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Aritra Konar

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6 papers

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AAAI Conference 2026 Conference Paper

A Scalable and Exact Relaxation for Densest k-Subgraph via Error Bounds

Ya Liu
Junbin Liu
Wing-Kin Ma
Aritra Konar

Given an undirected graph and a size parameter k, the Densest k-Subgraph (DkS) problem extracts the subgraph on k vertices with the largest number of induced edges. While DkS is NP--hard and difficult to approximate, penalty-based continuous relaxations of the problem have recently enjoyed practical success for real-world instances of DkS. In this work, we propose a scalable and exact continuous penalization approach for DkS using the error bound principle, which enables the design of suitable penalty functions. Notably, we develop new theoretical guarantees ensuring that both the global and local optima of the penalized problem match those of the original problem. The proposed penalized reformulation enables the use of first-order continuous optimization methods. In particular, we develop a non-convex proximal gradient algorithm, where the non-convex proximal operator can be computed in closed form, resulting in low per-iteration complexity. We also provide convergence analysis of the algorithm. Experiments on large-scale instances of the DkS problem and one of its variants, the Densest (k1, k2) Bipartite Subgraph (Dk1k2BS) problem, demonstrate that our method achieves a favorable balance between computation cost and solution quality.

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TMLR Journal 2026 Journal Article

Differentially Private and Scalable Estimation of the Network Principal Component

Alireza Khayatian
Anil Vullikanti
Aritra Konar

Computing the principal component (PC) of the adjacency matrix of an undirected graph has several applications ranging from identifying key vertices for influence maximization and controlling diffusion processes, to discovering densely interconnected vertex subsets. However, many networked datasets are sensitive, which necessitates private computation of the PC for use in the aforementioned applications. Differential privacy has emerged as the gold standard in privacy-preserving data analysis, but existing DP algorithms for private PC suffer from low accuracy due to large noise injection or high complexity. Motivated by the large gap between the local and global sensitivities of the PC on real-graphs, we consider instance-specific mechanisms for privately computing the PC under edge-DP. These mechanisms guarantee privacy for all datasets, but provide good utility on ``well-behaved'' datasets by injecting smaller amounts of noise. More specifically, we consider the Propose-Test-Release (PTR) framework. Although computationally expensive in general, we design a novel approach for implementing a PTR variant in the same time as computation of a non-private PC, while offering good utility. Our framework tests in a differentially-private manner whether a given graph is ``well-behaved'' or not, and then tests whether its private to release a noisy PC with small noise. As a consequence, this also leads to the first DP algorithm for the Densest-$k$-subgraph problem, a key graph mining primitive. We run our method on diverse real-world networks, with the largest having 3 million vertices, and compare its utility to a pre-existing baseline based on the private power method (PPM). Although PTR requires a slightly larger privacy budget, on average, it achieves a 180-fold improvement in runtime over PPM.

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AAAI Conference 2025 Conference Paper

Densest k-Subgraph Mining via a Provably Tight Relaxation

Qiheng Lu
Nicholas D Sidiropoulos
Aritra Konar

Given an unweighted, undirected, and simple graph, the Densest k-Subgraph (DkS) problem aims to find a subgraph of k vertices that has the maximum average induced degree. In this paper, we consider an equivalent reformulation of the DkS problem via diagonal loading. On relaxing the combinatorial constraint of the reformulated problem, we show that the resulting non-convex, continuous relaxation is tight under certain conditions by leveraging an extension of the Motzkin-Straus theorem. We utilize two projection-free approaches to solve the relaxed problem: one based on the Frank-Wolfe algorithm and the other on explicit constraint parameterization. We compare their performance to state-of-the-art baselines across various benchmarks. Our empirical results show that the Frank-Wolfe-based algorithm proposed in this paper outperforms existing baselines in terms of subgraph density and computational complexity.

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TMLR Journal 2025 Journal Article

Fairness-Aware Dense Subgraph Discovery

Emmanouil Kariotakis
Nicholas D Sidiropoulos
Aritra Konar

Dense subgraph discovery (DSD) is a key graph mining primitive with myriad applications including finding densely connected communities which are diverse in their vertex composition. In such a context, it is desirable to extract a dense subgraph that provides fair representation of the diverse subgroups that constitute the vertex set while incurring a small loss in terms of subgraph density. Existing methods for promoting fairness in DSD have important limitations - the associated formulations are NP-hard in the worst case and they do not provide flexible notions of fairness, making it non-trivial to analyze the inherent trade-off between density and fairness. In this paper, we introduce two tractable formulations for fair DSD, each offering a different notion of fairness. Our methods provide a structured and flexible approach to incorporate fairness, accommodating varying fairness levels. We introduce the fairness-induced relative loss in subgraph density as a price of fairness measure to quantify the associated trade-off. We are the first to study such a notion in the context of detecting fair dense subgraphs. Extensive experiments on real-world datasets demonstrate that our methods not only match but frequently outperform existing solutions, sometimes incurring even less than half the subgraph density loss compared to prior art, while achieving the target fairness levels. Importantly, they excel in scenarios that previous methods fail to adequately handle, i.e., those with extreme subgroup imbalances, highlighting their effectiveness in extracting fair and dense solutions.

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AAAI Conference 2024 Conference Paper

Optimal Quasi-clique: Hardness, Equivalence with Densest-k-Subgraph, and Quasi-partitioned Community Mining

Aritra Konar
Nicholas D. Sidiropoulos

Dense subgraph discovery (DSD) is a key primitive in graph mining that typically deals with extracting cliques and near-cliques. In this paper, we revisit the optimal quasi-clique (OQC) formulation for DSD and establish that it is NP--hard. In addition, we reveal the hitherto unknown property that OQC can be used to explore the entire spectrum of densest subgraphs of all distinct sizes by appropriately varying a single hyperparameter, thereby forging an intimate link with the classic densest-k-subgraph problem (DkS). We corroborate these findings on real-world graphs by applying the simple greedy algorithm for OQC with improved hyperparameter tuning, to quickly generate high-quality approximations of the size-density frontier. Our findings indicate that OQC not only extracts high quality (near)-cliques, but also large and loosely-connected subgraphs that exhibit well defined local community structure. The latter discovery is particularly intriguing, since OQC is not explicitly geared towards community detection.

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AAAI Conference 2022 Conference Paper

The Triangle-Densest-K-Subgraph Problem: Hardness, Lovász Extension, and Application to Document Summarization

Aritra Konar
Nicholas D. Sidiropoulos

We introduce the triangle-densest-k-subgraph problem (TDkS) for undirected graphs: given a size parameter k, compute a subset of k vertices that maximizes the number of induced triangles. The problem corresponds to the simplest generalization of the edge based densest-k-subgraph problem (DkS) to the case of higher-order network motifs. We prove that TDkS is NP-hard and is not amenable to efficient approximation, in the worst-case. By judiciously exploiting the structure of the problem, we propose a relaxation algorithm for the purpose of obtaining high-quality, sub-optimal solutions. Our approach utilizes the fact that the cost function of TDkS is submodular to construct a convex relaxation for the problem based on the Lovász extension for submodular functions. We demonstrate that our approaches attain state-of-theart performance on real-world graphs and can offer substantially improved exploration of the optimal density-size curve compared to sophisticated approximation baselines for DkS. We use document summarization to showcase why TDkS is a useful generalization of DkS.

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