Title: Corporate Needs You to Find the Difference: Revisiting Submodular and Supermodular Ratio Optimization Problems
Venue: NeurIPS 2025
Accepted: yes
Keywords: densest subgraph

densest supermodular set

submodular function minimization.
Abstract: We consider the following question: given a submodular or supermodular set function $f:2^V \to \mathbb{R}$, how should one minimize or maximize its average value $f(S)/|S|$ over non-empty subsets $S\subseteq V$? This problem generalizes several well-known objectives including Densest Subgraph (DSG), Densest Supermodular Set (DSS), and Submodular Function Minimization (SFM). Motivated by recent applications [39, 31], we formalize two new broad problems: the Unrestricted Sparsest Submodular Set (USSS) and Unrestricted Densest Supermodular Set (UDSS) which allow negative and non-monotone functions.

Using classical results we observe that DSS, SFM, USSS, UDSS, and 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. Via this perspective we show that a recent converging heuristic for DSS, \textsc{SuperGreedy++} [15, 29], 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 [39, 31] that \textsc{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 for example that it can act as an effective minimum $st$ cut algorithm. 

On the empirical front, we explore the utility of several different algorithms including Fujishige-Wolfe min-norm point algorithm for recent problems. We conduct over 400 experiments across seven problem types and large-scale synthetic and real-world datasets (up to $\approx 100$ million edges). Our results reveal that methods historically considered inefficient, such as convex-programming methods, flow-based solvers, and Fujishige-Wolfe’s algorithm, outperform state-of-the-art task-specific baselines by orders of magnitude on concrete problems like HNSN [39]. These findings challenge prevailing assumptions and demonstrate that with the right framing, general optimization algorithms can be both scalable and state-of-the-art for supermodular and submodular ratio problems.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors study a broad range of combinatorial optimization problems related to submodularity and supermodularity, particularly focusing on submodular and supermodular ratio optimization problems. The authors introduce two novel optimization problems: Unrestricted Sparsest Submodular Set (USSS, Problem 5) and Unrestricted Densest Supermodular Set (UDSS, Problem 7). In USSS/UDSS, given a universe $V$ and a normalized submodular/supermodular function $f$, we are asked to find a nonempty $S\\subseteq V$ that minimizes/maximizes $f(S)/|S|$. USSS is a generalization of the recently-introduced graph optimization problem called Heavy Nodes in a Small Neighborhood (HNSN, Problem 4), while UDSS is a generalization of Densest Supermodular Set (DSS, Problem 3), which is also known as a generalization of the well-known graph optimization problem called Densest Subgraph (DSG, Problem 2) and its variant called Anchored Densest Subgraph (ADS, Problem 6). In addition to the above ratio optimization problems, the authors consider two classic problems: Submodular Function Minimization (SFM, Problem 1) and Minimum Norm Point (MNP, Problem 8). The main contribution of this paper is to establish a novel algorithmic foundation for the above problems, especially focusing on USSS, UDSS, DSS, SFM, and MNP. To this end, the authors first prove that the five problems are equivalent in terms of exact optimization, meaning that any exact algorithm for one problem is applicable to any other problem. Specifically, they construct a strongly polynomial-time (at most $O(n\\log n)$ time) reduction between any two problems. Note that some of the reductions are known or trivial (e.g., DSS to SFM) but many of them are not trivial at all. Furthermore, the authors also study the relationships among the problems in terms of approximation. Specifically, they demonstrate that MNP plays a key role in terms of additive approximation by proving that an additive approximation algorithm to MNP can be used to obtain additive approximate solutions to any other problems. Note that the additive approximation is unavoidable, since some of the problems induce negative objective values. Computational experiments using a vast amount of instances for the problems demonstrate that some algorithm for one problem that have never been considered as potential algorithms for the other problems outperform state-of-the-art methods specifically designed for those problems.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:** \n\nThis is a mathematically solid paper, with a lot of originality and significance. The authors succeed in establishing a novel algorithmic foundation for the diverse problems, including USSS, UDSS, DSS, SFM, and MNP, by constructing a strongly polynomial-time reduction between any two problems and approximation-preserving reductions from USSS, UDSS (DSS), and SFM to MNP. These types of reductions are quite beneficial, since each of the problems (or its special case) has very actively been studied in the literature and the reductions enable algorithmic cross-over, applying an algorithm developed for one problem to the others. For instance, the classic Fujishige--Wolfe's algorithm for MNP is now applicable to the other problems and the authors empirically demonstrate that when applying it to the recent HNSN, a special case of USSS, it significantly outperforms state-of-the-art methods. On the other hand, the recently-invented algorithm for DSS called SuperGreedy++ is now applicable to the classic SFM, and the authors again empirically demonstrate that SuperGreedy++ often performs better than state-of-the-art methods. The experiments