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We simulate active Brownian particles (ABPs) with soft-repulsive interactions subjected to a four-roll-mill flow. In the absence of flow, this system exhibits motility-induced phase separation (MIPS). To investigate the interplay between MIPS and flow-induced mixing, we introduce dimensionless parameters: a scaled time, $\tau$, and a scaled velocity, ${\rm v}$, characterizing the ratio of ABP to fluid time and velocity scales, respectively. The parameter space defined by $\tau$ and ${\rm v}$ reveals three distinct ABP distribution regimes. At low velocities ${\rm v} \ll 1$, flow dominates, leading to a homogeneous mixture. Conversely, at high velocities ${\rm v} \gg 1$, motility prevails, resulting in MIPS. In the intermediate regime (${\rm v} \sim 1$), the system's behavior depends on $\tau$. For $\tau <1$, a moderately mixed homogeneous phase emerges, while for $\tau >1$, a novel phase, termed flow-induced phase separation (FIPS), arises due to the combined effects of flow topology and ABP motility and size. To characterize these phases, we analyze drift velocity, diffusivity, mean-squared displacement, giant number fluctuations, radial distribution function, and cluster-size distribution., Comment: 10 pages, 9 figures

Recent results suggest that highly active, chaotic, non-equilibrium states of living fluids might share much in common with high Reynolds number, inertial turbulence. We now show, by using a hydrodynamical model, the onset of intermittency and the consequent multiscaling of Eulerian and Lagrangian structure functions as a function of the bacterial activity. Our results bridge the worlds of low and high Reynolds number flows as well as open up intriguing possibilities of what makes flows intermittent.

We initiate the study of centralized algorithms for welfare-maximizing allocation of goods to buyers subject to average-value constraints. We show that this problem is NP-hard to approximate beyond a factor of $\frac{e}{e-1}$, and provide a $\frac{4e}{e-1}$-approximate offline algorithm. For the online setting, we show that no non-trivial approximations are achievable under adversarial arrivals. Under i.i.d. arrivals, we present a polytime online algorithm that provides a constant approximation of the optimal (computationally-unbounded) online algorithm. In contrast, we show that no constant approximation of the ex-post optimum is achievable by an online algorithm.

We study online contention resolution schemes (OCRSs) and prophet inequalities for non-product distributions. Specifically, when the active set is sampled according to a pairwise-independent (PI) distribution, we show a $(1-o_k(1))$-selectable OCRS for uniform matroids of rank $k$, and $\Omega(1)$-selectable OCRSs for laminar, graphic, cographic, transversal, and regular matroids. These imply prophet inequalities with the same ratios when the set of values is drawn according to a PI distribution. Our results complement recent work of Dughmi, Kalayci, and Patel (STOC '24) showing that no $\omega(1/k)$-selectable OCRS exists in the PI setting for general matroids of rank $k$., Comment: Contains new results on t-wise independent CRSs

Even for simple arithmetic tasks like integer addition, it is challenging for Transformers to generalize to longer sequences than those encountered during training. To tackle this problem, we propose position coupling, a simple yet effective method that directly embeds the structure of the tasks into the positional encoding of a (decoder-only) Transformer. Taking a departure from the vanilla absolute position mechanism assigning unique position IDs to each of the tokens, we assign the same position IDs to two or more "relevant" tokens; for integer addition tasks, we regard digits of the same significance as in the same position. On the empirical side, we show that with the proposed position coupling, a small (1-layer) Transformer trained on 1 to 30-digit additions can generalize up to 200-digit additions (6.67x of the trained length). On the theoretical side, we prove that a 1-layer Transformer with coupled positions can solve the addition task involving exponentially many digits, whereas any 1-layer Transformer without positional information cannot entirely solve it. We also demonstrate that position coupling can be applied to other algorithmic tasks such as addition with multiple summands, Nx2 multiplication, copy/reverse, and a two-dimensional task., Comment: 73 pages, 20 figures, 90 tables

