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The classical work of (Arora et al., 1999) provides a scheme that gives, for any $\epsilon>0$, a polynomial time $1-\epsilon$ approximation algorithm for dense instances of a family of $\mathcal{NP}$-hard problems, such as Max-CUT and Max-$k$-SAT. In this paper we extend and speed up this scheme using a logarithmic number of one-bit predictions. We propose a learning augmented framework which aims at finding fast algorithms which guarantees approximation consistency, smoothness and robustness with respect to the prediction error. We provide such algorithms, which moreover use predictions parsimoniously, for dense instances of various optimization problems.

$D^2$-sampling is a fundamental component of sampling-based clustering algorithms such as $k$-means++. Given a dataset $V \subset \mathbb{R}^d$ with $N$ points and a center set $C \subset \mathbb{R}^d$, $D^2$-sampling refers to picking a point from $V$ where the sampling probability of a point is proportional to its squared distance from the nearest center in $C$. Starting with empty $C$ and iteratively $D^2$-sampling and updating $C$ in $k$ rounds is precisely $k$-means++ seeding that runs in $O(Nkd)$ time and gives $O(\log{k})$-approximation in expectation for the $k$-means problem. We give a quantum algorithm for (approximate) $D^2$-sampling in the QRAM model that results in a quantum implementation of $k$-means++ that runs in time $\tilde{O}(\zeta^2 k^2)$. Here $\zeta$ is the aspect ratio (i.e., largest to smallest interpoint distance), and $\tilde{O}$ hides polylogarithmic factors in $N, d, k$. It can be shown through a robust approximation analysis of $k$-means++ that the quantum version preserves its $O(\log{k})$ approximation guarantee. Further, we show that our quantum algorithm for $D^2$-sampling can be 'dequantized' using the sample-query access model of Tang (PhD Thesis, Ewin Tang, University of Washington, 2023). This results in a fast quantum-inspired classical implementation of $k$-means++, which we call QI-$k$-means++, with a running time $O(Nd) + \tilde{O}(\zeta^2k^2d)$, where the $O(Nd)$ term is for setting up the sample-query access data structure. Experimental investigations show promising results for QI-$k$-means++ on large datasets with bounded aspect ratio. Finally, we use our quantum $D^2$-sampling with the known $ D^2$-sampling-based classical approximation scheme (i.e., $(1+\varepsilon)$-approximation for any given $\varepsilon>0$) to obtain the first quantum approximation scheme for the $k$-means problem with polylogarithmic running time dependence on $N$.

The Heilbronn triangle problem asks for the placement of $n$ points in a unit square that maximizes the smallest area of a triangle formed by any three of those points. In $1972$, Schmidt considered a natural generalization of this problem. He asked for the placement of $n$ points in a unit square that maximizes the smallest area of the convex hull formed by any four of those points. He showed a lower bound of $\Omega(n^{-3/2})$, which was improved to $\Omega(n^{-3/2}\log{n})$ by Leffman. A trivial upper bound of $3/n$ could be obtained, and Schmidt asked if this could be improved asymptotically. However, despite several efforts, no asymptotic improvement over the trivial upper bound was known for the last $50$ years, and the problem started to get the tag of being notoriously hard. Szemer{\'e}di posed the question of whether one can, at least, improve the constant in this trivial upper bound. In this work, we answer this question by proving an upper bound of $2/n+o(1/n)$. We also extend our results to any convex hulls formed by $k\geq 4$ points.

Differentially private computation often begins with a bound on some $d$-dimensional statistic's $\ell_p$ sensitivity. For pure differential privacy, the $K$-norm mechanism can improve on this approach using a norm tailored to the statistic's sensitivity space. Writing down a closed-form description of this optimal norm is often straightforward. However, running the $K$-norm mechanism reduces to uniformly sampling the norm's unit ball; this ball is a $d$-dimensional convex body, so general sampling algorithms can be slow. Turning to concentrated differential privacy, elliptic Gaussian noise offers similar improvement over spherical Gaussian noise. Once the shape of this ellipse is determined, sampling is easy; however, identifying the best such shape may be hard. This paper solves both problems for the simple statistics of sum, count, and vote. For each statistic, we provide a sampler for the optimal $K$-norm mechanism that runs in time $\tilde O(d^2)$ and derive a closed-form expression for the optimal shape of elliptic Gaussian noise. The resulting algorithms all yield meaningful accuracy improvements while remaining fast and simple enough to be practical. More broadly, we suggest that problem-specific sensitivity space analysis may be an overlooked tool for private additive noise.

