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We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time $\widetilde{O}(n + t\sqrt{p_{\max}})$, where $n$ is the number of items, $t$ is the knapsack capacity, and $p_{\max}$ is the maximum item profit. This improves over the $\widetilde{O}(n + t \, p_{\max})$-time algorithm based on the convolution and prediction technique by Bateni et al.~(STOC 2018). Moreover, we give some evidence, based on a strengthening of the Min-Plus Convolution Hypothesis, that our running time might be optimal. Our algorithm uses two new technical tools, which might be of independent interest. First, we generalize the $\widetilde{O}(n^{1.5})$-time algorithm for bounded monotone min-plus convolution by Chi et al.~(STOC 2022) to the \emph{rectangular} case where the range of entries can be different from the sequence length. Second, we give a reduction from general knapsack instances to \emph{balanced} instances, where all items have nearly the same profit-to-weight ratio, up to a constant factor. Using these techniques, we can also obtain algorithms that run in time $\widetilde{O}(n + OPT\sqrt{w_{\max}})$, $\widetilde{O}(n + (nw_{\max}p_{\max})^{1/3}t^{2/3})$, and $\widetilde{O}(n + (nw_{\max}p_{\max})^{1/3} OPT^{2/3})$, where $OPT$ is the optimal total profit and $w_{\max}$ is the maximum item weight.

Given a graph $G=(V,E)$ and an integer $k\in \mathbb{N}$, we investigate the 2-Eigenvalue Vertex Deletion (2-EVD) problem. The objective is to remove at most $k$ vertices such that the adjacency matrix of the resulting graph has at most two eigenvalues. It is established that the adjacency matrix of a graph has at most two eigenvalues if and only if the graph is a collection of equal-sized cliques. Thus, the 2-Eigenvalue Vertex Deletion amounts to removing a set of at most $k$ vertices to transform the graph into a collection of equal-sized cliques. The 2-Eigenvalue Edge Editing (2-EEE), 2-Eigenvalue Edge Deletion (2-EED) and 2-Eigenvalue Edge Addition (2-EEA) problems are defined analogously. We present a kernel of size $\mathcal{O}(k^{3})$ for $2$-EVD, along with an FPT algorithm with a running time of $\mathcal{O}^{*}(2^{k})$. For the problem $2$-EEE, we provide a kernel of size $\mathcal{O}(k^{2})$. Additionally, we present linear kernels of size $5k$ and $6k$ for $2$-EEA and $2$-EED respectively. For the $2$-EED, we also construct an algorithm with running time $\mathcal{O}^{*}(1.47^{k})$ . These results address open questions posed by Misra et al. (ISAAC 2023) regarding the complexity of these problems when parameterized by the solution size.

We present an algorithm for min-cost flow in graphs with $n$ vertices and $m$ edges, given a tree decomposition of width $\tau$ and size $S$, and polynomially bounded, integral edge capacities and costs, running in $\widetilde{O}(m\sqrt{\tau} + S)$ time. This improves upon the previous fastest algorithm in this setting achieved by the bounded-treewidth linear program solver by [Dong-Lee-Ye,21] and [Gu-Song,22], which runs in $\widetilde{O}(m \tau^{(\omega+1)/2})$ time, where $\omega \approx 2.37$ is the matrix multiplication exponent. Our approach leverages recent advances in structured linear program solvers and robust interior point methods (IPM). For general graphs where treewidth is trivially bounded by $n$, the algorithm runs in $\widetilde{O}(m \sqrt n)$ time, which is the best-known result without using the Lee-Sidford barrier or $\ell_1$ IPM, demonstrating the surprising power of robust interior point methods. As a corollary, we obtain a $\widetilde{O}(\operatorname{tw}^3 \cdot m)$ time algorithm to compute a tree decomposition of width $O(\operatorname{tw}\cdot \log(n))$, given a graph with $m$ edges.