are quite thorough, where a vast amount of instances for the problems are used and a lot of baseline methods are implemented and tested. Although the writing has some potential to be improved, the quality and clarity are generally satisfactory. \n\n**Weaknesses:**\n\n- The writing can be improved. Currently, the authors introduce USSS and UDSS as generalizations of the recently-introduced problems. However, this way makes the structure of the introduction a bit too complicated, where many special cases that are not directly studied in the methodological part are intertwined with the other main problems. The reviewer would suggest presenting the main problems first and then motivating them by putting some existing special cases. Considering the theoretical importance of the contribution, the reviewer believes the above presentation strategy still works while improves the readability a lot. The structure of the abstract can be modified in a similar way. \n\n- It would be better to include proof sketches in the main body to make the readers get the intuition about the technical contents better. \n\n- The selection of experiments presented in the main body should be justified. In the reviewer's understanding, the results for HNSN highlight the usefulness of the classic algorithms for the recent problem, while the results for Minimum $st$ Cut highlight the usefulness of the recently-invented algorithms for the classic problem. \n\n- There are some missing references. ADS was originally introduced by Dai et al. (SIGMOD 2022) not by [31]. Nagano et al. (ICML 2011) studied an MNP-based algorithm for Densest $k$-Subgraph, which seems relevant to the present paper. \n\n- There are several undefined notations, typos, and other minor things, listed below.\n\nMissing references: \n- Dai, Qiao, Chang: Anchored Densest Subgraph, SIGMOD 2022.\n- Nagano, Kawahara, Aihara: Size-Constrained Submodular Minimization through Minimum Norm Base, ICML 2011.\n\nTypos etc.:\n- Line 9: MNP is undefined. \n- Line 93: deg is undefined. \n- Line 155: $n$ is undefined. \n- Line 167: no data-structures -> no special data-structures\n- Line 188: It would be better to highlight that the approximation-preserving reductions are different from the above reductions. \n- Line 212: that each have -> each of which has \n- Lines 222 & 247: Put a period. \n- Line 226: $x(S)$ is undefined.\n- Line 241: that that -> that\n- Line 274: $n=|V|$ is redundant. \n- Theorem 14: What is $d$? Should it be $x$? The operation $V-u$ is undefined. \n- Line 310: Should $\\delta(v)\\subseteq S$ be $\\delta(v)\\cap S \\neq \\emptyset$?\n- Figures 1b and 1d are not mentioned (in the main body). \n- Line 384: Veldt -> Huang \n- Line 401: Make a section for conclusion and limitations.\n\nQUESTIONS:\n- Is it possible to summarize the time complexities of the algorithms for different problems. For example, the reviewer is interested in the time complexity of SuperGreedy++ when applying it to SFM. Does it give a competitive complexity with state-of-the-art? The runtime is empirically thoroughly evaluated, but the reviewer believes this kind of work is useful. \n- Isn't it worth studying restricted version of USSS, say SSS?\n\nLIMITATIONS:\nyes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-06-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a unified theoretical and algorithmic framework connecting several submodular and supermodular ratio optimization problems, including Densest Subgraph (DSS), Heavy Nodes in a Small Neighborhood (HNSN),  Unrestricted Sparsest Submodular Set (USSS), and Submodular Function Minimization (SFM), Minimum Norm Point (MNP), etc. The authors establish strong polynomial-time equivalences among these problems and demonstrate that their approximation guarantees can be mutually preserved via reductions to the Minimum Norm Point (MNP) problem. This equivalence enables algorithmic cross-pollination: methods like SUPERGREEDY++, Frank-Wolfe, and Fujishige-Wolfe’s MNP algorithm—originally designed for specific settings—are shown to serve as universal solvers across all problem classes.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1.\tThe connection between several submodular and supermodular ratio optimization problems is very interesting and important. \n\n2.\tThe theoretical insights are compelling, particularly the formalization of how approximations for MNP transfer to other problems, which broadens the applicability of existing algorithms. \n\n3.\tThe experimental results show the theoretical results can be applied in practice. \n\nWeakness: The paper discusses many problems and algorithms, so due to the space limitation, it is difficult to cover every detail in the main body.\n\nQUESTIONS:\n1.\tThe paper states that DSG is polynomial time solvable. What about the rest problems mentioned in the paper? Like DSS, HNSN, USSS, etc.\n\n2.\tIt seems that the condition of Theorem 13 is slightly different with the approximation result for MNP, no matter that we talk about additive approximation or multiplicative approximation. In my understand, the direct implication of the condition is that $||\\hat{x}||_2\\leq ||x^*||_2+\\epsilon^2/||\\hat{x}||_2$. So it might be useful to write down approximation solutions of MNP itself based on this condition in the theorem.\n\n3.\tI want to confirm whether the following understanding is correct. When we say the algorithm (SUPERGREEDY++, Frank-Wolfe, or Fujishige-Wolfe) can solve the problems mentioned in the paper efficiently and with good approximation, we mean that we should firstly transfer the problem into MNP , then use the algorithm to solve the MNP problem and finally transfer the solution back. It does not mean we can obtain good result if we apply the algorithm directly to the problem (suppose we have some natural way to apply the algorithm).