arXiv subjects

Adam Polak

Publications and source records attributed to Adam Polak.

36 records · Page 2Linked to original sources

arXiv2022

Tight Vector Bin Packing with Few Small Items via Fast Exact Matching in Multigraphs

We solve the Bin Packing problem in $O^*(2^k)$ time, where $k$ is the number of items less or equal to one third of the bin capacity. This parameter measures the distance from the polynomially solvable case of only large (i.e., greater than one third) items. Our algorithm is actually designed to work for a more general Vector Bin Packing problem, in which items are multidimensional vectors. We improve over the previous fastest $O^*(k! \cdot 4^k)$ time algorithm. Our algorithm works by reducing the problem to finding an exact weight perfect matching in a (multi-)graph with $O^*(2^k)$ edges, whose weights are integers of the order of $O^*(2^k)$. To solve the matching problem in the desired time, we give a variant of the classic Mulmuley-Vazirani-Vazirani algorithm with only a linear dependence on the edge weights and the number of edges, which may be of independent interest. Moreover, we give a tight lower bound, under the Strong Exponential Time Hypothesis (SETH), showing that the constant $2$ in the base of the exponent cannot be further improved for Vector Bin Packing. Our techniques also lead to improved algorithms for Vector Multiple Knapsack, Vector Bin Covering, and Perfect Matching with Hitting Constraints.

cs.DS↗

arXiv2022

Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost

We study the basic problem of assigning memoryless workers to tasks with dynamically changing demands. Given a set of $w$ workers and a multiset $T \subseteq[t]$ of $|T|=w$ tasks, a memoryless worker-task assignment function is any function $ϕ$ that assigns the workers $[w]$ to the tasks $T$ based only on the current value of $T$. The assignment function $ϕ$ is said to have switching cost at most $k$ if, for every task multiset $T$, changing the contents of $T$ by one task changes $ϕ(T)$ by at most $k$ worker assignments. The goal of memoryless worker task assignment is to construct an assignment function with the smallest possible switching cost. In past work, the problem of determining the optimal switching cost has been posed as an open question. There are no known sub-linear upper bounds, and after considerable effort, the best known lower bound remains 4 (ICALP 2020). We show that it is possible to achieve polylogarithmic switching cost. We give a construction via the probabilistic method that achieves switching cost $O(\log w \log (wt))$ and an explicit construction that achieves switching cost $\operatorname{polylog} (wt)$. We also prove a super-constant lower bound on switching cost: we show that for any value of $w$, there exists a value of $t$ for which the optimal switching cost is $w$. Thus it is not possible to achieve a switching cost that is sublinear strictly as a function of $w$. Finally, we present an application of the worker-task assignment problem to a metric embeddings problem. In particular, we use our results to give the first low-distortion embedding from sparse binary vectors into low-dimensional Hamming space.

cs.DS↗

arXiv2021

Learning-Augmented Dynamic Power Management with Multiple States via New Ski Rental Bounds

We study the online problem of minimizing power consumption in systems with multiple power-saving states. During idle periods of unknown lengths, an algorithm has to choose between power-saving states of different energy consumption and wake-up costs. We develop a learning-augmented online algorithm that makes decisions based on (potentially inaccurate) predicted lengths of the idle periods. The algorithm's performance is near-optimal when predictions are accurate and degrades gracefully with increasing prediction error, with a worst-case guarantee almost identical to the optimal classical online algorithm for the problem. A key ingredient in our approach is a new algorithm for the online ski rental problem in the learning augmented setting with tight dependence on the prediction error. We support our theoretical findings with experiments.

cs.DS↗

arXiv2021

Nearly-Tight and Oblivious Algorithms for Explainable Clustering

We study the problem of explainable clustering in the setting first formalized by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). A $k$-clustering is said to be explainable if it is given by a decision tree where each internal node splits data points with a threshold cut in a single dimension (feature), and each of the $k$ leaves corresponds to a cluster. We give an algorithm that outputs an explainable clustering that loses at most a factor of $O(\log^2 k)$ compared to an optimal (not necessarily explainable) clustering for the $k$-medians objective, and a factor of $O(k \log^2 k)$ for the $k$-means objective. This improves over the previous best upper bounds of $O(k)$ and $O(k^2)$, respectively, and nearly matches the previous $Ω(\log k)$ lower bound for $k$-medians and our new $Ω(k)$ lower bound for $k$-means. The algorithm is remarkably simple. In particular, given an initial not necessarily explainable clustering in $\mathbb{R}^d$, it is oblivious to the data points and runs in time $O(dk \log^2 k)$, independent of the number of data points $n$. Our upper and lower bounds also generalize to objectives given by higher $\ell_p$-norms.

