Theory of Computing Reporten7th Eastern Great Lakes (EaGL) Theory of Computation Workshophttp://cstheory-events.org/2024/08/15/7th-eastern-great-lakes-eagl-theory-of-computation-workshop/
https://cstheory-events.org/2024/08/15/7th-eastern-great-lakes-eagl-theory-of-computation-workshop/
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<p>October 5-6, 2024 University at Buffalo, Buffalo, NY https://www.cs.rochester.edu/u/shossei2/eagl2024website/index.html Registration deadline: September 15, 2024 This regional theory workshop features talks by distinguished scientists and a student poster session. All interested in theoretical computer science are welcome to attend. Registration is required.</p>
<p class="authors">By shacharlovett</p>
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2024-08-15 14:16:26 UTCCS Theory EventsQuantifying over Optimum Answer Setshttp://arxiv.org/abs/2408.07697v1
http://arxiv.org/abs/2408.07697v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Giuseppe+Mazzotta">Giuseppe Mazzotta</a>, <a href="https://dblp.uni-trier.de/search?q=Francesco+Ricca">Francesco Ricca</a>, <a href="https://dblp.uni-trier.de/search?q=Mirek+Truszczynski">Mirek Truszczynski</a></p>Answer Set Programming with Quantifiers (ASP(Q)) has been introduced to
provide a natural extension of ASP modeling to problems in the polynomial
hierarchy (PH). However, ASP(Q) lacks a method for encoding in an elegant and
compact way problems requiring a polynomial number of calls to an oracle in
$\Sigma_n^p$ (that is, problems in $\Delta_{n+1}^p$). Such problems include, in
particular, optimization problems. In this paper we propose an extension of
ASP(Q), in which component programs may contain weak constraints. Weak
constraints can be used both for expressing local optimization within
quantified component programs and for modeling global optimization criteria. We
showcase the modeling capabilities of the new formalism through various
application scenarios. Further, we study its computational properties obtaining
complexity results and unveiling non-obvious characteristics of ASP(Q) programs
with weak constraints.</body></html>
2024-08-15 00:00:00 UTCarXiv: Computational ComplexityExact values of generic subrankhttp://arxiv.org/abs/2408.07550v1
http://arxiv.org/abs/2408.07550v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Pawe%C5%82+Pielasa">Paweł Pielasa</a>, <a href="https://dblp.uni-trier.de/search?q=Matou%C5%A1+%C5%A0afr%C3%A1nek">Matouš Šafránek</a>, <a href="https://dblp.uni-trier.de/search?q=Anatoli+Shatsila">Anatoli Shatsila</a></p>In this article we prove the subrank of a generic tensor in
$\mathbb{C}^{n,n,n}$ to be $Q(n) = \lfloor\sqrt{3n - 2}\rfloor$ by providing a
lower bound to the known upper bound. More generally, we find the generic
subrank of tensors of all orders and dimensions. This answers two open
questions posed in arXiv:2205.15168v2. Finally, we compute dimensions of
varieties of tensors of subrank at least $r$.</body></html>
2024-08-15 00:00:00 UTCarXiv: Computational ComplexityOracle without Optimal Proof Systems outside Nondeterministic
Subexponential Timehttp://arxiv.org/abs/2408.07408v1
http://arxiv.org/abs/2408.07408v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Fabian+Egidy">Fabian Egidy</a>, <a href="https://dblp.uni-trier.de/search?q=Christian+Gla%C3%9Fer">Christian Glaßer</a></p>We study the existence of optimal proof systems for sets outside of
$\mathrm{NP}$. Currently, no set $L \notin \mathrm{NP}$ is known that has
optimal proof systems. Our main result shows that this is not surprising,
because we can rule out relativizable proofs of optimality for all sets outside
$\mathrm{NTIME}(t)$ where $t$ is slightly superpolynomial. We construct an
oracle $O$, such that for any set $L \subseteq \Sigma^*$ at least one of the
following two properties holds: $L$ does not have optimal proof systems
relative to $O$. $L \in \mathrm{UTIME}^O(2^{2(\log
n)^{8+4\log(\log(\log(n)))}})$. The runtime bound is slightly superpolynomial.
So there is no relativizable proof showing that a complex set has optimal proof
systems. Hence, searching for non-trivial optimal proof systems with
relativizable methods can only be successful (if at all) in a narrow range
above $\mathrm{NP}$.</body></html>
2024-08-15 00:00:00 UTCarXiv: Computational ComplexityExact Trajectory Similarity Search With N-tree: An Efficient Metric
Index for kNN and Range Querieshttp://arxiv.org/abs/2408.07650v1
http://arxiv.org/abs/2408.07650v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Ralf+Hartmut+G%C3%BCting">Ralf Hartmut Güting</a>, <a href="https://dblp.uni-trier.de/search?q=Suvam+Kumar+Das">Suvam Kumar Das</a>, <a href="https://dblp.uni-trier.de/search?q=Fabio+Vald%C3%A9s">Fabio Valdés</a>, <a href="https://dblp.uni-trier.de/search?q=Suprio+Ray">Suprio Ray</a></p>Similarity search is the problem of finding in a collection of objects those
that are similar to a given query object. It is a fundamental problem in modern
applications and the objects considered may be as diverse as locations in
space, text documents, images, twitter messages, or trajectories of moving
objects.
