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1. Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries

   https://doi.org/10.4230/LIPIcs.ICALP.2021.53  

   Chen, Yu; Khanna, Sanjeev; Nagda, Sanjeev (July 2021, 48th International Colloquium on Automata, Languages, and Programming)

   null (Ed.)

   The problem of sparsifying a graph or a hypergraph while approximately preserving its cut structure has been extensively studied and has many applications. In a seminal work, Benczúr and Karger (1996) showed that given any n-vertex undirected weighted graph G and a parameter ε ∈ (0,1), there is a near-linear time algorithm that outputs a weighted subgraph G' of G of size Õ(n/ε²) such that the weight of every cut in G is preserved to within a (1 ± ε)-factor in G'. The graph G' is referred to as a (1 ± ε)-approximate cut sparsifier of G. Subsequent recent work has obtained a similar result for the more general problem of hypergraph cut sparsifiers. However, all known sparsification algorithms require Ω(n + m) time where n denotes the number of vertices and m denotes the number of hyperedges in the hypergraph. Since m can be exponentially large in n, a natural question is if it is possible to create a hypergraph cut sparsifier in time polynomial in n, independent of the number of edges. We resolve this question in the affirmative, giving the first sublinear time algorithm for this problem, given appropriate query access to the hypergraph. Specifically, we design an algorithm that constructs a (1 ± ε)-approximate cut sparsifier of a hypergraph H(V,E) in polynomial time in n, independent of the number of hyperedges, when given access to the hypergraph using the following two queries: 1) given any cut (S, ̄S), return the size |δ_E(S)| (cut value queries); and 2) given any cut (S, ̄S), return a uniformly at random edge crossing the cut (cut edge sample queries). Our algorithm outputs a sparsifier with Õ(n/ε²) edges, which is essentially optimal. We then extend our results to show that cut value and cut edge sample queries can also be used to construct hypergraph spectral sparsifiers in poly(n) time, independent of the number of hyperedges. We complement the algorithmic results above by showing that any algorithm that has access to only one of the above two types of queries can not give a hypergraph cut sparsifier in time that is polynomial in n. Finally, we show that our algorithmic results also hold if we replace the cut edge sample queries with a pair neighbor sample query that for any pair of vertices, returns a random edge incident on them. In contrast, we show that having access only to cut value queries and queries that return a random edge incident on a given single vertex, is not sufficient. 

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2. Counting and Enumerating Optimum Cut Sets for Hypergraph k -Partitioning Problems for Fixed k

   https://doi.org/10.1287/moor.2022.0259  

   Beideman, Calvin; Chandrasekaran, Karthekeyan; Wang, Weihang (December 2023, Mathematics of Operations Research)

   We consider the problem of enumerating optimal solutions for two hypergraph k-partitioning problems, namely, Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. The input in hypergraph k-partitioning problems is a hypergraph [Formula: see text] with positive hyperedge costs along with a fixed positive integer k. The goal is to find a partition of V into k nonempty parts [Formula: see text]—known as a k-partition—so as to minimize an objective of interest. (1) If the objective of interest is the maximum cut value of the parts, then the problem is known as Minmax-Hypergraph-k-Partition. A subset of hyperedges is a minmax-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Minmax-Hypergraph-k-Partition. (2) If the objective of interest is the total cost of hyperedges crossing the k-partition, then the problem is known as Hypergraph-k-Cut. A subset of hyperedges is a min-k-cut-set if it is the subset of hyperedges crossing an optimum k-partition for Hypergraph-k-Cut. We give the first polynomial bound on the number of minmax-k-cut-sets and a polynomial-time algorithm to enumerate all of them in hypergraphs for every fixed k. Our technique is strong enough to also enable an [Formula: see text]-time deterministic algorithm to enumerate all min-k-cut-sets in hypergraphs, thus improving on the previously known [Formula: see text]-time deterministic algorithm, in which n is the number of vertices and p is the size of the hypergraph. The correctness analysis of our enumeration approach relies on a structural result that is a strong and unifying generalization of known structural results for Hypergraph-k-Cut and Minmax-Hypergraph-k-Partition. We believe that our structural result is likely to be of independent interest in the theory of hypergraphs (and graphs). Funding: All authors were supported by NSF AF 1814613 and 1907937. 