Given a (multi)graph $G$ which contains a bipartite subgraph with $\rho$ edges, what is the largest triangle-free subgraph of $G$ that can be found efficiently? We present an SDP-based algorithm that finds one with at least $0.8823 \rho$ edges, thus improving on the subgraph with $0.878 \rho$ edges obtained by the classic Max-Cut algorithm of Goemans and Williamson. On the other hand, by a reduction from Hastad's 3-bit PCP we show that it is NP-hard to find a triangle-free subgraph with $(25 / 26 + \epsilon) \rho \approx (0.961 + \epsilon) \rho$ edges. As an application, we classify the Maximum Promise Constraint Satisfaction Problem MaxPCSP($G$,$H$) for all bipartite $G$: Given an input (multi)graph $X$ which admits a $G$-colouring satisfying $\rho$ edges, find an $H$-colouring of $X$ that satisfies $\rho$ edges. This problem is solvable in polynomial time, apart from trivial cases, if $H$ contains a triangle, and is NP-hard otherwise.

Let $R \cup B$ be a set of $n$ points in $\mathbb{R}^2$, and let $k \in 1..n$. Our goal is to compute a line that "best" separates the "red" points $R$ from the "blue" points $B$ with at most $k$ outliers. We present an efficient semi-online dynamic data structure that can maintain whether such a separator exists. Furthermore, we present efficient exact and approximation algorithms that compute a linear separator that is guaranteed to misclassify at most $k$, points and minimizes the distance to the farthest outlier. Our exact algorithm runs in $O(nk + n \log n)$ time, and our $(1+\varepsilon)$-approximation algorithm runs in $O(\varepsilon^{-1/2}((n + k^2) \log n))$ time. Based on our $(1+\varepsilon)$-approximation algorithm we then also obtain a semi-online data structure to maintain such a separator efficiently.

Sorting has a natural generalization where the input consists of: (1) a ground set $X$ of size $n$, (2) a partial oracle $O_P$ specifying some fixed partial order $P$ on $X$ and (3) a linear oracle $O_L$ specifying a linear order $L$ that extends $P$. The goal is to recover the linear order $L$ on $X$ using the fewest number of linear oracle queries. In this problem, we measure algorithmic complexity through three metrics: oracle queries to $O_L$, oracle queries to $O_P$, and the time spent. Any algorithm requires worst-case $\log_2 e(P)$ linear oracle queries to recover the linear order on $X$. Kahn and Saks presented the first algorithm that uses $\Theta(\log e(P))$ linear oracle queries (using $O(n^2)$ partial oracle queries and exponential time). The state-of-the-art for the general problem is by Cardinal, Fiorini, Joret, Jungers and Munro who at STOC'10 manage to separate the linear and partial oracle queries into a preprocessing and query phase. They can preprocess $P$ using $O(n^2)$ partial oracle queries and $O(n^{2.5})$ time. Then, given $O_L$, they uncover the linear order on $X$ in $\Theta(\log e(P))$ linear oracle queries and $O(n + \log e(P))$ time -- which is worst-case optimal in the number of linear oracle queries but not in the time spent. For $c \geq 1$, our algorithm can preprocess $O_P$ using $O(n^{1 + \frac{1}{c}})$ queries and time. Given $O_L$, we uncover $L$ using $\Theta(c \log e(P))$ queries and time. We show a matching lower bound, as there exist positive constants $(\alpha, \beta)$ where for any constant $c \geq 1$, any algorithm that uses at most $\alpha \cdot n^{1 + \frac{1}{c}}$ preprocessing must use worst-case at least $\beta \cdot c \log e(P)$ linear oracle queries. Thus, we solve the problem of sorting under partial information through an algorithm that is asymptotically tight across all three metrics.

We consider the problem of sampling from the posterior distribution of a $d$-dimensional coefficient vector $\boldsymbol{\theta}$, given linear observations $\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\theta}+\boldsymbol{\varepsilon}$. In general, such posteriors are multimodal, and therefore challenging to sample from. This observation has prompted the exploration of various heuristics that aim at approximating the posterior distribution. In this paper, we study a different approach based on decomposing the posterior distribution into a log-concave mixture of simple product measures. This decomposition allows us to reduce sampling from a multimodal distribution of interest to sampling from a log-concave one, which is tractable and has been investigated in detail. We prove that, under mild conditions on the prior, for random designs, such measure decomposition is generally feasible when the number of samples per parameter $n/d$ exceeds a constant threshold. We thus obtain a provably efficient (polynomial time) sampling algorithm in a regime where this was previously not known. Numerical simulations confirm that the algorithm is practical, and reveal that it has attractive statistical properties compared to state-of-the-art methods.