cs.DS↗

arXiv2021

Consumer Wearables and Affective Computing for Wellbeing Support

Wearables equipped with pervasive sensors enable us to monitor physiological and behavioral signals in our everyday life. We propose the WellAff system able to recognize affective states for wellbeing support. It also includes health care scenarios, in particular patients with chronic kidney disease (CKD) suffering from bipolar disorders. For the need of a large-scale field study, we revised over 50 off-the-shelf devices in terms of usefulness for emotion, stress, meditation, sleep, and physical activity recognition and analysis. Their usability directly comes from the types of sensors they possess as well as the quality and availability of raw signals. We found there is no versatile device suitable for all purposes. Using Empatica E4 and Samsung Galaxy Watch, we have recorded physiological signals from 11 participants over many weeks. The gathered data enabled us to train a classifier that accurately recognizes strong affective states.

cs.HC↗

arXiv2021

Robust Learning-Augmented Caching: An Experimental Study

Effective caching is crucial for the performance of modern-day computing systems. A key optimization problem arising in caching -- which item to evict to make room for a new item -- cannot be optimally solved without knowing the future. There are many classical approximation algorithms for this problem, but more recently researchers started to successfully apply machine learning to decide what to evict by discovering implicit input patterns and predicting the future. While machine learning typically does not provide any worst-case guarantees, the new field of learning-augmented algorithms proposes solutions that leverage classical online caching algorithms to make the machine-learned predictors robust. We are the first to comprehensively evaluate these learning-augmented algorithms on real-world caching datasets and state-of-the-art machine-learned predictors. We show that a straightforward method -- blindly following either a predictor or a classical robust algorithm, and switching whenever one becomes worse than the other -- has only a low overhead over a well-performing predictor, while competing with classical methods when the coupled predictor fails, thus providing a cheap worst-case insurance.

cs.LG↗

arXiv2021

Knapsack and Subset Sum with Small Items

Knapsack and Subset Sum are fundamental NP-hard problems in combinatorial optimization. Recently there has been a growing interest in understanding the best possible pseudopolynomial running times for these problems with respect to various parameters. In this paper we focus on the maximum item size $s$ and the maximum item value $v$. We give algorithms that run in time $O(n + s^3)$ and $O(n + v^3)$ for the Knapsack problem, and in time $\tilde{O}(n + s^{5/3})$ for the Subset Sum problem. Our algorithms work for the more general problem variants with multiplicities, where each input item comes with a (binary encoded) multiplicity, which succinctly describes how many times the item appears in the instance. In these variants $n$ denotes the (possibly much smaller) number of distinct items. Our results follow from combining and optimizing several diverse lines of research, notably proximity arguments for integer programming due to Eisenbrand and Weismantel (TALG 2019), fast structured $(\min,+)$-convolution by Kellerer and Pferschy (J. Comb. Optim. 2004), and additive combinatorics methods originating from Galil and Margalit (SICOMP 1991).

cs.DS↗

arXiv2021

Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths

One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in $n^{3-o(1)}$ time, and it is widely open whether it has an $O(n^{3-ε})$ time algorithm for $ε> 0$. To better understand APSP, one often strives to obtain subcubic time algorithms for structured instances of APSP and problems equivalent to it, such as the Min-Plus matrix product. A natural structured version of Min-Plus product is Monotone Min-Plus product which has been studied in the context of the Batch Range Mode [SODA'20] and Dynamic Range Mode [ICALP'20] problems. This paper improves the known algorithms for Monotone Min-Plus Product and for Batch and Dynamic Range Mode, and establishes a connection between Monotone Min-Plus Product and the Single Source Replacement Paths (SSRP) problem on an $n$-vertex graph with potentially negative edge weights in $\{-M, \ldots, M\}$. SSRP with positive integer edge weights bounded by $M$ can be solved in $\tilde{O}(Mn^ω)$ time, whereas the prior fastest algorithm for graphs with possibly negative weights [FOCS'12] runs in $O(M^{0.7519} n^{2.5286})$ time, the current best running time for directed APSP with small integer weights. Using Monotone Min-Plus Product, we obtain an improved $O(M^{0.8043} n^{2.4957})$ time SSRP algorithm, showing that SSRP with constant negative integer weights is likely easier than directed unweighted APSP, a problem that is believed to require $n^{2.5-o(1)}$ time. Complementing our algorithm for SSRP, we give a reduction from the Bounded-Difference Min-Plus Product problem studied by Bringmann et al. [FOCS'16] to negative weight SSRP. This reduction shows that it might be difficult to obtain an $\tilde{O}(M n^ω)$ time algorithm for SSRP with negative weight edges, thus separating the problem from SSRP with only positive weight edges.

