The densest subgraph problem has received significant attention, both in theory and in practice, due to its applications in problems such as community detection, social network analysis, and spam detection. Due to the high cost of obtaining exact solutions, much attention has focused on designing approximate densest subgraph algorithms. However, existing approaches are not able to scale to massive graphs with billions of edges.

Dataset Versioning is extremely important for ensuring the reproducibility of results, tracking data changes over time, maintaining quality measures, enabling collaboration, and ensuring legal compliance. In this work, we study the cost efficient data versioning problem, where the goal is to optimize the storage and reconstruction (retrieval) costs of data versions, given a graph of datasets as nodes and edges capturing edit/delta information. One central variant we study is MINSUM RETRIEVAL (MSR) where the goal is to minimize the total retrieval costs, while keeping the storage costs bounded. This problem (along with its variants) was introduced by Bhattacherjee et al. [VLDB’15]. While such problems are frequently encountered in collaborative tools (e.g., version control systems and data analysis pipelines), to the best of our knowledge, no existing research studies the theoretical aspects of these problems.We established, in the full version of this work1, that the previous best heuristic, LMG (introduced in [VLDB’15]) can perform arbitrarily badly in a simple worst case. Moreover, we show that it is hard to get o(n)-approximation for MSR on general graphs even if we relax the storage constraints by an O(log n) factor. Similar hardness results are shown for other variants. Meanwhile, we propose poly-time approximation schemes for tree-like graphs, motivated by the fact that the graphs arising in practice from typical edit operations are often not arbitrary. As version graphs typically have low treewidth, we further develop new algorithms for bounded treewidth graphs.Furthermore, we propose two new heuristics and evaluate them empirically. First, we extend LMG by considering more potential "moves", to propose a new heuristic LMG-All. LMG-All consistently outperforms LMG while having comparable run time on a wide variety of datasets, i.e., version graphs. Secondly, we apply our tree algorithms on the minimum-storage arborescence of an instance, yielding algorithms that are qualitatively better than all previous heuristics for MSR, as well as for another variant BOUNDEDMIN RETRIEVAL (BMR).

In data platforms with thousands of data tables available for exploration, users often need to retrieve some data based on limited knowledge of the data sources and schema. The task that automates retrieving attributes from an online data lake given a set of entities from users is called “entity augmentation”. The key for successful entity augmentation is an accurate construction of semantic relationships between data tables. Current techniques either focus on retrieving categorical values or numerical values with pre-defined rules. Further, they assume there are meta-data available for each table, such as texts and tags. In this paper, we introduce a semantic graph for numerical data augmentation that (i) matches columns with similar semantic relationships without any meta-data from the tables; (ii) infer the conversion rules among different numerical columns based on the values. The approach is designed to be highly scalable and parallel for large-scale data lakes with millions of large datasets. We also propose efficient algorithms to construct the semantic graph on a distributed computing environment (i.e. Spark) and conduct numerical data augmentation using the graph. Through comprehensive experiments on real-world datasets, the approach is shown to (1) achieve better accuracy on semantic matches and value conversions and (2) scales to the tractable computation time on large-scale data. Finally, we also present an interface to apply the semantic graph for real-world scenarios.

Individual preference (IP) stability, introduced by Ahmadi et al. (ICML 2022), is a natural clustering objective inspired by stability and fairness constraints. A clustering is $\alpha$-IP stable if the average distance of every data point to its own cluster is at most $\alpha$ times the average distance to any other cluster. Unfortunately, determining if a dataset admits a $1$-IP stable clustering is NP-Hard. Moreover, before this work, it was unknown if an $o(n)$-IP stable clustering always exists, as the prior state of the art only guaranteed an $O(n)$-IP stable clustering. We close this gap in understanding and show that an $O(1)$-IP stable clustering always exists for general metrics, and we give an efficient algorithm which outputs such a clustering. We also introduce generalizations of IP stability beyond average distance and give efficient near optimal algorithms in the cases where we consider the maximum and minimum distances within and between clusters.

