Title: New Parallel and Streaming Algorithms for Directed Densest Subgraph
Venue: NeurIPS 2025
Accepted: yes
Keywords: semi-streaming

massively parallel computation

directed densest subgraph

approximation algorithms
Abstract: Finding dense subgraphs is a fundamental problem with applications to community detection, clustering, and data mining. Our work focuses on finding approximate densest subgraphs in directed graphs in computational models for processing massive data. We consider two such models: Massively Parallel Computation (MPC) and semi-streaming. We show how to find a $(2+\varepsilon)$-approximation in $\tilde{O}(\sqrt{\log n})$ MPC rounds with sublinear memory per machine. This improves the state-of-the-art results by Bahmani et al. (WAW 2014) and Mitrovic \& Pan (ICML 2024). Moreover, we show how to find an $O(\log n)$-approximation in a single pass in semi-streaming. This is in stark contrast to prior work, which implies $\tilde{\Omega}(n^{1/6})$-approximation for a single pass; a better approximation is known only for randomized streams (Mitrovi\'c \& Pan). This is the first deterministic single-pass semi-streaming algorithm for the densest subgraph problem, both for undirected and directed graphs. Our semi-streaming approach is also an insertion-only dynamic algorithm, attaining the first directed densest subgraph algorithm with $O(\log^2 n)$ worst-case update time while using sub-linear memory. We empirically evaluate our approaches in two ways. First, we illustrate that our single-pass semi-streaming algorithm performs much better than the theoretical guarantee. Specifically, its approximation on temporal datasets matches the $(2+\varepsilon)$-approximation of an $O(\log n)$-pass algorithm by Bahmani et al. (VLDB 2012). Second, we demonstrate that our MPC algorithm requires fewer rounds than prior work.


Peer reviews:
comments: SUMMARY:
This paper studies the directed densest-subgraph (DS) problem in the Massively Parallel Computation (MPC) model and the streaming model. Let $n$ and $m$ be the numbers of vertices and edges of the input directed graph, respectively.

- In the MPC model, the authors present a $\tilde{O}(\sqrt{\log n})$-round algorithm that returns a $(2+\epsilon)$-approximation while using sub-linear (i.e., $O(n^{\delta})$ for some $\delta<1$) memory per machine. This result improves previous works which needs either $\Omega(\log n)$ rounds or $\tilde{\Omega}(n)$ memory, but at the cost of a looser approximation ratio.
- In the streaming model, this paper introduces the first single-pass deterministic streaming algorithm for directed DS on arbitrary streams. It uses $\tilde{O}(n)$ memory and achieves an $O(\log ⁡n)$-approximation.

STRENGTHS AND WEAKNESSES:
**Strengths**: This is a theoretical paper. These results are well-motivated, and strengthen existing bounds in meaningful ways. Moreover, the paper is clearly structured and well written.

**Weaknesses  and suggestions:** Please provide more specifics in the Introduction about the MPC algorithm, including the total number of machines used and the scheme used to partition the graph across them. In addition, I recommend adding two summary tables: one for existing MPC algorithms and another for streaming algorithms for tackling directed DS. Each table should list the key metrics for every algorithm (e.g., round complexity, memory per machine, and approximation ratio).

QUESTIONS:
See "weaknesses"