cs.DS↗

arXiv2021

Euler Meets GPU: Practical Graph Algorithms with Theoretical Guarantees

The Euler tour technique is a classical tool for designing parallel graph algorithms, originally proposed for the PRAM model. We ask whether it can be adapted to run efficiently on GPU. We focus on two established applications of the technique: (1) the problem of finding lowest common ancestors (LCA) of pairs of nodes in trees, and (2) the problem of finding bridges in undirected graphs. In our experiments, we compare theoretically optimal algorithms using the Euler tour technique against simpler heuristics supposed to perform particularly well on typical instances. We show that the Euler tour-based algorithms not only fulfill their theoretical promises and outperform practical heuristics on hard instances, but also perform on par with them on easy instances.

cs.DS↗

arXiv2020

Monochromatic Triangles, Intermediate Matrix Products, and Convolutions

The most studied linear algebraic operation, matrix multiplication, has surprisingly fast $O(n^ω)$ time algorithms for $ω<2.373$. On the other hand, the $(\min,+)$ matrix product which is at the heart of many fundamental graph problems such as APSP, has received only minor improvements over its brute-force cubic running time and is widely conjectured to require $n^{3-o(1)}$ time. There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems. For instance, the Min-Max matrix product, the Minimum Witness matrix product, APSP in directed unweighted graphs and determining whether an edge-colored graph contains a monochromatic triangle, can all be solved in $\tilde O(n^{(3+ω)/2})$ time. A similar phenomenon occurs for convolution problems, where analogous intermediate problems can be solved in $\tilde O(n^{1.5})$ time. Can one improve upon the running times for these intermediate problems, in either the matrix product or the convolution world? Or, alternatively, can one relate these problems to each other and to other key problems in a meaningful way? This paper makes progress on these questions by providing a network of fine-grained reductions. We show for instance that APSP in directed unweighted graphs and Minimum Witness product can be reduced to both the Min-Max product and a variant of the monochromatic triangle problem. We also show that a natural convolution variant of monochromatic triangle is fine-grained equivalent to the famous 3SUM problem. As this variant is solvable in $O(n^{1.5})$ time and 3SUM is in $O(n^2)$ time (and is conjectured to require $n^{2-o(1)}$ time), our result gives the first fine-grained equivalence between natural problems of different running times.

cs.CC↗

arXiv2020

Online Coloring of Short Intervals

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in $[1,σ]$. For $σ=1$ it is the class of unit interval graphs, and for $σ=\infty$ the class of all interval graphs. Our focus is on intermediary classes. We present a $(1+σ)$-competitive algorithm, which beats the state of the art for $1 < σ< 2$, and proves that the problem we study can be strictly easier than online coloring of general interval graphs. On the lower bound side, we prove that no algorithm is better than $5/3$-competitive for any $σ>1$, nor better than $7/4$-competitive for any $σ>2$, and that no algorithm beats the $5/2$ asymptotic competitive ratio for all, arbitrarily large, values of $σ$. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than $2$.

math.CO↗

arXiv2020

Equivalences between triangle and range query problems

We define a natural class of range query problems, and prove that all problems within this class have the same time complexity (up to polylogarithmic factors). The equivalence is very general, and even applies to online algorithms. This allows us to obtain new improved algorithms for all of the problems in the class. We then focus on the special case of the problems when the queries are offline and the number of queries is linear. We show that our range query problems are runtime-equivalent (up to polylogarithmic factors) to counting for each edge $e$ in an $m$-edge graph the number of triangles through $e$. This natural triangle problem can be solved using the best known triangle counting algorithm, running in $O(m^{2ω/(ω+1)}) \leq O(m^{1.41})$ time. Moreover, if $ω=2$, the $O(m^{2ω/(ω+1)})$ running time is known to be tight (within $m^{o(1)}$ factors) under the 3SUM Hypothesis. In this case, our equivalence settles the complexity of the range query problems. Our problems constitute the first equivalence class with this peculiar running time bound. To better understand the complexity of these problems, we also provide a deeper insight into the family of triangle problems, in particular showing black-box reductions between triangle listing and per-edge triangle detection and counting. As a byproduct of our reductions, we obtain a simple triangle listing algorithm matching the state-of-the-art for all regimes of the number of triangles. We also give some not necessarily tight, but still surprising reductions from variants of matrix products, such as the $(\min,\max)$-product.