Algorithms with predictions have attracted much attention in the last years across various domains, including variants of facility location, as a way to surpass traditional worst-case analyses. We study the $k$-facility location mechanism design problem, where the $n$ agents are strategic and might misreport their location. Unlike previous models, where predictions are for the $k$ optimal facility locations, we receive $n$ predictions for the locations of each of the agents. However, these predictions are only "mostly" and "approximately" correct (or MAC for short) -- i.e., some $\delta$-fraction of the predicted locations are allowed to be arbitrarily incorrect, and the remainder of the predictions are allowed to be correct up to an $\varepsilon$-error. We make no assumption on the independence of the errors. Can such predictions allow us to beat the current best bounds for strategyproof facility location? We show that the $1$-median (geometric median) of a set of points is naturally robust under corruptions, which leads to an algorithm for single-facility location with MAC predictions. We extend the robustness result to a "balanced" variant of the $k$ facilities case. Without balancedness, we show that robustness completely breaks down, even for the setting of $k=2$ facilities on a line. For this "unbalanced" setting, we devise a truthful random mechanism that outperforms the best known result of Lu et al. [2010], which does not use predictions. En route, we introduce the problem of "second" facility location (when the first facility's location is already fixed). Our findings on the robustness of the $1$-median and more generally $k$-medians may be of independent interest, as quantitative versions of classic breakdown-point results in robust statistics.

We study the approximability of the MaxCut problem in the presence of predictions. Specifically, we consider two models: in the noisy predictions model, for each vertex we are given its correct label in $\{-1,+1\}$ with some unknown probability $1/2 + \epsilon$, and the other (incorrect) label otherwise. In the more-informative partial predictions model, for each vertex we are given its correct label with probability $\epsilon$ and no label otherwise. We assume only pairwise independence between vertices in both models. We show how these predictions can be used to improve on the worst-case approximation ratios for this problem. Specifically, we give an algorithm that achieves an $\alpha + \widetilde{\Omega}(\epsilon^4)$-approximation for the noisy predictions model, where $\alpha \approx 0.878$ is the MaxCut threshold. While this result also holds for the partial predictions model, we can also give a $\beta + \Omega(\epsilon)$-approximation, where $\beta \approx 0.858$ is the approximation ratio for MaxBisection given by Raghavendra and Tan. This answers a question posed by Ola Svensson in his plenary session talk at SODA'23., Comment: 18 pages

This paper introduces a novel segmentation framework that integrates a classifier network with a reverse HRNet architecture for efficient image segmentation. Our approach utilizes a ResNet-50 backbone, pretrained in a semi-supervised manner, to generate feature maps at various scales. These maps are then processed by a reverse HRNet, which is adapted to handle varying channel dimensions through 1x1 convolutions, to produce the final segmentation output. We strategically avoid fine-tuning the backbone network to minimize memory consumption during training. Our methodology is rigorously tested across several benchmark datasets including Mapillary Vistas, Cityscapes, CamVid, COCO, and PASCAL-VOC2012, employing metrics such as pixel accuracy and mean Intersection over Union (mIoU) to evaluate segmentation performance. The results demonstrate the effectiveness of our proposed model in achieving high segmentation accuracy, indicating its potential for various applications in image analysis. By leveraging the strengths of both the ResNet-50 and reverse HRNet within a unified framework, we present a robust solution to the challenges of image segmentation.

Simulations of elastic turbulence, the chaotic flow of highly elastic and inertialess polymer solutions, are plagued by numerical difficulties: The chaotically advected polymer conformation tensor develops extremely large gradients and can loose its positive definiteness, which triggers numerical instabilities. While efforts to tackle these issues have produced a plethora of specialized techniques -- tensor decompositions, artificial diffusion, and shock-capturing advection schemes -- we still lack an unambiguous route to accurate and efficient simulations. In this work, we show that even when a simulation is numerically stable, maintaining positive-definiteness and displaying the expected chaotic fluctuations, it can still suffer from errors significant enough to distort the large-scale dynamics and flow-structures. Focusing on two-dimensional simulations of the Oldroyd-B and FENE-P equations, we first compare two decompositions of the conformation tensor: symmetric square root (SSR) and Cholesky with a logarithmic transformation (Cholesky-log). While both simulations yield chaotic flows, only the Cholesky-log preserves the pattern of the forcing, i.e., its vortical cells remain ordered in a lattice as opposed to the vortices of the SSR simulations which shrink, expand and reorient constantly. To identify the accurate simulation, we appeal to a hitherto overlooked mathematical bound on the determinant of the conformation tensor, which unequivocally rejects the SSR simulation. Importantly, the accuracy of the Cholesky-log simulation is shown to arise from the logarithmic transformation. We then consider local artificial diffusion, a potential low-cost alternative to high-order advection schemes, and find unfortunately that it significantly modifies the dynamics. We end with an example, showing how the spurious large-scale motions identified here contaminate predictions of scalar mixing., Comment: 24 pages, 11 figures