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the directed densest subgraph (DS) problem under the MPC model and the semi-streaming model. In this problem, given a directed graph $(V, E)$, the goal is to find subsets S, T of V that maximize $\\frac{|E(S, T)|}{\\sqrt{|S| \\cdot |T|}}$. The algorithms for both models are built on top of a simple \"base\" DS algorithm, which combines the standard iterated peeling procedure with a new stopping rule. In the MPC, the base algorithm is simulate via graph exponentiation. To apply the known simulation techniques, the authors design graph sparsification steps (e.g., freezing high-degree vertices and sampling edges) to control the neighborhood size of the graph. In semi-streaming, the authors make a key observation that the base algorithm can be simulated by maintaining a state vector recording how many peeling iterations each vertex survives. The MPC algorithm achieves $(2+\\varepsilon)$-approximation in $\\tilde{O}(\\sqrt{\\log n})$ rounds, using sublinear memory per machine. This matches the undirected bound and improves over the previous directed memory bound of $\\tilde{\\Theta}(n)$. The semi-streaming algorithm provides the first bound for arbitrary streams.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**:\n\n1. I like the general presentation of this paper: it starts with giving a simple ''base algorithm'' that is later used as a skeleton for both the MPC and the semi-streaming algorithm. \n\\\nAlthough most proofs are delayed to the appendix due to the page limit, I think the high-level ideas of both algorithms and analysis are explained clearly in the main body of the paper. When describing the algorithms the authors carefully discuss the intuition behind each ingredient of the algorithm. And when outlining the analysis, they detailedly explain how lemmas interlock with each other and together lead to the main result.\n\n2. While both algorithms are derived from the simple \"base algorithm\", lifting it to the MPC / semi-streaming settings involves ideas that feels nice and non-trivial.\n\n- In the MPC case, to apply the graph exponentiation technique for simulation, the authors carefully control the size of neighborhoods by designing steps to freeze high-degree vertices and sample edges appropriately.\n\n- In the semi-streaming case, I find the observation of reducing the simulation to computing a vertex-vector which records the peeling iteration for each vertex interesting. And I'm particularly surprised by how the algorithm contains a seemingly crude ''degree resetting'' procedure but still manage to achieve a log(n)-approximation.\n\n**Weakness**:\n\n- The main weakness that I can think of is both approximation factors achieved by this paper, $(2+\\epsilon)$ and $O(\\log n)$, can be a bit unsatisfying. However, as mentioned in the **Summary** above, these factors are already noteworthy improvements over states of the arts. So I find the contribution of this paper clear.\n\nQUESTIONS:\n1. It seems that the factor 2 in the (2+eps)-approximation MPC comes from the fact that the subgraph condition gives only 2-approximation. \n- Is this relates to the fact that to handle directed graph, this paper ''duplicates'' the vertices to produce an undirected bipartite graph? \n- Does the existing (1+eps)-approximation algorithm for *directed* graph use a similar subgraph condition?\n\n2. In the context of semi-streaming, for undirected DS, [EHW15] provide a single-pass algorithm that guarantees a $(1+\\varepsilon)$-approximation. The authors mention that their algorithm for directed DS can also be adapted to approximating the undirected DS. I assume this results in worse approximation factor compared to [EHW15], but do the authors observe any other advantage over them?\n\nLIMITATIONS:\nYes, the limitations are clearly discussed. The authors also point out multiple potential future directions.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents new algorithms for finding approximate densest subgraphs in directed graphs within Massively Parallel Computation (MPC) and semi-streaming models. The proposed approaches achieve a $(2+\\varepsilon)$-approximation in $\\tilde{O}(\\sqrt{\\log n})$ MPC rounds with sublinear memory per machine and an $O(\\log n)$-approximation in a single pass in the semi-streaming setting, significantly improving upon prior results. Empirical evaluations demonstrate that the algorithms are efficient in practice, with the semi-streaming method achieving better-than-theoretical guarantees on temporal datasets and the MPC algorithm requiring fewer rounds than existing methods.\n\nSTRENGTHS AND WEAKNESSES:\nS1. The paper introduces novel algorithms for the directed densest subgraph problem that significantly improve upon existing methods, particularly in the Massively Parallel Computation and semi-streaming models, offering better approximation guarantees and efficiency.\nS2. It provides strong theoretical results, including a $(2+\\varepsilon)$-approximation in $\\tilde{O}(\\sqrt{\\log n})$  MPC rounds and an $O(\\log n)$-approximation in a single pass for semi-streaming, which bridge the gap between directed and undirected densest subgraph problems.\nS3. The empirical evaluations show that the proposed algorithms perform well in practice, often exceeding theoretical expectations, especially in the semi-streaming context.\n\nW1. Some steps in Algorithm 2 involve complex sampling and degree-thresholding mechanisms, and the explanations—although sufficiently technical—could benefit from more intuitive discussion, especially regarding relation to graph sparsification and the effect of 'freezing' high-degree vertices.\nW2. Most experimental comparisons are offered for the semi-streaming algorithm. While this supports the main claims, the MPC algorithm’s empirical performance is only briefly mentioned and detailed results are relegated to the appendix, limiting the main-paper evidence for the MPC improvements.\n\nQUESTIONS:\nQ1. Please add more experiments regarding the MPC case.\nQ2. Please add more explaination regarding the proposed algorithms to improve the readability.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes new algorithms for approximating directed densest subgraphs in Massively Parallel Computation (MPC) and semi-streaming models. Specifically, in the MPC model, the paper provides a $(2+\\epsilon)$-approximation in $\\tilde{O}(\\sqrt{\\log n})$ rounds in the sublinear memory regime, which improves the best-known results and bridges the gap between the directed and undirected cases. In the semi-streaming model, the paper provides a (deterministic) $O(\\log n)$-approximation using a single pass, which is the first (deterministic) single-pass semi-streaming algorithm for this problem. Both algorithms are based on an elegant peeling-based algorithm, but its efficient implementation in MPC and streaming models are non-trivial, and involves several novel insights developed in this paper. Last but not least, the paper empirically evaluates the effectiveness of the proposed algorithms on several large real-world datasets.