Designing dynamic algorithms against an adaptive adversary whose performance match the ones assuming an oblivious adversary is a major research program in the field of dynamic graph algorithms. One of the prominent examples whose oblivious-vs-adaptive gap remains maximally large is the \emph{fully dynamic spanner} problem; there exist algorithms assuming an oblivious adversary with near-optimal size-stretch trade-off using only $\operatorname{polylog}(n)$ update time [Baswana, Khurana, and Sarkar TALG'12; Forster and Goranci STOC'19; Bernstein, Forster, and Henzinger SODA'20], while against an adaptive adversary, even when we allow infinite time and only count recourse (i.e. the number of edge changes per update in the maintained spanner), all previous algorithms with stretch at most $\log^{5}(n)$ require at least $Ω(n)$ amortized recourse [Ausiello, Franciosa, and Italiano ESA'05]. In this paper, we completely close this gap with respect to recourse by showing algorithms against an adaptive adversary with near-optimal size-stretch trade-off and recourse. More precisely, for any $k\ge1$, our algorithm maintains a $(2k-1)$-spanner of size $O(n^{1+1/k}\log n)$ with $O(\log n)$ amortized recourse, which is optimal in all parameters up to a $O(\log n)$ factor. As a step toward algorithms with small update time (not just recourse), we show another algorithm that maintains a $3$-spanner of size $\tilde O(n^{1.5})$ with $\operatorname{polylog}(n)$ amortized recourse \emph{and} simultaneously $\tilde O(\sqrt{n})$ worst-case update time.

In the k-edge-connected spanning subgraph (kECSS) problem, our goal is to compute a minimum-cost sub-network that is resilient against up to k link failures: Given an n-node m-edge graph with a cost function on the edges, our goal is to compute a minimum-cost k-edge-connected spanning subgraph. This NP-hard problem generalizes the minimum spanning tree problem and is the "uniform case" of a much broader class of survival network design problems (SNDP). A factor of two has remained the best approximation ratio for polynomial-time algorithms for the whole class of SNDP, even for a special case of 2ECSS. The fastest 2-approximation algorithm is however rather slow, taking O(mn k) time [Khuller, Vishkin, STOC'92]. A faster time complexity of O(n²) can be obtained, but with a higher approximation guarantee of (2k-1) [Gabow, Goemans, Williamson, IPCO'93]. Our main contribution is an algorithm that (1+ε)-approximates the optimal fractional solution in Õ(m/ε²) time (independent of k), which can be turned into a (2+ε) approximation algorithm that runs in time Õ(m/(ε²) + {k²n^{1.5}}/ε²) for (integral) kECSS; this improves the running time of the aforementioned results while keeping the approximation ratio arbitrarily close to a factor of two.

In this paper, we propose a natural notion of individual preference (IP) stability for clustering, which asks that every data point, on average, is closer to the points in its own cluster than to the points in any other cluster. Our notion can be motivated from several perspectives, including game theory and algorithmic fairness. We study several questions related to our proposed notion. We first show that deciding whether a given data set allows for an IP-stable clustering in general is NP-hard. As a result, we explore the design of efficient algorithms for finding IP-stable clusterings in some restricted metric spaces. We present a polytime algorithm to find a clustering satisfying exact IP-stability on the real line, and an efficient algorithm to find an IP-stable 2-clustering for a tree metric. We also consider relaxing the stability constraint, i.e., every data point should not be too far from its own cluster compared to any other cluster. For this case, we provide polytime algorithms with different guarantees. We evaluate some of our algorithms and several standard clustering approaches on real data sets.

In this paper, we study a primal and dual relationship about triangles: For any graph G, let v(G) be the maximum number of edge-disjoint triangles in G, and τ(G) be the minimum subset F of edges such that G \ F is triangle-free. It is easy to see that v(G) ≤ τ(G) ≤ 3v(G), and in fact, this rather obvious inequality holds for a much more general primal-dual relation between k-hyper matching and covering in hypergraphs. Tuza conjectured in 1981 that τ(G) ≤ 2v(G), and this question has received attention from various groups of researchers in discrete mathematics, settling various special cases such as planar graphs and generalized to bounded maximum average degree graphs, some cases of minor-free graphs, and very dense graphs. Despite these efforts, the conjecture in general graphs has remained wide open for almost four decades. In this paper, we provide a proof of a non-trivial consequence of the conjecture; that is, for every k ≥ 2, there exist a (multi)-set F ⊆ E(G): |F| ≤ 2kv(G) such that each triangle in G overlaps at least k elements in F. Our result can be seen as a strengthened statement of Krivelevich's result on the fractional version of Tuza's conjecture (and we give some examples illustrating this.) The main technical ingredient of our result is a charging argument, that locally identifies edges in F based on a local view of the packing solution. This idea might be useful in further studying the primal-dual relations in general and the Tuza's conjecture in particular.