It has been observed in recent years that transformers have problems with length generalization for certain types of reasoning and arithmetic tasks. In particular, the performance of a transformer model trained on tasks (say addition) up to a certain length (e.g., 5 digit numbers) drops sharply when applied to longer instances of the same problem. This work proposes an approach based on task hinting towards addressing length generalization. Our key idea is that while training the model on task-specific data, it is helpful to simultaneously train the model to solve a simpler but related auxiliary task as well. We study the classical sorting problem as a canonical example to evaluate our approach. We design a multitask training framework and show that task hinting significantly improve length generalization. For sorting we show that it is possible to train models on data consisting of sequences having length at most $20$, and improve the test accuracy on sequences of length $100$ from less than 1% (for standard training) to more than 92% (via task hinting). Our study uncovers several interesting aspects of length generalization. We observe that while several auxiliary tasks may seem natural a priori, their effectiveness in improving length generalization differs dramatically. We further use probing and visualization-based techniques to understand the internal mechanisms via which the model performs the task, and propose a theoretical construction consistent with the observed learning behaviors of the model. Based on our construction, we show that introducing a small number of length dependent parameters into the training procedure can further boost the performance on unseen lengths. Finally, we also show the efficacy of our task hinting based approach beyond sorting, giving hope that these techniques will be applicable in broader contexts.

Maintaining a maximum bipartite matching online while minimizing recourse/augmentations is a well studied problem, motivated by content delivery, job scheduling, and hashing. A breakthrough result of Bernstein, Holm, and Rotenberg (\emph{SODA 2018}) resolved this problem up to a logarithmic factors. However, we may need a richer class of combinatorial constraints (e.g., matroid constraints) to model other problems in scheduling and resource allocation. We consider the problem of maintaining a maximum independent set of an arbitrary matroid $\mathcal{M}$ and a partition matroid $\mathcal{P}$ in the online setting. Specifically, at each timestep $t$ one part $P_t$ of the partition matroid (i.e., a subset of elements) is revealed: we must now select at most one of these newly-revealed elements, but can exchange some of the previously selected elements for new ones from previous parts, to maintain a maximum independent set on the elements seen thus far. The goal is to minimize the number of augmentations/changes done by our algorithm. If $\mathcal{M}$ is also a partition matroid, we recover the problem of maintaining a maximum bipartite matching online with recourse as a special case. In our work, we allow arbitrary matroids $\mathcal{M}$, and so we can model broader classes of problems. Our main result is an $O(n \log^2 n)$-competitive algorithm, where $n$ is the rank of the largest common base; this matches the current best quantitative bound for the bipartite matching special case. Our result builds substantively on the breakthrough result of Bernstein, Holm, and Rotenberg for maintaining bipartite matchings: a key contribution of our work is to make connections to market equilibria and prices, and our use of properties of these equilibria in submodular utility allocation markets to prove our bound on the number of augmentations.

In this paper, we study the weighted $k$-server problem on the uniform metric in both the offline and online settings. We start with the offline setting. In contrast to the (unweighted) $k$-server problem which has a polynomial-time solution using min-cost flows, there are strong computational lower bounds for the weighted $k$-server problem, even on the uniform metric. Specifically, we show that assuming the unique games conjecture, there are no polynomial-time algorithms with a sub-polynomial approximation factor, even if we use $c$-resource augmentation for $c < 2$. Furthermore, if we consider the natural LP relaxation of the problem, then obtaining a bounded integrality gap requires us to use at least $\ell$ resource augmentation, where $\ell$ is the number of distinct server weights. We complement these results by obtaining a constant-approximation algorithm via LP rounding, with a resource augmentation of $(2+\epsilon)\ell$ for any constant $\epsilon > 0$. In the online setting, an $\exp(k)$ lower bound is known for the competitive ratio of any randomized algorithm for the weighted $k$-server problem on the uniform metric. In contrast, we show that $2\ell$-resource augmentation can bring the competitive ratio down by an exponential factor to only $O(\ell^2 \log \ell)$. Our online algorithm uses the two-stage approach of first obtaining a fractional solution using the online primal-dual framework, and then rounding it online., Comment: This paper will appear in the proceedings of APPROX 2023