In the study of Hilbert schemes, the integer partition $\lambda$ helps researchers identify some geometric and combinatorial properties of the scheme in question. To aid researchers in extracting such information from a Hilbert polynomial, we describe an efficient algorithm which can identify if $p(x)\in\mathbb{Q}[x]$ is a Hilbert polynomial and if so, recover the integer partition $\lambda$ associated with it.

We study the categorical structure of the Grothendieck construction of an indexed category $\mathcal{L}:\mathcal{C}^{op}\to\mathbf{CAT}$ and characterise fibred limits, colimits, and monoidal structures. Next, we give sufficient conditions for the monoidal closure of the total category $\Sigma_\mathcal{C} \mathcal{L}$ of a Grothendieck construction of an indexed category $\mathcal{L}:\mathcal{C}^{op}\to\mathbf{CAT}$. Our analysis is a generalization of G\"odel's Dialectica interpretation, and it relies on a novel notion of $\Sigma$-tractable monoidal structure. As we will see, $\Sigma$-tractable coproducts simultaneously generalize cocartesian coclosed structures, biproducts and extensive coproducts. We analyse when the closed structure is fibred -- usually it is not.

We examine sorting algorithms for $n$ elements whose basic operation is comparing $t$ elements simultaneously (a $t$-comparator). We focus on algorithms that use only a single round or two rounds -- comparisons performed in the second round depend on the outcomes of the first round comparators. We design deterministic and randomized algorithms. In the deterministic case, we show an interesting relation to design theory (namely, to 2-Steiner systems), which yields a single-round optimal algorithm for $n=t^{2^k}$ with any $k\ge 1$ and a variety of possible values of $t$. For some values of $t$, however, no algorithm can reach the optimal (information-theoretic) bound on the number of comparators. For this case (and any other $n$ and $t$), we show an algorithm that uses at most three times as many comparators as the theoretical bound. We also design a randomized Las-Vegas two-rounds sorting algorithm for any $n$ and $t$. Our algorithm uses an asymptotically optimal number of $O(\max(\frac{n^{3/2}}{t^2},\frac{n}{t}))$ comparators, with high probability, i.e., with probability at least $1-1/n$. The analysis of this algorithm involves the gradual unveiling of randomness, using a novel technique which we coin the binary tree of deferred randomness.

M${}^{\natural}$-concave functions, a.k.a. gross substitute valuation functions, play a fundamental role in many fields, including discrete mathematics and economics. In practice, perfect knowledge of M${}^{\natural}$-concave functions is often unavailable a priori, and we can optimize them only interactively based on some feedback. Motivated by such situations, we study online M${}^{\natural}$-concave function maximization problems, which are interactive versions of the problem studied by Murota and Shioura (1999). For the stochastic bandit setting, we present $O(T^{-1/2})$-simple regret and $O(T^{2/3})$-regret algorithms under $T$ times access to unbiased noisy value oracles of M${}^{\natural}$-concave functions. A key to proving these results is the robustness of the greedy algorithm to local errors in M${}^{\natural}$-concave function maximization, which is one of our main technical results. While we obtain those positive results for the stochastic setting, another main result of our work is an impossibility in the adversarial setting. We prove that, even with full-information feedback, no algorithms that run in polynomial time per round can achieve $O(T^{1-c})$ regret for any constant $c > 0$ unless $\mathsf{P} = \mathsf{NP}$. Our proof is based on a reduction from the matroid intersection problem for three matroids, which would be a novel idea in the context of online learning.

The $K$-medoids problem is a challenging combinatorial clustering task, widely used in data analysis applications. While numerous algorithms have been proposed to solve this problem, none of these are able to obtain an exact (globally optimal) solution for the problem in polynomial time. In this paper, we present EKM: a novel algorithm for solving this problem exactly with worst-case $O\left(N^{K+1}\right)$ time complexity. EKM is developed according to recent advances in transformational programming and combinatorial generation, using formal program derivation steps. The derived algorithm is provably correct by construction. We demonstrate the effectiveness of our algorithm by comparing it against various approximate methods on numerous real-world datasets. We show that the wall-clock run time of our algorithm matches the worst-case time complexity analysis on synthetic datasets, clearly outperforming the exponential time complexity of benchmark branch-and-bound based MIP solvers. To our knowledge, this is the first, rigorously-proven polynomial time, practical algorithm for this ubiquitous problem.