In this paper we are motivated by the latter application. Trajectories are
recorded movements of mobile objects such as vehicles, animals, public
transportation, or parts of the human body. We propose a novel distance
function called DistanceAvg to capture the similarity of such movements. To be
practical, it is necessary to provide indexing for this distance measure.
Fortunately we do not need to start from scratch. A generic and unifying
approach is metric space, which organizes the set of objects solely by a
distance (similarity) function with certain natural properties. Our function
DistanceAvg is a metric.
Although metric indexes have been studied for decades and many such
structures are available, they do not offer the best performance with
trajectories. In this paper we propose a new design, which outperforms the best
existing indexes for kNN queries and is equally good for range queries. It is
especially suitable for expensive distance functions as they occur in
trajectory similarity search. In many applications, kNN queries are more
practical than range queries as it may be difficult to determine an appropriate
search radius. Our index provides exact result sets for the given distance
function.</body></html>
2024-08-15 00:00:00 UTCarXiv: Data Structures and AlgorithmsProphet Inequalities: Competing with the Top $\ell$ Items is Easyhttp://arxiv.org/abs/2408.07616v1
http://arxiv.org/abs/2408.07616v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Mathieu+Molina">Mathieu Molina</a>, <a href="https://dblp.uni-trier.de/search?q=Nicolas+Gast">Nicolas Gast</a>, <a href="https://dblp.uni-trier.de/search?q=Patrick+Loiseau">Patrick Loiseau</a>, <a href="https://dblp.uni-trier.de/search?q=Vianney+Perchet">Vianney Perchet</a></p>We explore a novel variant of the classical prophet inequality problem, where
the values of a sequence of items are drawn i.i.d. from some distribution, and
an online decision maker must select one item irrevocably. We establish that
the competitive ratio between the expected optimal performance of the online
decision maker compared to that of a prophet, who uses the average of the top
$\ell$ items, must be greater than $\ell/c_{\ell}$, with $c_{\ell}$ the
solution to an integral equation. We prove that this lower bound is larger than
$1-1/(\exp(\ell)-1)$. This implies that the bound converges exponentially fast
to $1$ as $\ell$ grows. In particular, the bound for $\ell=2$ is $2/c_{2}
\approx 0.966$ which is much closer to $1$ than the classical bound of $0.745$
for $\ell=1$. Additionally, the proposed algorithm can be extended to a more
general scenario, where the decision maker is permitted to select $k$ items.
This subsumes the $k$ multi-unit i.i.d. prophet problem and provides the
current best asymptotic guarantees, as well as enables broader understanding in
the more general framework. Finally, we prove a nearly tight competitive ratio
when only static threshold policies are allowed.</body></html>
2024-08-15 00:00:00 UTCarXiv: Data Structures and AlgorithmsFast and Accurate Algorithms to Calculate Expected Modularity in
Probabilistic Networkshttp://arxiv.org/abs/2408.07161v1
http://arxiv.org/abs/2408.07161v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Xin+Shen">Xin Shen</a>, <a href="https://dblp.uni-trier.de/search?q=Matteo+Magnani">Matteo Magnani</a>, <a href="https://dblp.uni-trier.de/search?q=Christian+Rohner">Christian Rohner</a>, <a href="https://dblp.uni-trier.de/search?q=Fiona+Skerman">Fiona Skerman</a></p>Modularity maximization is a widely used community detection technique for
deterministic networks. However, little research has been performed to develop
efficient modularity calculation algorithms for probabilistic networks.