Given a set of points in $d$-dimensional space, an explainable clustering is one where the clusters are specified by a tree of axis-aligned threshold cuts. Dasgupta et al. (ICML 2020) posed the question of the price of explainability: the worst-case ratio between the cost of the best explainable clusterings to that of the best clusterings. We show that the price of explainability for $k$-medians is at most $1+H_{k-1}$; in fact, we show that the popular Random Thresholds algorithm has exactly this price of explanability, matching the known lower bound constructions. We complement our tight analysis of this particular algorithm by constructing instances where the price of explanability (using any algorithm) is at least $(1-o(1)) \ln k$, showing that our result is best possible, up to lower-order terms. We also improve the price of explanability for the $k$-means problem to $O(k \ln \ln k)$ from the previous $O(k \ln k)$, considerably closing the gap to the lower bounds of $\Omega(k)$. Finally, we study the algorithmic question of finding the best explainable clustering: We show that explainable $k$-medians and $k$-means cannot be approximated better than $O(\ln k)$, under standard complexity-theoretic conjectures. This essentially settles the approximability of explainable $k$-medians and leaves open the intriguing possibility to get significantly better approximation algorithms for $k$-means than its price of explainability.

In the stochastic set cover problem (Grandoni et al., FOCS '08), we are given a collection $\mathcal{S}$ of $m$ sets over a universe $\mathcal{U}$ of size $N$, and a distribution $D$ over elements of $\mathcal{U}$. The algorithm draws $n$ elements one-by-one from $D$ and must buy a set to cover each element on arrival; the goal is to minimize the total cost of sets bought during this process. A universal algorithm a priori maps each element $u \in \mathcal{U}$ to a set $S(u)$ such that if $U \subseteq \mathcal{U}$ is formed by drawing $n$ times from distribution $D$, then the algorithm commits to outputting $S(U)$. Grandoni et al. gave an $O(\log mN)$-competitive universal algorithm for this stochastic set cover problem. We improve unilaterally upon this result by giving a simple, polynomial time $O(\log mn)$-competitive universal algorithm for the more general prophet version, in which $U$ is formed by drawing from $n$ different distributions $D_1, \ldots, D_n$. Furthermore, we show that we do not need full foreknowledge of the distributions: in fact, a single sample from each distribution suffices. We show similar results for the 2-stage prophet setting and for the online-with-a-sample setting. We obtain our results via a generic reduction from the single-sample prophet setting to the random-order setting; this reduction holds for a broad class of minimization problems that includes all covering problems. We take advantage of this framework by giving random-order algorithms for non-metric facility location and set multicover; using our framework, these automatically translate to universal prophet algorithms.

Consider an agent exploring an unknown graph in search of some goal state. As it walks around the graph, it learns the nodes and their neighbors. The agent only knows where the goal state is when it reaches it. How do we reach this goal while moving only a small distance? This problem seems hopeless, even on trees of bounded degree, unless we give the agent some help. This setting with ''help'' often arises in exploring large search spaces (e.g., huge game trees) where we assume access to some score/quality function for each node, which we use to guide us towards the goal. In our case, we assume the help comes in the form of distance predictions: each node $v$ provides a prediction $f(v)$ of its distance to the goal vertex. Naturally if these predictions are correct, we can reach the goal along a shortest path. What if the predictions are unreliable and some of them are erroneous? Can we get an algorithm whose performance relates to the error of the predictions? In this work, we consider the problem on trees and give deterministic algorithms whose total movement cost is only $O(OPT + \Delta \cdot ERR)$, where $OPT$ is the distance from the start to the goal vertex, $\Delta$ the maximum degree, and the $ERR$ is the total number of vertices whose predictions are erroneous. We show this guarantee is optimal. We then consider a ''planning'' version of the problem where the graph and predictions are known at the beginning, so the agent can use this global information to devise a search strategy of low cost. For this planning version, we go beyond trees and give an algorithms which gets good performance on (weighted) graphs with bounded doubling dimension.

In the Set Cover problem, we are given a set system with each set having a weight, and we want to find a collection of sets that cover the universe, whilst having low total weight. There are several approaches known (based on greedy approaches, relax-and-round, and dual-fitting) that achieve a $H_k \approx \ln k + O(1)$ approximation for this problem, where the size of each set is bounded by $k$. Moreover, getting a $\ln k - O(\ln \ln k)$ approximation is hard. Where does the truth lie? Can we close the gap between the upper and lower bounds? An improvement would be particularly interesting for small values of $k$, which are often used in reductions between Set Cover and other combinatorial optimization problems. We consider a non-oblivious local-search approach: to the best of our knowledge this gives the first $H_k$-approximation for Set Cover using an approach based on local-search. Our proof fits in one page, and gives a integrality gap result as well. Refining our approach by considering larger moves and an optimized potential function gives an $(H_k - \Omega(\log^2 k)/k)$-approximation, improving on the previous bound of $(H_k - \Omega(1/k^8))$ (\emph{R.\ Hassin and A.\ Levin, SICOMP '05}) based on a modified greedy algorithm., Comment: To appear in SOSA '23

