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1. Separating the NP-Hardness of the Grothendieck Problem from the Little-Grothendieck Problem

   https://doi.org/10.4230/LIPIcs.ITCS.2022.22  

   Bhattiprolu, Vijay and (January 2022, Leibniz international proceedings in informatics)

   Braverman, Mark (Ed.)

   Grothendieck’s inequality [Grothendieck, 1953] states that there is an absolute constant K > 1 such that for any n× n matrix A, ‖A‖_{∞→1} := max_{s,t ∈ {± 1}ⁿ}∑_{i,j} A[i,j]⋅s(i)⋅t(j) ≥ 1/K ⋅ max_{u_i,v_j ∈ S^{n-1}}∑_{i,j} A[i,j]⋅⟨u_i,v_j⟩. In addition to having a tremendous impact on Banach space theory, this inequality has found applications in several unrelated fields like quantum information, regularity partitioning, communication complexity, etc. Let K_G (known as Grothendieck’s constant) denote the smallest constant K above. Grothendieck’s inequality implies that a natural semidefinite programming relaxation obtains a constant factor approximation to ‖A‖_{∞ → 1}. The exact value of K_G is yet unknown with the best lower bound (1.67…) being due to Reeds and the best upper bound (1.78…) being due to Braverman, Makarychev, Makarychev and Naor [Braverman et al., 2013]. In contrast, the little Grothendieck inequality states that under the assumption that A is PSD the constant K above can be improved to π/2 and moreover this is tight. The inapproximability of ‖A‖_{∞ → 1} has been studied in several papers culminating in a tight UGC-based hardness result due to Raghavendra and Steurer (remarkably they achieve this without knowing the value of K_G). Briet, Regev and Saket [Briët et al., 2015] proved tight NP-hardness of approximating the little Grothendieck problem within π/2, based on a framework by Guruswami, Raghavendra, Saket and Wu [Guruswami et al., 2016] for bypassing UGC for geometric problems. This also remained the best known NP-hardness for the general Grothendieck problem due to the nature of the Guruswami et al. framework, which utilized a projection operator onto the degree-1 Fourier coefficients of long code encodings, which naturally yielded a PSD matrix A. We show how to extend the above framework to go beyond the degree-1 Fourier coefficients, using the global structure of optimal solutions to the Grothendieck problem. As a result, we obtain a separation between the NP-hardness results for the two problems, obtaining an inapproximability result for the Grothendieck problem, of a factor π/2 + ε₀ for a fixed constant ε₀ > 0. 

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2. On Approximability of 𝓁₂² Min-Sum Clustering

   https://doi.org/10.4230/lipics.socg.2025.62  

   S, Karthik C; Lee, Euiwoong; Rabani, Yuval; Schwiegelshohn, Chris; Zhou, Samson (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Aichholzer, Oswin; Wang, Haitao (Ed.)

   The 𝓁₂² min-sum k-clustering problem is to partition an input set into clusters C_1,…,C_k to minimize ∑_{i=1}^k ∑_{p,q ∈ C_i} ‖p-q‖₂². Although 𝓁₂² min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate 𝓁₂² min-sum k-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the 𝓁₂² min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving a fast PTAS for 𝓁₂² min-sum k-clustering. Specifically, our algorithm runs in time O(n^{1+o(1)}d⋅ 2^{(k/ε)^O(1)}), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error α ∈ [0,1/2). We give a polynomial-time algorithm that outputs a (1+γα)/(1-α)²-approximation to 𝓁₂² min-sum k-clustering, for a fixed constant γ > 0. 

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3. Linear space streaming lower bounds for approximating CSPs

   https://doi.org/10.1145/3519935.3519983  

   Chou, Chi-Ning; Golovnev, Alexander; Sudan, Madhu; Velingker, Ameya; Velusamy, Santhoshini (June 2022, {STOC} '22: 54th Annual {ACM} {SIGACT} Symposium on Theory of Computing)

   Leonardi, Stefano; Gupta, Anupam (Ed.)