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1: Computing densest subgraphs in directed graphs is a fundamental problem and would be of interest to the NeurIPS community. The paper is clearly structured and very well written.\n\n2: The paper provides new algorithms in the MPC and semi-streaming models, improving upon the best-known results. The analyses are non-trivial, technically sound, and involve several novel insights.\n\n3: The paper conducts empirical evaluations on several large real-world networks, demonstrating the effectiveness and efficiency of the proposed algorithms.\n\nWeaknesses:\n\n1: The proposed semi-streaming algorithm appears to work only in insertion-only streams. It would strengthen the work to extend it to dynamic streams, which represent a more general setting.\n\n2: More comprehensive evaluations could be conducted in the experiments, e.g. the space usage of the algorithms.\n\n3: Some minor issues:\n\n-- Line 63: show -> shows\n\n-- Line 64: require -> requires\n\n-- In addition, it would be helpful to include a table summarizing existing results and the new results proposed in this paper for clearer comparison.\n\nQUESTIONS:\nQ1: Algorithm 3 resets the vertex degree to 0 when it moves to a higher level, assuming the worst case. Could this be improved to achieve a better approximation under the assumption that the stream arrives in random order?\n\nQ2: Can the semi-streaming algorithm (i.e. Algorithm 3) be extended to work in dynamic streams, where both edge insertions and deletions are allowed?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of approximating the densest subgraph in directed graphs within the MPC and streaming models. There are three main categories of known results for the densest subgraph problem (both directed and undirected) in these computational settings.\n\nFirst, for undirected graphs, Mitrovic and Pan \\cite{MP24} proposed an $O(\\sqrt{\\log n})$-round MPC algorithm that requires $\\tilde{O}(n)$ space per machine, which is linear in the number of nodes.\n\nSecond, in the streaming model, Esfandiari, Hajiaghayi, and Woodruff \\cite{EHW15} (SPAA 2015) developed streaming algorithms that handle both insertions and deletions of edges. Their method samples $\\tilde{O}(n)$ edges uniformly at random and then applies an offline densest subgraph algorithm to the sampled set at the end of the stream.\n\nFor directed graphs, Mitrovic and Pan \\cite{MP24} also presented an MPC algorithm with $O(\\log n)$ rounds. The streaming algorithm of \\cite{EHW15} can be generalized to the directed case, but this requires $O(n^{1.5})$ space, which is no longer near-linear. Alternatively, the method of \\cite{MP24} can produce a streaming algorithm with near-linear space complexity, but only if the input edges arrive in random order (i.e., the stream is randomly permuted).\n\nIn contrast, this paper presents new MPC and streaming algorithms for the directed densest subgraph problem that outperform the prior results of \\cite{EHW15} and \\cite{MP24}. In the MPC model, the authors propose an $O(\\sqrt{\\log n})$-round algorithm. However, throughout the paper (including Result 1 and Theorem 4.2), the authors repeatedly state that their algorithm uses “sublinear” space per machine without specifying the exact bound. This makes it hard to properly evaluate the result, and the exact space complexity should be clearly stated.\n\nThe second main contribution, described in Result 2 and Theorem 5.1, is a single-pass deterministic semi-streaming algorithm achieving an $O(\\log n)$-approximation for the directed densest subgraph. Although the approximation factor is not constant, this result is interesting because it appears to be the first deterministic streaming algorithm for this problem.\n\nHere are two additional suggestions to improve the paper:\n\n- The authors should cite and discuss the work of Monika Henzinger (STOC 2015), which is a pioneering contribution in this area and should be acknowledged.\n\n- The paper introduces a non-standard term for the model where edges arrive in random order; the common name for this setting is random order streams. Using the standard terminology would make the presentation clearer.\n\n**Summary**\n\nOverall, the paper presents interesting and potentially impactful results for the directed densest subgraph problem in MPC and streaming models. However, the presentation could be significantly improved by clearly stating the exact space complexity of the algorithms, structuring the related work discussion more carefully, and aligning terminology with standard usage in the literature. Addressing these points would make the contribution clearer and strengthen the paper.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- The paper studies the directed densest subgraph problem, which is an important and timely topic in streaming and MPC models.\n- Proposes an $O(\\sqrt{\\log n})$-round MPC algorithm, improving over the previous $O(\\log n)$-round result by \\cite{MP24}.\n- Presents what appears to be the first deterministic semi-streaming algorithm for approximating the densest subgraph in directed graphs, which could be of independent theoretical interest.\n\n**Weaknesses**\n\n- The paper does not clearly specify the exact space per machine used in the MPC algorithm, repeatedly describing it only as “sublinear,” which makes the result hard to evaluate and compare.\n\n- The discussion of related work is limited and does not include important prior contributions (e.g., Henzinger STOC 2015).\n\n- The paper uses non-standard terminology for “random order streams,” which could confuse readers.\n\nQUESTIONS:\n**Questions and Suggestions**\n\n- Could the authors clearly specify the exact space complexity per machine used by the MPC algorithm? Using only the term “sublinear” is too vague; please state the precise asymptotic bound (e.g., $\\tilde{O}(n^\\alpha)$ for some explicit $\\alpha < 1$).\n\n- Can the authors elaborate on the technical novelty compared to prior work, especially \\cite{MP24} and \\cite{EHW15}? What new ideas or analysis allow achieving the improved number of MPC rounds or the deterministic streaming result?\n\n- Please discuss the pioneering work by Monika Henzinger (STOC 2015), and explain how this paper relates to or builds on that contribution.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-07-05"}