We investigate how a shepherd should move in order to effectively herd and guide a flock of agents towards a target. Using a detailed agent-based model (ABM) for the members of the flock, we pose and solve an optimization problem for the shepherd that has to simultaneously work to keep the flock cohesive while coercing it towards a prescribed project. We find that three distinct strategies emerge as potential solutions as a function of just two parameters: the ratio of herd size to shepherd repulsion length and the ratio of herd speed to shepherd speed. We term these as: (i) mustering, in which the shepherd circles the herd to ensure compactness, (ii) droving, in which the shepherd chases the herd in a desired direction, and (iii) driving, a hitherto unreported strategy where the flock surrounds a shepherd that drives it from within. A minimal dynamical model for the size, shape and position of the herd captures the effective behavior of the ABM, and further allows us to characterize the different herding strategies in terms of the behavior of the shepherd that librates (mustering), oscillates (droving) or moves steadily (driving). All together, our study yields a simple and intuitive classification of herding strategies that ought to be of general interest in the context of controlling the collective behavior of active matter., Comment: A couple paragraphs removed for brevity

The configuration balancing problem with stochastic requests generalizes many well-studied resource allocation problems such as load balancing and virtual circuit routing. In it, we have $m$ resources and $n$ requests. Each request has multiple possible configurations, each of which increases the load of each resource by some amount. The goal is to select one configuration for each request to minimize the makespan: the load of the most-loaded resource. In our work, we focus on a stochastic setting, where we only know the distribution for how each configuration increases the resource loads, learning the realized value only after a configuration is chosen. We develop both offline and online algorithms for configuration balancing with stochastic requests. When the requests are known offline, we give a non-adaptive policy for configuration balancing with stochastic requests that $O(\frac{\log m}{\log \log m})$-approximates the optimal adaptive policy. In particular, this closes the adaptivity gap for this problem as there is an asymptotically matching lower bound even for the very special case of load balancing on identical machines. When requests arrive online in a list, we give a non-adaptive policy that is $O(\log m)$ competitive. Again, this result is asymptotically tight due to information-theoretic lower bounds for very special cases (e.g., for load balancing on unrelated machines). Finally, we show how to leverage adaptivity in the special case of load balancing on related machines to obtain a constant-factor approximation offline and an $O(\log \log m)$-approximation online. A crucial technical ingredient in all of our results is a new structural characterization of the optimal adaptive policy that allows us to limit the correlations between its decisions.

We study the classic problem of minimizing the expected total completion time of jobs on $m$ identical machines in the setting where the sizes of the jobs are stochastic. Specifically, the size of each job is a random variable whose distribution is known to the algorithm, but whose realization is revealed only after the job is scheduled. While minimizing the total completion time is easy in the deterministic setting, the stochastic problem has long been notorious: all known algorithms have approximation ratios that either depend on the variances, or depend linearly on the number of machines. We give an $\widetilde{O}(\sqrt{m})$-approximation for stochastic jobs which have Bernoulli processing times. This is the first approximation for this problem that is both independent of the variance in the job sizes, and is sublinear in the number of machines $m$. Our algorithm is based on a novel reduction from minimizing the total completion time to a natural makespan-like objective, which we call the weighted free time. We hope this free time objective will be useful in further improvements to this problem, as well as other stochastic scheduling problems.

The sense of touch is fundamental to human dexterity. When mimicked in robotic touch, particularly by use of soft optical tactile sensors, it suffers from distortion due to motion-dependent shear. This complicates tactile tasks like shape reconstruction and exploration that require information about contact geometry. In this work, we pursue a semi-supervised approach to remove shear while preserving contact-only information. We validate our approach by showing a match between the model-generated unsheared images with their counterparts from vertically tapping onto the object. The model-generated unsheared images give faithful reconstruction of contact-geometry otherwise masked by shear, along with robust estimation of object pose then used for sliding exploration and full reconstruction of several planar shapes. We show that our semi-supervised approach achieves comparable performance to its fully supervised counterpart across all validation tasks with an order of magnitude less supervision. The semi-supervised method is thus more computational and labeled sample-efficient. We expect it will have broad applicability to wide range of complex tactile exploration and manipulation tasks performed via a shear-sensitive sense of touch., Comment: 7 pages, 3 figures, accepted for publication in IROS 2022

