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1. Quotient sparsification for submodular functions

   https://doi.org/10.1137/1.9781611977912.187  

   Quanrud, Kent (January 2024, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024)

   Woodruff, David P. (Ed.)

   Graph sparsification has been an important topic with many structural and algorithmic consequences. Recently hypergraph sparsification has come to the fore and has seen exciting progress. In this paper we take a fresh perspective and show that they can be both be derived as corollaries of a general theorem on sparsifying matroids and monotone submodular functions. Quotients of matroids and monotone submodular functions generalize k-cuts in graphs and hypergraphs. We show that a weighted ground set of a monotone submodular function f can be sparsified while approximately preserving the weight of every quotient of f with high probability in randomized polynomial time. This theorem conceptually unifies cut sparsifiers for undirected graphs [BK15] with other interesting applications. One basic application is to reduce the number of elements in a matroid while preserving the weight of every quotient of the matroid. For hypergraphs, the theorem gives an alternative approach to the hypergraph cut sparsifiers obtained recently in [CKN20], that also preserves all k-cuts. Another application is to reduce the number of points in a set system while preserving the weight of the union of every collection of sets. We also present algorithms that sparsify hypergraphs and set systems in nearly linear time, and sparsify matroids in nearly linear time and queries in the rank oracle model. 

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2. 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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3. Approximating Nash Social Welfare under Submodular Valuations through (Un)Matchings

   https://doi.org/10.1137/1.9781611975994.163  

   Garg, Jugal; Kulkarni, Pooja; Kulkarni, Rucha (January 2020, Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms)

   We study the problem of approximating maximum Nash social welfare (NSW) when allocating m indivisible items among n asymmetric agents with submodular valuations. The NSW is a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents' valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with factor independent of m is known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations. In this paper, we extend our understanding of the NSW problem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations respectively. Both algorithms are simple to understand and involve non-trivial modifications of a greedy repeated matchings approach. Allocations of high valued items are done separately by un-matching certain items and re-matching them, by processes that are different in both algorithms. We show that these approaches achieve approximation factors of O(n) and O(n log n) for additive and submodular case respectively, which is independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item (EF1). Furthermore, we show that the NSW problem under submodular valuations is strictly harder than all currently known settings with an e/(e-1) factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of e/(e-1), hence resolving it completely. 

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4. Approximating Nash Social Welfare under Submodular Valuations through (Un)Matchings

   https://doi.org/10.1145/3613452  

   Garg, Jugal; Kulkarni, Pooja; Kulkarni, Rucha (October 2023, ACM Transactions on Algorithms)

   We study the problem of approximating maximum Nash social welfare (NSW) when allocatingmindivisible items amongnasymmetric agents with submodular valuations. TheNSWis a well-established notion of fairness and efficiency, defined as the weighted geometric mean of agents’ valuations. For special cases of the problem with symmetric agents and additive(-like) valuation functions, approximation algorithms have been designed using approaches customized for these specific settings, and they fail to extend to more general settings. Hence, no approximation algorithm with a factor independent ofmwas known either for asymmetric agents with additive valuations or for symmetric agents beyond additive(-like) valuations before this work. In this article, we extend our understanding of theNSWproblem to far more general settings. Our main contribution is two approximation algorithms for asymmetric agents with additive and submodular valuations. Both algorithms are simple to understand and involve non-trivial modifications of a greedy repeated matchings approach. Allocations of high-valued items are done separately by un-matching certain items and re-matching them by different processes in both algorithms. We show that these approaches achieve approximation factors ofO(n) andO(nlogn) for additive and submodular cases, independent of the number of items. For additive valuations, our algorithm outputs an allocation that also achieves the fairness property of envy-free up to one item (EF1). Furthermore, we show that theNSWproblem under submodular valuations is strictly harder than all currently known settings with an\(\frac{\mathrm{e}}{\mathrm{e}-1}\)factor of the hardness of approximation, even for constantly many agents. For this case, we provide a different approximation algorithm that achieves a factor of\(\frac{\mathrm{e}}{\mathrm{e}-1}\), hence resolving it completely. 

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5. The Power of Subsampling in Submodular Maximization

   https://doi.org/10.1287/moor.2021.1172  

   Harshaw, Christopher; Kazemi, Ehsan; Feldman, Moran; Karbasi, Amin (January 2021, Mathematics of Operations Research)

   We propose subsampling as a unified algorithmic technique for submodular maximization in centralized and online settings. The idea is simple: independently sample elements from the ground set and use simple combinatorial techniques (such as greedy or local search) on these sampled elements. We show that this approach leads to optimal/state-of-the-art results despite being much simpler than existing methods. In the usual off-line setting, we present SampleGreedy, which obtains a [Formula: see text]-approximation for maximizing a submodular function subject to a p-extendible system using [Formula: see text] evaluation and feasibility queries, where k is the size of the largest feasible set. The approximation ratio improves to p + 1 and p for monotone submodular and linear objectives, respectively. In the streaming setting, we present Sample-Streaming, which obtains a [Formula: see text]-approximation for maximizing a submodular function subject to a p-matchoid using O(k) memory and [Formula: see text] evaluation and feasibility queries per element, and m is the number of matroids defining the p-matchoid. The approximation ratio improves to 4p for monotone submodular objectives. We empirically demonstrate the effectiveness of our algorithms on video summarization, location summarization, and movie recommendation tasks. 

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