   We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on n variables taking values in {0,…,q−1}, we prove that improving over the trivial approximability by a factor of q requires Ω(n) space even on instances with O(n) constraints. We also identify a broad subclass of problems for which any improvement over the trivial approximability requires Ω(n) space. The key technical core is an optimal, q−(k−1)-inapproximability for the Max k-LIN-mod q problem, which is the Max CSP problem where every constraint is given by a system of k−1 linear equations mod q over k variables. Our work builds on and extends the breakthrough work of Kapralov and Krachun (Proc. STOC 2019) who showed a linear lower bound on any non-trivial approximation of the MaxCut problem in graphs. MaxCut corresponds roughly to the case of Max k-LIN-mod q with k=q=2. For general CSPs in the streaming setting, prior results only yielded Ω(√n) space bounds. In particular no linear space lower bound was known for an approximation factor less than 1/2 for any CSP. Extending the work of Kapralov and Krachun to Max k-LIN-mod q to k>2 and q>2 (while getting optimal hardness results) is the main technical contribution of this work. Each one of these extensions provides non-trivial technical challenges that we overcome in this work. 

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4. Black-Box k-to-1-PCA Reductions: Theory and Applications

   Jambulapati, A; Kumar, S; Li, J; Pandey, S; Pensia, A; Tian, K (June 2024, https://doi.org/10.48550/arXiv.2403.03905)

   The k-principal component analysis (k-PCA) problem is a fundamental algorithmic primitive that is widely-used in data analysis and dimensionality reduction applications. In statistical settings, the goal of k-PCA is to identify a top eigenspace of the covariance matrix of a distribution, which we only have black-box access to via samples. Motivated by these settings, we analyze black-box deflation methods as a framework for designing k-PCA algorithms, where we model access to the unknown target matrix via a black-box 1-PCA oracle which returns an approximate top eigenvector, under two popular notions of approximation. Despite being arguably the most natural reduction-based approach to k-PCA algorithm design, such black-box methods, which recursively call a 1-PCA oracle k times, were previously poorly-understood. Our main contribution is significantly sharper bounds on the approximation parameter degradation of deflation methods for k-PCA. For a quadratic form notion of approximation we term ePCA (energy PCA), we show deflation methods suffer no parameter loss. For an alternative well-studied approximation notion we term cPCA (correlation PCA), we tightly characterize the parameter regimes where deflation methods are feasible. Moreover, we show that in all feasible regimes, k-cPCA deflation algorithms suffer no asymptotic parameter loss for any constant k. We apply our framework to obtain state-of-the-art k-PCA algorithms robust to dataset contamination, improving prior work in sample complexity by a 𝗉𝗈𝗅𝗒(k) factor. 

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5. Balancing Notions of Equity: Trade-offs Between Fair Portfolio Sizes and Achievable Guarantees

   https://doi.org/10.1137/1.9781611978322.33  

   Gupta, Swati; Moondra, Jai; Singh, Mohit (January 2025, Society for Industrial and Applied Mathematics)

   Motivated by fairness concerns, we study existence and computation of portfolios, defined as: given anoptimization problem with feasible solutions D, a class C of fairness objective functions, a set X of feasible solutions is an α-approximate portfolio if for each objective f in C, there is an α-approximation for f in X. We study the trade-off between the size |X| of the portfolio and its approximation factor αfor various combinatorial problems, such as scheduling, covering, and facility location, and choices of C as top-k, ordered and symmetric monotonic norms. Our results include: (i) an α-approximate portfolio of size O(log d*log(α/4))for ordered norms and lower bounds of size \Omega( log dlog α+log log d)for the problem of scheduling identical jobs on d unidentical machines, (ii) O(log n)-approximate O(log n)-sized portfolios for facility location on n points for symmetric monotonic norms, and (iii) logO(d)^(r^2)-size O(1)-approximate portfolios for ordered norms and O(log d)-approximate for symmetric monotonic norms for covering polyhedra with a constant rnumber of constraints. The latter result uses our novel OrderAndCount framework that obtains an exponentialimprovement in portfolio sizes compared to current state-of-the-art, which may be of independent interest. 

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