We use the mean-bacterial-velocity model to investigate the \textit{irreversibility} of two-dimensional (2D) \textit{bacterial turbulence} and to compare it with its 2D fluid-turbulence counterpart. We carry out extensive direct numerical simulations of Lagrangian tracer particles that are advected by the velocity field in this model. Our work uncovers an important, qualitative way in which irreversibility in bacterial turbulence is different from its fluid-turbulence counterpart: For large positive (or large but negative) values of the \textit{friction} (or \textit{activity}) parameter, the probability distribution functions of energy increments, along tracer trajectories, or the power are \textit{positively} skewed; so irreversibility in bacterial turbulence can lead, on average, to \textit{particles gaining energy faster than they lose it}, which is the exact opposite of what is observed for tracers in 2D fluid turbulence.

We study the problem of solving Packing Integer Programs (PIPs) in the online setting, where columns in $[0,1]^d$ of the constraint matrix are revealed sequentially, and the goal is to pick a subset of the columns that sum to at most $B$ in each coordinate while maximizing the objective. Excellent results are known in the secretary setting, where the columns are adversarially chosen, but presented in a uniformly random order. However, these existing algorithms are susceptible to adversarial attacks: they try to "learn" characteristics of a good solution, but tend to over-fit to the model, and hence a small number of adversarial corruptions can cause the algorithm to fail. In this paper, we give the first robust algorithms for Packing Integer Programs, specifically in the recently proposed Byzantine Secretary framework. Our techniques are based on a two-level use of online learning, to robustly learn an approximation to the optimal value, and then to use this robust estimate to pick a good solution. These techniques are general and we use them to design robust algorithms for PIPs in the prophet model as well, specifically in the Prophet-with-Augmentations framework. We also improve known results in the Byzantine Secretary framework: we make the non-constructive results algorithmic and improve the existing bounds for single-item and matroid constraints., Comment: Appears in SODA 2022

We show how to round any half-integral solution to the subtour-elimination relaxation for the TSP, while losing a less-than-1.5 factor. Such a rounding algorithm was recently given by Karlin, Klein, and Oveis Gharan based on sampling from max-entropy distributions. We build on an approach of Haddadan and Newman to show how sampling from the matroid intersection polytope, and a new use of max-entropy sampling, can give better guarantees.

We study the $k$-server problem with time-windows. In this problem, each request $i$ arrives at some point $v_i$ of an $n$-point metric space at time $b_i$ and comes with a deadline $e_i$. One of the $k$ servers must be moved to $v_i$ at some time in the interval $[b_i, e_i]$ to satisfy this request. We give an online algorithm for this problem with a competitive ratio of ${\rm polylog} (n,\Delta)$, where $\Delta$ is the aspect ratio of the metric space. Prior to our work, the best competitive ratio known for this problem was $O(k \cdot {\rm polylog}(n))$ given by Azar et al. (STOC 2017). Our algorithm is based on a new covering linear program relaxation for $k$-server on HSTs. This LP naturally corresponds to the min-cost flow formulation of $k$-server, and easily extends to the case of time-windows. We give an online algorithm for obtaining a feasible fractional solution for this LP, and a primal dual analysis framework for accounting the cost of the solution. Together, they yield a new $k$-server algorithm with poly-logarithmic competitive ratio, and extend to the time-windows case as well. Our principal technical contribution lies in thinking of the covering LP as yielding a {\em truncated} covering LP at each internal node of the tree, which allows us to keep account of server movements across subtrees. We hope that this LP relaxation and the algorithm/analysis will be a useful tool for addressing $k$-server and related problems., Comment: Full version of paper appearing in FOCS 2021

We give a polynomial-time algorithm for OnlineSetCover with a competitive ratio of $O(\log mn)$ when the elements are revealed in random order, essentially matching the best possible offline bound of $O(\log n)$ and circumventing the $\Omega(\log m \log n)$ lower bound known in adversarial order. We also extend the result to solving pure covering IPs when constraints arrive in random order. The algorithm is a multiplicative-weights-based round-and-solve approach we call LearnOrCover. We maintain a coarse fractional solution that is neither feasible nor monotone increasing, but can nevertheless be rounded online to achieve the claimed guarantee (in the random order model). This gives a new offline algorithm for SetCover that performs a single pass through the elements, which may be of independent interest., Comment: minor correction to potential argument