Given a weighted graph $G$, a minimum weight $\alpha$-spanner is a least-weight subgraph $H\subseteq G$ that preserves minimum distances between all node pairs up to a factor of $\alpha$. There are many results on heuristics and approximation algorithms, including a recent investigation of their practical performance [20]. Exact approaches, in contrast, have long been denounced as impractical: The first exact ILP (integer linear program) method [48] from 2004 is based on a model with exponentially many path variables, solved via column generation. A second approach [2], modeling via arc-based multicommodity flow, was presented in 2019. In both cases, only graphs with 40-100 nodes were reported to be solvable. In this paper, we briefly report on a theoretical comparison between these two models from a polyhedral point of view, and then concentrate on improvements and engineering aspects. We evaluate their performance in a large-scale empirical study. We report that our tuned column generation approach, based on multicriteria shortest path computations, is able to solve instances with over 16000 nodes within 13 minutes. Furthermore, now knowing optimal solutions for larger graphs, we are able to investigate the quality of the strongest known heuristic on reasonably sized instances for the first time.

We introduce an Outlier-Efficient Modern Hopfield Model (termed $\mathrm{OutEffHop}$) and use it to address the outlier inefficiency problem of {training} gigantic transformer-based models. Our main contribution is a novel associative memory model facilitating \textit{outlier-efficient} associative memory retrievals. Interestingly, this memory model manifests a model-based interpretation of an outlier-efficient attention mechanism (${\rm Softmax}_1$): it is an approximation of the memory retrieval process of $\mathrm{OutEffHop}$. Methodologically, this allows us to introduce novel outlier-efficient Hopfield layers as powerful alternatives to traditional attention mechanisms, with superior post-quantization performance. Theoretically, the Outlier-Efficient Modern Hopfield Model retains and improves the desirable properties of standard modern Hopfield models, including fixed point convergence and exponential storage capacity. Empirically, we demonstrate the efficacy of the proposed model across large-scale transformer-based and Hopfield-based models (including BERT, OPT, ViT, and STanHop-Net), benchmarking against state-of-the-art methods like $\mathtt{Clipped\_Softmax}$ and $\mathtt{Gated\_Attention}$. Notably, $\mathrm{OutEffHop}$ achieves an average reduction of 22+\% in average kurtosis and 26+\% in the maximum infinity norm of model outputs across four models. Code is available at \href{//github.com/MAGICS-LAB/OutEffHop}{GitHub}; models are on \href{//huggingface.co/collections/magicslabnu/outeffhop-6610fcede8d2cda23009a98f}{Hugging Face Hub}; future updates are on \href{//arxiv.org/abs/2404.03828}{arXiv}.

In the Maximum Independent Set of Hyperrectangles problem, we are given a set of $n$ (possibly overlapping) $d$-dimensional axis-aligned hyperrectangles, and the goal is to find a subset of non-overlapping hyperrectangles of maximum cardinality. For $d=1$, this corresponds to the classical Interval Scheduling problem, where a simple greedy algorithm returns an optimal solution. In the offline setting, for $d$-dimensional hyperrectangles, polynomial time $(\log n)^{O(d)}$-approximation algorithms are known. However, the problem becomes notably challenging in the online setting, where the input objects (hyperrectangles) appear one by one in an adversarial order, and on the arrival of an object, the algorithm needs to make an immediate and irrevocable decision whether or not to select the object while maintaining the feasibility. Even for interval scheduling, an $\Omega(n)$ lower bound is known on the competitive ratio. To circumvent these negative results, in this work, we study the online maximum independent set of axis-aligned hyperrectangles in the random-order arrival model, where the adversary specifies the set of input objects which then arrive in a uniformly random order. Starting from the prototypical secretary problem, the random-order model has received significant attention to study algorithms beyond the worst-case competitive analysis. Surprisingly, we show that the problem in the random-order model almost matches the best-known offline approximation guarantees, up to polylogarithmic factors. In particular, we give a simple $(\log n)^{O(d)}$-competitive algorithm for $d$-dimensional hyperrectangles in this model, which runs in $\tilde{O_d}(n)$ time. Our approach also yields $(\log n)^{O(d)}$-competitive algorithms in the random-order model for more general objects such as $d$-dimensional fat objects and ellipsoids. Furthermore, our guarantees hold with high probability.