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3. Augmentations in Hypergraph Contrastive Learning: Fabricated and Generative

   Wei, Tianxin; You, Yuning; Chen, Tianlong; Shen, Yang; He, Jingrui; Wang, Zhangyang (January 2022, Advances in neural information processing systems)

   Koyejo S.; Mohamed S.; Agarwal A.; Belgrave D.; Cho K.; Oh A. (Ed.)

   This paper targets at improving the generalizability of hypergraph neural networks in the low-label regime, through applying the contrastive learning approach from images/graphs (we refer to it as HyperGCL). We focus on the following question: How to construct contrastive views for hypergraphs via augmentations? We provide the solutions in two folds. First, guided by domain knowledge, we fabricate two schemes to augment hyperedges with higher-order relations encoded, and adopt three vertex augmentation strategies from graph-structured data. Second, in search of more effective views in a data-driven manner, we for the first time propose a hypergraph generative model to generate augmented views, and then an end-to-end differentiable pipeline to jointly learn hypergraph augmentations and model parameters. Our technical innovations are reflected in designing both fabricated and generative augmentations of hypergraphs. The experimental findings include: (i) Among fabricated augmentations in HyperGCL, augmenting hyperedges provides the most numerical gains, implying that higher-order information in structures is usually more downstream-relevant; (ii) Generative augmentations do better in preserving higher-order information to further benefit generalizability; (iii) HyperGCL also boosts robustness and fairness in hypergraph representation learning. Codes are released at https://github.com/weitianxin/HyperGCL. 

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4. Hypergraph Partitioning with Embeddings

   https://doi.org/10.1109/TKDE.2020.3017120  

   Sybrandt, Justin; Shaydulin, Ruslan; Safro, Ilya (December 2020, IEEE Transactions on Knowledge and Data Engineering)

   null (Ed.)

   HYPERGRAPHS provide the formalism needed to solve problems consisting of interconnected item sets. Similar to a traditional graph, the hypergraph has the added generalization that “hyperedges” may connect any number of nodes. Domains such as very-large-scale integration for creating integrated circuits [1], machine learning [2], [3], [4], parallel algorithms [5], combinatorial scientific computing [6], and social network analysis [7], [8] all contain significant and challenging instances of hypergraph problems. One important problem, Hypergraph partitioning, involves dividing the nodes of a hypergraph among k similarly-sized disjoint sets while reducing the number of hyperedges that span multiple partitions. In the context of load balancing, this is the problem of dividing logical threads (nodes) that share data dependencies (hyperedges) among available machines (partitions) in order to balance the number of threads per machine and minimize communication overhead. However, hypergraph partitioning is both NP-Hard to solve [9] and approximate [10]. 

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5. Relative Survivable Network Design

   Dinitz, Michael; Koranteng, Ama; Kortsarz, Guy (September 2022, Leibniz international proceedings in informatics)

   One of the most important and well-studied settings for network design is edge-connectivity requirements. This encompasses uniform demands such as the Minimum k-Edge-Connected Spanning Subgraph problem (k-ECSS), as well as nonuniform demands such as the Survivable Network Design problem. A weakness of these formulations, though, is that we are not able to ask for fault-tolerance larger than the connectivity. Taking inspiration from recent definitions and progress in graph spanners, we introduce and study new variants of these problems under a notion of relative fault-tolerance. Informally, we require not that two nodes are connected if there are a bounded number of faults (as in the classical setting), but that two nodes are connected if there are a bounded number of faults and the two nodes are connected in the underlying graph post-faults. That is, the subgraph we build must "behave" identically to the underlying graph with respect to connectivity after bounded faults. We define and introduce these problems, and provide the first approximation algorithms: a (1+4/k)-approximation for the unweighted relative version of k-ECSS, a 2-approximation for the weighted relative version of k-ECSS, and a 27/4-approximation for the special case of Relative Survivable Network Design with only a single demand with a connectivity requirement of 3. To obtain these results, we introduce a number of technical ideas that may of independent interest. First, we give a generalization of Jain’s iterative rounding analysis that works even when the cut-requirement function is not weakly supermodular, but instead satisfies a weaker definition we introduce and term local weak supermodularity. Second, we prove a structure theorem and design an approximation algorithm utilizing a new decomposition based on important separators, which are structures commonly used in fixed-parameter algorithms that have not commonly been used in approximation algorithms. 

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