Particularly, it is challenging to efficiently calculate expected modularity
when all possible worlds are considered. To address this problem, we propose
two algorithms, namely $\mathrm{PWP}^{\mathrm{EMOD}}$ and
$\mathrm{APWP}^{\mathrm{EMOD}}$, partitioning the possible worlds based on
their modularities to significantly reduce the number of probability
calculations. We evaluate the accuracy and time efficiency of our algorithms
through comprehensive experiments.</body></html>
2024-08-15 00:00:00 UTCarXiv: Data Structures and AlgorithmsMy Reading Burdenhttps://scottaaronson.blog/?p=8217
https://scottaaronson.blog/?p=8217
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<p>Want some honesty about how I (mis)spend my time? These days, my daily routine includes reading all of the following:</p>
<ul class="wp-block-list">
<li>The comments on this blog (many of which I then answer)</li>
<li>
<a href="https://www.washingtonpost.com/">The Washington Post</a> (“The Post Most” takes me about an hour a day, every day)</li>
<li><a href="https://www.nytimes.com/">The New York Times</a></li>
<li>Scott Alexander’s <a href="https://www.astralcodexten.com/">Astral Codex Ten</a>
</li>
<li>Zvi Mowshowitz’s <a href="https://thezvi.substack.com/">Don’t Worry About the Vase</a> (Zvi is so superhumanly prolific that reading him easily takes 12 hours per week … but it’s all good stuff!)</li>
<li>Peter Woit’s <a href="https://www.math.columbia.edu/~woit/wordpress/">Not Even Wrong</a> (rarely updated anymore, thankfully for my reading time!)</li>
<li>
<a href="https://www.quantamagazine.org/">Quanta</a> (how could I forget?)</li>
<li><a href="https://quillette.com/">Quillette</a></li>
<li><a href="https://www.thefp.com/">The Free Press</a></li>
<li><a href="https://mosaicmagazine.com/">Mosaic</a></li>
<li><a href="https://www.tabletmag.com/">Tablet</a></li>
<li><a href="https://www.commentary.org/">Commentary</a></li>
<li><a href="https://twitter.com/paulg">Paul Graham’s Twitter</a></li>
<li><a href="https://twitter.com/DavidDeutschOxf">David Deutsch’s Twitter</a></li>
<li><a href="https://twitter.com/ESYudkowsky">Eliezer Yudkowsky’s Twitter</a></li>
<li>Updates and comments from my Facebook friends (this can easily take a couple hours per day)</li>
<li>The <a href="https://arxiv.org/list/quant-ph/new">quant-ph arXiv</a> (I scan maybe 50 titles and abstracts per day, and if any papers are relevant to me, read at least their introductions)</li>
<li>The science fiction and fantasy novels I read with my kids</li>
<li>Whichever other books I’m currently reading</li>
</ul>
<p>Many of these materials contain lists of links to <em>other</em> articles, or tweet threads, some of which then take me hours to read in themselves. This is not counting podcasts or movies or TV shows.</p>
<p>While I read unusually quickly, I’d estimate that my reading burden is now at eight hours per day, seven days per week. I haven’t finished reading by the time my kids are back from school or day camp. Now let’s add in <em>my actual job</em> (or two jobs, although the OpenAI one is ending this month, and I start teaching again in two weeks). Add in answering emails (including from fans and advice-seekers), giving lectures, meeting grad students and undergrads, doing Zoom calls, filling out forms, consulting, going on podcasts, reviewing papers, taking care of my kids, eating, shopping, personal hygiene.</p>
<p>As often as not, when the day is done, it’s not just that I’ve achieved nothing of lasting value—it’s that I’ve <em>never even started</em> with research, writing, or any long-term projects. This contrasts with my twenties, when obsessively working on research problems and writing up the results could easily fill my day.</p>
<p>The solution seems obvious: <strong>stop reading so much.</strong> Cut back to a few hours per day, tops. But it’s hard. The rapid scale-up of AI is a once-in-the-history-of-civilization story that I feel astounded to be living through and compelled to follow, and just keeping up with the highlights is almost a full-time job in itself. The threat to democracy from Trump, Putin, Xi, Maduro, and the world’s other authoritarians is another story that I feel unable to look away from.</p>
<p>Since October 7, though, the once-again-precarious situation of Jews everywhere on earth has become, on top of everything else it is, the #1 drain on my time. It would be one thing if I limited myself to thoughtful analyses, but I can easily lose hours per day doomscrolling through the infinite firehose of strident anti-Zionism (and often, simple unconcealed Jew-hatred) that one finds for example on Twitter, Facebook, and the comment sections of <em>Washington Post</em> articles. Every time someone calls the “Zios” land-stealing baby-killers who deserve to die, my brain insists that they’re addressing me personally. So I stop to ponder the psychology of each individual commenter before moving on to the next, struggle to see the world from their eyes. Would explaining the complex realities of the conflict change this person’s mind? What about introducing them to my friends and relatives in Israel who never knew any other home and want nothing but peace, coexistence, and a two-state solution?</p>
