieves a tight $(1/2-\varepsilon)$-approximation guarantee using $\tilde{O}(\varepsilon^{-1})$ adaptive rounds and a linear number of function evaluations. No previously known algorithm for this problem achieves an approximation ratio better than $1/3$ using less than $Ω(n)$ rounds of adaptivity, where $n$ is the size of the ground set. Moreover, our algorithm easily extends to the maximization of a non-negative continuous DR-submodular function subject to a box constraint and achieves a tight $(1/2-\varepsilon)$-approximation guarantee for this problem while keeping the same adaptive and query complexities.

Moran Feldman, Amin Karbasi, Ehsan Kazemi

In this paper, we develop the first one-pass streaming algorithm for submodular maximization that does not evaluate the entire stream even once. By carefully subsampling each element of data stream, our algorithm enjoys the tightest approximation guarantees in various settings while having the smallest memory footprint and requiring the lowest number of function evaluations. More specifically, for a monotone submodular function and a $p$-matchoid constraint, our randomized algorithm achieves a $4p$ approximation ratio (in expectation) with $O(k)$ memory and $O(km/p)$ queries per element ($k$ is the size of the largest feasible solution and $m$ is the number of matroids used to define the constraint). For the non-monotone case, our approximation ratio increases only slightly to $4p+2-o(1)$. To the best or our knowledge, our algorithm is the first that combines the benefits of streaming and subsampling in a novel way in order to truly scale submodular maximization to massive machine learning problems. To showcase its practicality, we empirically evaluated the performance of our algorithm on a video summarization application and observed that it outperforms the state-of-the-art algorithm by up to fifty fold, while maintaining practically the same utility.

Lin Chen, Moran Feldman, Amin Karbasi

Submodular functions are a broad class of set functions, which naturally arise in diverse areas. Many algorithms have been suggested for the maximization of these functions. Unfortunately, once the function deviates from submodularity, the known algorithms may perform arbitrarily poorly. Amending this issue, by obtaining approximation results for set functions generalizing submodular functions, has been the focus of recent works. One such class, known as weakly submodular functions, has received a lot of attention. A key result proved by Das and Kempe (2011) showed that the approximation ratio of the greedy algorithm for weakly submodular maximization subject to a cardinality constraint degrades smoothly with the distance from submodularity. However, no results have been obtained for maximization subject to constraints beyond cardinality. In particular, it is not known whether the greedy algorithm achieves any non-trivial approximation ratio for such constraints. In this paper, we prove that a randomized version of the greedy algorithm (previously used by Buchbinder et al. (2014) for a different problem) achieves an approximation ratio of $(1 + 1/γ)^{-2}$ for the maximization of a weakly submodular function subject to a general matroid constraint, where $γ$ is a parameter measuring the distance of the function from submodularity. Moreover, we also experimentally compare the performance of this version of the greedy algorithm on real world problems against natural benchmarks, and show that the algorithm we study performs well also in practice. To the best of our knowledge, this is the first algorithm with a non-trivial approximation guarantee for maximizing a weakly submodular function subject to a constraint other than the simple cardinality constraint. In particular, it is the first algorithm with such a guarantee for the important and broad class of matroid constraints.

Moran Feldman, Christopher Harshaw, Amin Karbasi

It is known that greedy methods perform well for maximizing monotone submodular functions. At the same time, such methods perform poorly in the face of non-monotonicity. In this paper, we show - arguably, surprisingly - that invoking the classical greedy algorithm $O(\sqrt{k})$-times leads to the (currently) fastest deterministic algorithm, called Repeated Greedy, for maximizing a general submodular function subject to $k$-independent system constraints. Repeated Greedy achieves $(1 + O(1/\sqrt{k}))k$ approximation using $O(nr\sqrt{k})$ function evaluations (here, $n$ and $r$ denote the size of the ground set and the maximum size of a feasible solution, respectively). We then show that by a careful sampling procedure, we can run the greedy algorithm only once and obtain the (currently) fastest randomized algorithm, called Sample Greedy, for maximizing a submodular function subject to $k$-extendible system constraints (a subclass of $k$-independent system constrains). Sample Greedy achieves $(k + 3)$-approximation with only $O(nr/k)$ function evaluations. Finally, we derive an almost matching lower bound, and show that no polynomial time algorithm can have an approximation ratio smaller than $ k + 1/2 - \varepsilon$. To further support our theoretical results, we compare the performance of Repeated Greedy and Sample Greedy with prior art in a concrete application (movie recommendation). We consistently observe that while Sample Greedy achieves practically the same utility as the best baseline, it performs at least two orders of magnitude faster.

Ethan R. Elenberg, Alexandros G. Dimakis, Moran Feldman et al.

In many machine learning applications, it is important to explain the predictions of a black-box classifier. For example, why does a deep neural network assign an image to a particular class? We cast interpretability of black-box classifiers as a combinatorial maximization problem and propose an efficient streaming algorithm to solve it subject to cardinality constraints. By extending ideas from Badanidiyuru et al. [2014], we provide a constant factor approximation guarantee for our algorithm in the case of random stream order and a weakly submodular objective function. This is the first such theoretical guarantee for this general class of functions, and we also show that no such algorithm exists for a worst case stream order. Our algorithm obtains similar explanations of Inception V3 predictions $10$ times faster than the state-of-the-art LIME framework of Ribeiro et al. [2016].