\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work mainly proves five classes of problems to be equivalent, and also that approximate solutions to MNP can be transformed into approximate solutions for these five problems. This suggests that the method originally specifically on one of theseo another of these problems (namely, become somewhat problems can be transformed int universal), and even achieve a better performance. It also provides a possible way to connect the independent methods research together in a unified framework. To evaluate the discussion in the theoretical parts, the authors also conducted numerical experiments on a large-scale dataset, where basically convincing results are presented. Many interesting discoveries are discussed in the paper, while some of them are left as open questions in the limitations section.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1. The text is clear and understandable, and the work is clearly discussed in the paper.\n2. The idea of this paper is very interesting and of significance; it proves the equivalence of the problem types in Theorem 12, thereby creating an opportunity to unify the previously independent lines of research on these problems.\n3. The experiments are well-conducted and are basically sufficient to numerically prove the ideas discussed in the theoretical part.\n\nWeakness:\n1. Some important limitations are left as open questions\n2. Some abbreviations are used repeatedly\n\nQUESTIONS:\n1. According to the experimental results, the authors claim that “MNP and FW excel on HNSN, they both underperform here, underscoring how problem structure affects solver efficiency.” (Line 371, “here” here refers to Minimum s-t Cut). If HNSN and MNP problems are already proven to be equivalent in Theorem 12 (Line 12), then why would the performances of two methos MNP and FW differ that much on two somehow equivalent problems? Can you provide any anaylsis of how exactly the problem structure is affecting the efficiency here?\n\n2. According to the authors’ experimental results, in some problems the general methods can beat sota problem-specific methods. I'm quite curious about this result, does this mean the problem-specific method in fact does not take advantage of that much useful “specific information” in the problem? Is it possible to directly design problem-specific methods with better performance?\n\n3. Some abbreviations are used repeatedly. e.g. MNP refers to both an algorithm (Fujishige-Wolfe Minimum Norm Point Algorithm, e.g. Line 372) and a problem class (MINIMUM NORM POINT Problem, e.g. Line 244). It’s not a major problem but can somehow affect the clarity of the paper, we expect that authors can do a little modification here.\n\nLIMITATIONS:\nYes. The authors have already discussed the main limitations from Line 401 to Line 405, though I’m also quite curious about the answers to these limitations.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nOptimization problems involving submodular and supermodular \nfunctions have many applications in computer science. The paper\naddresses an important version of these problems, called the ratio\noptimization problem where the goal is to optimize the ratio f(S)/|S|,\nwhere S is the solution set and f is the submodular or supermodular\nfunction. The paper points out that the ratio optimization problem models\na number of optimization problems studied in the literature. It is shown\nthat many of these problems are equivalent to each other (with respect to\noptimal solutions) through polynomial time reductions. It is also shown\nthat traditional approximation methods (namely SuperGreedy++ and\nMinimum Norm Point) yield good approximations for all of the problems\nconsidered. Extensive experimental results are presented for the\nproblems considered.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n(a) The topic of the paper (optimizing submodular and supermodular\nfunctions) is of great interest to researchers because of numerous\napplications.\n\n(b) The paper nicely unifies known results for many problems and\nshows that traditional approximation methods provide efficient\napproximations for the problems. (The result for minimum st cut was\na pleasant surprise to this reviewer.)\n\n(c) The experimental results are comprehensive.\n\n(d) The paper is written very well.\n\nWeakness:  This reviewer can't identify any major weakness. A (very)\nminor weakness is that the the reader needs to keep track of a large\nnumber of acronyms. A simple way to overcome this minor weakness\nis to include a table of acronyms (in a longer version of the paper).\n\nQUESTIONS:\n(1) In the definition of Problem 4 (lines 82--85), it seems to this\nreviewer that the summation should be over the nodes in the solution\nset S (rather than the set R of all nodes). This may just be a typo.\nPlease clarify.\n\n(2) Lines 127--128: You mention that \"... deemed impractical for\nlarge graphs due to requiring an expensive binary search procedure\".\nCan you please explain? (Usually, binary search is not expensive.)\n\n(3) As stated in Theorem 14, the number of iterations\ndepends on the maximum value of the submodular or supermodular function.\nSo, it seems to this reviewer that SuperGreedy++ is a pseudo-polynomial time\nalgorithm rather than a polynomial time algorithm. Can you please clarify?\n\nA minor suggestion: In several places (e.g., lines 98, 105)\nthe paper has sentences of the form \"[39] showed that ...\" or\n\"indeed [31] state ...\". It is better to revise them to something like\n\"Li et al. [39} showed that ...\".\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-03"}