cs.CC↗

arXiv2020

Emotion Recognition Using Wearables: A Systematic Literature Review Work in progress

Wearables like smartwatches or wrist bands equipped with pervasive sensors enable us to monitor our physiological signals. In this study, we address the question whether they can help us to recognize our emotions in our everyday life for ubiquitous computing. Using the systematic literature review, we identified crucial research steps and discussed the main limitations and problems in the domain.

cs.HC↗

arXiv2017

On an extremal problem for poset dimension

Let $f(n)$ be the largest integer such that every poset on $n$ elements has a $2$-dimensional subposet on $f(n)$ elements. What is the asymptotics of $f(n)$? It is easy to see that $f(n)\geqslant n^{1/2}$. We improve the best known upper bound and show $f(n)=\mathcal{O}(n^{2/3})$. For higher dimensions, we show $f_d(n)=\mathcal{O}\left(n^\frac{d}{d+1}\right)$, where $f_d(n)$ is the largest integer such that every poset on $n$ elements has a $d$-dimensional subposet on $f_d(n)$ elements.

math.CO↗

arXiv2017

Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called $k$-LCIS: Given $k$ integer sequences $X_1,\dots,X_k$ of length at most $n$, the task is to determine the length of the longest common subsequence of $X_1,\dots,X_k$ that is also strictly increasing. Especially for the case of $k=2$ (called LCIS for short), several algorithms have been proposed that require quadratic time in the worst case. Assuming the Strong Exponential Time Hypothesis (SETH), we prove a tight lower bound, specifically, that no algorithm solves LCIS in (strongly) subquadratic time. Interestingly, the proof makes no use of normalization tricks common to hardness proofs for similar problems such as LCS. We further strengthen this lower bound (1) to rule out $O((nL)^{1-\varepsilon})$ time algorithms for LCIS, where $L$ denotes the solution size, (2) to rule out $O(n^{k-\varepsilon})$ time algorithms for $k$-LCIS, and (3) to follow already from weaker variants of SETH. We obtain the same conditional lower bounds for the related Longest Common Weakly Increasing Subsequence problem.

cs.CC↗

arXiv2017

Why is it hard to beat $O(n^2)$ for Longest Common Weakly Increasing Subsequence?

The Longest Common Weakly Increasing Subsequence problem (LCWIS) is a variant of the classic Longest Common Subsequence problem (LCS). Both problems can be solved with simple quadratic time algorithms. A recent line of research led to a number of matching conditional lower bounds for LCS and other related problems. However, the status of LCWIS remained open. In this paper we show that LCWIS cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis (SETH) is false. The ideas which we developed can also be used to obtain a lower bound based on a safer assumption of NC-SETH, i.e. a version of SETH which talks about NC circuits instead of less expressive CNF formulas.

cs.CC↗

arXiv2017

Flying, Hopping Pit-Bots for Cave and Lava Tube Exploration on the Moon and Mars

Wheeled ground robots are limited from exploring extreme environments such as caves, lava tubes and skylights. Small robots that utilize unconventional mobility through hopping, flying and rolling can overcome many roughness limitations and thus extend exploration sites of interest on Moon and Mars. In this paper we introduce a network of 3 kg, 0.30 m diameter ball robots (pit-bots) that can fly, hop and roll using an onboard miniature propulsion system. These pit-bots can be deployed from a lander or large rover. Each robot is equipped with a smartphone sized computer, stereo camera and laser rangefinder to per-form navigation and mapping. The ball robot can carry a payload of 1 kg or perform sample return. Our studies show a range of 5 km and 0.7 hours flight time on the Moon.

cs.RO↗

arXiv2016

Counting Triangles in Large Graphs on GPU

The clustering coefficient and the transitivity ratio are concepts often used in network analysis, which creates a need for fast practical algorithms for counting triangles in large graphs. Previous research in this area focused on sequential algorithms, MapReduce parallelization, and fast approximations. In this paper we propose a parallel triangle counting algorithm for CUDA GPU. We describe the implementation details necessary to achieve high performance and present the experimental evaluation of our approach. Our algorithm achieves 8 to 15 times speedup over the CPU implementation and is capable of finding 3.8 billion triangles in an 89 million edges graph in less than 10 seconds on the Nvidia Tesla C2050 GPU.

cs.DC↗