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1. Corporate Needs You to Find the Difference: Revisiting Submodular and Supermodular Ratio Optimization Problems

   Harb, Elfarouk; Yassin, Yousef; Chekuri, Chandra (December 2025, Proceedings of NeurIPS, 2025)

   We consider the following question: given a submodular/supermodular set function f : 2V → R, how should one minimize/maximize its average value f (S)/|S|over non-empty subsets S ⊆ V ? This problem generalizes several well-known objectives, including Densest Subgraph (DSG), Densest Supermodular Set (DSS), and Submodular Function Minimization (SFM). Motivated by recent applications [42, 34], we formalize two new broad problems: the Unrestricted Sparsest Submodular Set (USSS) and Unrestricted Densest Supermodular Set (UDSS), both of which allow negative and non-monotone functions. Using classical results, we show that DSS, SFM, USSS, UDSS, and Minimum Norm Point (MNP) are all equivalent under strongly polynomial-time reductions. This equivalence enables algorithmic cross-over: methods designed for one problem can be repurposed to solve others efficiently. In particular, we use the perspective of the minimum norm point in the base polyhedron of a sub/supermodular function, which, via Fujishige’s results, yields the dense decomposition as a byproduct. Through this perspective, we show that a recent converging heuristic for DSS, SUPERGREEDY++ [17, 32], and Wolfe’s minimum norm point algorithm are both universal solvers for all of these problems. On the theoretical front, we explain the observation made in recent work [42, 34] that SUPERGREEDY++ appears to work well even in settings beyond DSS. Surprisingly, we also show that this simple algorithm can be used for Submodular Function Minimization, including acting as a practical minimum s-t cut algorithm. On the empirical front, we explore the utility of several algorithms for recent problems. We conduct over 400 experiments across seven problem types and largescale synthetic and real-world datasets (up to ≈ 100 million edges). Our results reveal that methods historically considered inefficient, such as convex-programming methods, flow-based solvers, and Fujishige-Wolfe’s algorithm, outperform stateof-the-art task-specific baselines by orders of magnitude on concrete problems like HNSN [42]. These findings challenge prevailing assumptions and demonstrate that with the proper framing, general optimization algorithms can be both scalable and state-of-the-art for supermodular and submodular ratio problems. 

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2. Approximation algorithms for the vertex-weighted grade-of-service Steiner tree problem

   Ahmed, R.; Efrat, A.; Kobourov, S.; Krieger, S.; Spence, R. (January 2019, ArXiv.org)

   Given a graph G = (V, E) and a subset T ⊆ V of terminals, a Steiner tree of G is a tree that spans T. In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a minimum weight Steiner tree of G. Vertex-weighted problems have applications in network design and routing, where there are different costs for installing or maintaining facilities at different vertices. We study a natural generalization of the VST problem motivated by multi-level graph construction, the vertex-weighted grade-of-service Steiner tree problem (V-GSST), which can be stated as follows: given a graph G and terminals T, where each terminal v ∈ T requires a facility of a minimum grade of service R(v) ∈ {1, 2, . . . `}, compute a Steiner tree G0 by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in G 0 with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edge-weighted case, they have not been studied as well in the more general vertex-weighted case. We first describe a simple heuristic for the V-GSST problem whose approximation ratio depends on `, the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the V-GSST problem admits a (2 ln |T|)-approximation, where T is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multi-grade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service. Finally, we show that this problem is a special case of the directed Steiner tree problem and provide an integer linear programming (ILP) formulation for the V-GSST problem. 

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3. Densest Subgraph: Supermodularity, Iterative Peeling, and Flow

   https://doi.org/10.1137/1.9781611977073.64  

   Chekuri, Chandra; Quanrud, Kent; Torres, Manuel (January 2022, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms)