The vector-balancing problem is a fundamental problem in discrepancy theory: given T vectors in $[-1,1]^n$, find a signing $\sigma(a) \in \{\pm 1\}$ of each vector $a$ to minimize the discrepancy $\| \sum_{a} \sigma(a) \cdot a \|_{\infty}$. This problem has been extensively studied in the static/offline setting. In this paper we initiate its study in the fully-dynamic setting with recourse: the algorithm sees a stream of T insertions and deletions of vectors, and at each time must maintain a low-discrepancy signing, while also minimizing the amortized recourse (the number of times any vector changes its sign) per update. For general vectors, we show algorithms which almost match Spencer's $O(\sqrt{n})$ offline discrepancy bound, with ${O}(n\cdot poly\!\log T)$ amortized recourse per update. The crucial idea is to compute a basic feasible solution to the linear relaxation in a distributed and recursive manner, which helps find a low-discrepancy signing. To bound recourse we argue that only a small part of the instance needs to be re-computed at each update. Since vector balancing has also been greatly studied for sparse vectors, we then give algorithms for low-discrepancy edge orientation, where we dynamically maintain signings for 2-sparse vectors. Alternatively, this can be seen as orienting a dynamic set of edges of an n-vertex graph to minimize the absolute difference between in- and out-degrees at any vertex. We present a deterministic algorithm with $O(poly\!\log n)$ discrepancy and $O(poly\!\log n)$ amortized recourse. The core ideas are to dynamically maintain an expander-decomposition with low recourse and then to show that, as the expanders change over time, a natural local-search algorithm converges quickly (i.e., with low recourse) to a low-discrepancy solution. We also give strong lower bounds for local-search discrepancy minimization algorithms., Comment: 29 pages. Appears in SODA 2022

Sequential testing problems involve a complex system with several components, each of which is "working" with some independent probability. The outcome of each component can be determined by performing a test, which incurs some cost. The overall system status is given by a function $f$ of the outcomes of its components. The goal is to evaluate this function $f$ by performing tests at the minimum expected cost. While there has been extensive prior work on this topic, provable approximation bounds are mainly limited to simple functions like ``k-out-of-n'' and halfspaces. We consider significantly more general "score classification" functions, and provide the first constant factor approximation algorithm (improving over a previous logarithmic approximation ratio). Moreover, our policy is non adaptive: it just involves performing tests in an a priori fixed order. We also consider the related halfspace evaluation problem, where we want to evaluate some function on $d$ halfspaces (e.g., intersection of halfspaces). We show that our approach provides an $O(d^2\log d)$-approximation algorithm for this problem. Our algorithms also extend to the setting of "batched'' tests, where multiple tests can be performed simultaneously while incurring an extra setup cost. Finally, we perform computational experiments that demonstrate the practical performance of our algorithm for score classification. We observe that, for most instances, the cost of our algorithm is within $50\%$ of an information-theoretic lower bound on the optimal value., Comment: Full version of IPCO 2022 paper

We present a new local-search algorithm for the $k$-median clustering problem. We show that local optima for this algorithm give a $(2.836+\epsilon)$-approximation; our result improves upon the $(3+\epsilon)$-approximate local-search algorithm of Arya et al. [STOC 01]. Moreover, a computer-aided analysis of a natural extension suggests that this approach may lead to an improvement over the best-known approximation guarantee for the problem. The new ingredient in our algorithm is the use of a potential function based on both the closest and second-closest facilities to each client. Specifically, the potential is the sum over all clients, of the distance of the client to its closest facility, plus (a small constant times) the truncated distance to its second-closest facility. We move from one solution to another only if the latter can be obtained by swapping a constant number of facilities, and has a smaller potential than the former. This refined potential allows us to avoid the bad local optima given by Arya et al. for the local-search algorithm based only on the cost of the solution., Comment: To appear at SODA 22

We develop approximation algorithms for set-selection problems with deterministic constraints, but random objective values, i.e., stochastic probing problems. When the goal is to maximize the objective, approximation algorithms for probing problems are well-studied. On the other hand, few techniques are known for minimizing the objective, especially in the adaptive setting, where information about the random objective is revealed during the set-selection process and allowed to influence it. For minimization problems in particular, incorporating adaptivity can have a considerable effect on performance. In this work, we seek approximation algorithms that compare well to the optimal adaptive policy. We develop new techniques for adaptive minimization, applying them to a few problems of interest. The core technique we develop here is an approximate reduction from an adaptive expectation minimization problem to a set of adaptive probability minimization problems which we call threshold problems. By providing near-optimal solutions to these threshold problems, we obtain bicriteria adaptive policies. We apply this method to obtain an adaptive approximation algorithm for the MIN-ELEMENT problem, where the goal is to adaptively pick random variables to minimize the expected minimum value seen among them, subject to a knapsack constraint. This partially resolves an open problem raised in Goel et. al's "How to probe for an extreme value". We further consider three extensions on the MIN-ELEMENT problem, where our objective is the sum of the smallest k element-weights, or the weight of the min-weight basis of a given matroid, or where the constraint is not given by a knapsack but by a matroid constraint. For all three variations we explore, we develop adaptive approximation algorithms for their corresponding threshold problems, and prove their near-optimality via coupling arguments., Comment: Accepted to ITCS 2022