   The densest subgraph problem in a graph (\dsg), in the simplest form, is the following. Given an undirected graph $G=(V,E)$ find a subset $$S \subseteq V$$ of vertices that maximizes the ratio $|E(S)|/|S|$ where $E(S)$ is the set of edges with both endpoints in $$S$$. \dsg and several of its variants are well-studied in theory and practice and have many applications in data mining and network analysis. In this paper we study fast algorithms and structural aspects of \dsg via the lens of \emph{supermodularity}. For this we consider the densest supermodular subset problem (\dssp): given a non-negative supermodular function $$f: 2^V \rightarrow \mathbb{R}_+$, maximize $f(S)/|S|$$. For \dsg we describe a simple flow-based algorithm that outputs a $$(1-\eps)$-approximation in deterministic $$\tilde{O}(m/\eps)$$ time where $$m$$ is the number of edges. Our algorithm is the first to have a near-linear dependence on $$m$$ and $$1/\eps$$ and improves previous methods based on an LP relaxation. It generalizes to hypergraphs, and also yields a faster algorithm for directed \dsg. Greedy peeling algorithms have been very popular for \dsg and several variants due to their efficiency, empirical performance, and worst-case approximation guarantees. We describe a simple peeling algorithm for \dssp and analyze its approximation guarantee in a fashion that unifies several existing results. Boob et al.\ \cite{bgpstww-20} developed an \emph{iterative} peeling algorithm for \dsg which appears to work very well in practice, and made a conjecture about its convergence to optimality. We affirmatively answer their conjecture, and in fact prove that a natural generalization of their algorithm converges to a $$(1-\eps)$$-approximation for \emph{any} supermodular function $$f$$; the key to our proof is to consider an LP formulation that is derived via the \Lovasz extension of a supermodular function. For \dsg the bound on the number of iterations we prove is $$O(\frac{\Delta \ln |V|}{\lambda^*}\cdot \frac{1}{\eps^2})$ where $$\Delta$$ is the maximum degree and $$\lambda^*$$ is the optimum value. Our work suggests that iterative peeling can be an effective heuristic for several objectives considered in the literature. Finally, we show that the $$2$$-approximation for densest-at-least-$$k$$ subgraph \cite{ks-09} extends to the supermodular setting. We also give a unified analysis of the peeling algorithm for this problem, and via this analysis derive an approximation guarantee for a generalization of \dssp to maximize $$f(S)/g(|S|)$ for a concave function $$g$$. 

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4. Covering a Few Submodular Constraints and Applications

   https://doi.org/10.4230/lipics.approx/random.2025.25  

   Bajpai, Tanvi; Chekuri, Chandra; Kulkarni, Pooja (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Ene, Alina; Chattopadhyay, Eshan (Ed.)

   We consider the problem of covering multiple submodular constraints. Given a finite ground set N, a cost function c: N → ℝ_+, r monotone submodular functions f_1,f_2,…,f_r over N and requirements b_1,b_2,…,b_r the goal is to find a minimum cost subset S ⊆ N such that f_i(S) ≥ b_i for 1 ≤ i ≤ r. When r = 1 this is the well-known Submodular Set Cover problem. Previous work [Chekuri et al., 2022] considered the setting when r is large and developed bi-criteria approximation algorithms, and approximation algorithms for the important special case when each f_i is a weighted coverage function. These are fairly general models and capture several concrete and interesting problems as special cases. The approximation ratios for these problem are at least Ω(log r) which is unavoidable when r is part of the input. In this paper, motivated by some recent applications, we consider the problem when r is a fixed constant and obtain two main results. When the f_i are weighted coverage functions from a deletion-closed set system we obtain a (1+ε)(e/(e-1))(1+β)-approximation where β is the approximation ratio for the underlying set cover instances via the natural LP. Second, for covering multiple submodular constraints we obtain a randomized bi-criteria approximation algorithm that for any given integer α ≥ 1 outputs a set S such that f_i(S) ≥ (1-1/e^α-ε)b_i for each i ∈ [r] and 𝔼[c(S)] ≤ (1+ε)α ⋅ OPT. These results show that one can obtain nearly as good an approximation for any fixed r as what one would achieve for r = 1. We also demonstrate applications of our results to implicit covering problems such as fair facility location. 

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5. Sum-of-Squares Lower Bounds for Densest k-Subgraph

   https://doi.org/10.1145/3564246.3585221  

   Jones, Chris; Potechin, Aaron; Rajendran, Goutham; Xu, Jeff (June 2023, STOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of Computing)

   Given a graph and an integer k, Densest k-Subgraph is the algorithmic task of finding the subgraph on k vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an O(n^{1/4+ϵ})-factor approximation (for any ϵ>0) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an O(n^{1/4−ϵ})-factor approximation. In the average case, Densest k-Subgraph is a prototypical noisy inference task which is conjectured to exhibit a statistical-computational gap. In this work, we provide the strongest evidence yet of hardness for Densest k-Subgraph by showing matching lower bounds against the powerful Sum-of-Squares (SoS) algorithm, a meta-algorithm based on convex programming that achieves state-of-art algorithmic guarantees for many optimization and inference problems. For k ≤ n^1/2, we obtain a degree n^δ SoS lower bound for the hard regime as predicted by the log-density framework. To show this, we utilize the modern framework for proving SoS lower bounds on average-case problems pioneered by Barak et al. [FOCS '16]. A key issue is that small denser-than-average subgraphs in the input will greatly affect the value of the candidate pseudo-expectation operator around the subgraph. To handle this challenge, we devise a novel matrix factorization scheme based on the positive minimum vertex separator. We then prove an intersection tradeoff lemma to show that the error terms when using this separator are indeed small. 

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