Robotic touch, particularly when using soft optical tactile sensors, suffers from distortion caused by motion-dependent shear. The manner in which the sensor contacts a stimulus is entangled with the tactile information about the geometry of the stimulus. In this work, we propose a supervised convolutional deep neural network model that learns to disentangle, in the latent space, the components of sensor deformations caused by contact geometry from those due to sliding-induced shear. The approach is validated by reconstructing unsheared tactile images from sheared images and showing they match unsheared tactile images collected with no sliding motion. In addition, the unsheared tactile images give a faithful reconstruction of the contact geometry that is not possible from the sheared data, and robust estimation of the contact pose that can be used for servo control sliding around various 2D shapes. Finally, the contact geometry reconstruction in conjunction with servo control sliding were used for faithful full object reconstruction of various 2D shapes. The methods have broad applicability to deep learning models for robots with a shear-sensitive sense of touch., Comment: 15 pages, 6 figure, under review CORL 2021

We consider online scheduling to minimize weighted completion time on related machines, where each job consists of several tasks that can be concurrently executed. A job gets completed when all its component tasks finish. We obtain an $O(K^3 \log^2 K)$-competitive algorithm in the non-clairvoyant setting, where $K$ denotes the number of distinct machine speeds. The analysis is based on dual-fitting on a precedence-constrained LP relaxation that may be of independent interest., Comment: Preliminary version in APPROX 2021

In the stochastic submodular cover problem, the goal is to select a subset of stochastic items of minimum expected cost to cover a submodular function. Solutions in this setting correspond to sequential decision processes that select items one by one "adaptively" (depending on prior observations). While such adaptive solutions achieve the best objective, the inherently sequential nature makes them undesirable in many applications. We ask: how well can solutions with only a few adaptive rounds approximate fully-adaptive solutions? We give nearly tight answers for both independent and correlated settings, proving smooth tradeoffs between the number of adaptive rounds and the solution quality, relative to fully adaptive solutions. Experiments on synthetic and real datasets show qualitative improvements in the solutions as we allow more rounds of adaptivity; in practice, solutions with a few rounds of adaptivity are nearly as good as fully adaptive solutions., Comment: In proceedings of ICML 2021

The (non-uniform) sparsest cut problem is the following graph-partitioning problem: given a "supply" graph, and demands on pairs of vertices, delete some subset of supply edges to minimize the ratio of the supply edges cut to the total demand of the pairs separated by this deletion. Despite much effort, there are only a handful of nontrivial classes of supply graphs for which constant-factor approximations are known. We consider the problem for planar graphs, and give a $(2+\varepsilon)$-approximation algorithm that runs in quasipolynomial time. Our approach defines a new structural decomposition of an optimal solution using a "patching" primitive. We combine this decomposition with a Sherali-Adams-style linear programming relaxation of the problem, which we then round. This should be compared with the polynomial-time approximation algorithm of Rao (1999), which uses the metric linear programming relaxation and $\ell_1$-embeddings, and achieves an $O(\sqrt{\log n})$-approximation in polynomial time., Comment: To appear at STOC 2021

The current best algorithms for convex body chasing problem in online algorithms use the notion of the Steiner point of a convex set. In particular, the algorithm which always moves to the Steiner point of the request set is $O(d)$ competitive for nested convex body chasing, and this is optimal among memoryless algorithms [Bubeck et al. 2020]. A memoryless algorithm coincides with the notion of a selector in functional analysis. The Steiner point is noted for being Lipschitz with respect to the Hausdorff metric, and for achieving the minimal Lipschitz constant possible. It is natural to ask whether every selector with this Lipschitz property yields a competitive algorithm for nested convex body chasing. We answer this question in the negative by exhibiting a selector which yields a non-competitive algorithm for nested convex body chasing but is Lipschitz with respect to Hausdorff distance. Furthermore, we show that being Lipschitz with respect to an $L_p$-type analog to the Hausdorff distance is sufficient to guarantee competitiveness if and only if $p=1$.