<p>I naturally can’t say that all this compulsive reading makes me happy or fulfilled. Worse yet, I can’t even say it makes me feel <em>more informed</em>. What I suppose it does make me feel is … <em>excused</em>. If so much is being written daily about the biggest controversies in the world, then how can I be blamed for reading it rather than doing anything new?</p>
<p>At the risk of adding even more to the terrifying torrent of words, I’d like to hear from anyone who ever struggled with a similar reading addiction, and successfully overcame it. What worked for you?</p>
<p class="authors">By Scott</p>
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2024-08-14 19:41:18 UTCScott AaronsonFavorite Theorems: Random Oraclestag:blogger.com,1999:blog-3722233.post-4066846424758672535
https://blog.computationalcomplexity.org/2024/08/favorite-theorems-random-oracles.html
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<div style="text-align: left;"><a href="https://blog.computationalcomplexity.org/2024/07/favorite-theorems-extracting-ramsey.html">July Edition</a></div>
<div style="text-align: left;"><br></div>
<div style="text-align: left;">This months favorite theorem is a circuit result that implies the polynomial-time hierarchy is infinite relative to a random oracle, answering an open question that goes back to the 80's. </div>
<div style="text-align: center;"><br></div>
<div style="text-align: center;"><a href="https://doi.org/10.1145/3095799">An Average-Case Depth Hierarchy Theorem for Boolean Circuits</a></div>
<div style="text-align: center;">Johan Håstad, Benjamin Rossman, Rocco A. Servedio and Li-Yang Tan</div>
<div style="text-align: left;"><br></div>
<div style="text-align: left;">The authors show how to separate depth d from depth d+1 circuits for random inputs. As a corollary, the polynomial hierarchy is infinite with a random oracle, which means that if we choose an oracle R at random, with probability one, the k+1-st level of the polynomial-time hierarchy relative to R is different than the k-th level relative to R. </div>
<div style="text-align: left;"><br></div>
<div style="text-align: left;">Why should we care about random oracles? By the <a href="https://blog.computationalcomplexity.org/2018/08/the-zero-one-law-for-random-oracles.html">Kolmogorov zero-one law</a>, every complexity statement holds with probability zero or probability one with a random oracle, so for every statement either it or its negation holds with probability one. And since the countable intersection of measure-one sets is measure one, every complexity statement true relative to a random oracle are all simultaneously true relative to a random oracle, a kind of consistent world. With a random oracle, we have full derandomization, like BPP = P, AM = NP and PH in \(\oplus\mathrm{P}\). We have separations like P ≠ UP ≠ NP. We have results like NP doesn't have measure zero and SAT solutions can be found with non-adaptive queries to an NP oracle. And now we have that the PH is infinite simultaneously with all these other results. </div>
<div style="text-align: left;"><br></div>
<div style="text-align: left;">
<a href="https://blog.computationalcomplexity.org/2015/04/ph-infinite-under-random-oracle.html">More details</a> on this paper from a post I wrote back in 2015. </div>
<p class="authors">By Lance Fortnow</p>
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2024-08-14 13:04:00 UTCComputational ComplexityPolynomial 2D Green Coordinates for High-order Cageshttp://arxiv.org/abs/2408.06831v1
http://arxiv.org/abs/2408.06831v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Shibo+Liu">Shibo Liu</a>, <a href="https://dblp.uni-trier.de/search?q=Ligang+Liu">Ligang Liu</a>, <a href="https://dblp.uni-trier.de/search?q=Xiao-Ming+Fu">Xiao-Ming Fu</a></p>We propose conformal polynomial coordinates for 2D closed high-order cages,
which consist of polynomial curves of any order. The coordinates enable the
transformation of the input polynomial curves into polynomial curves of any
order. We extend the classical 2D Green coordinates to define our coordinates,
thereby leading to cage-aware conformal harmonic deformations. We extensively
test our method on various 2D deformations, allowing users to manipulate the
\Bezier control points to easily generate the desired deformation.</body></html>
2024-08-14 00:00:00 UTCarXiv: Computational GeometryStructure-preserving Planar Simplification for Indoor Environmentshttp://arxiv.org/abs/2408.06814v1
http://arxiv.org/abs/2408.06814v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Bishwash+Khanal">Bishwash Khanal</a>, <a href="https://dblp.uni-trier.de/search?q=Sanjay+Rijal">Sanjay Rijal</a>, <a href="https://dblp.uni-trier.de/search?q=Manish+Awale">Manish Awale</a>, <a href="https://dblp.uni-trier.de/search?q=Vaghawan+Ojha">Vaghawan Ojha</a></p>This paper presents a novel approach for structure-preserving planar
simplification of indoor scene point clouds for both simulated and real-world
environments. Initially, the scene point cloud undergoes preprocessing steps,
including noise reduction and Manhattan world alignment, to ensure robustness
and coherence in subsequent analyses. We segment each captured scene into
structured (walls-ceiling-floor) and non-structured (indoor objects) scenes.
Leveraging a RANSAC algorithm, we extract primitive planes from the input point
cloud, facilitating the segmentation and simplification of the structured
scene. The best-fitting wall meshes are then generated from the primitives,
followed by adjacent mesh merging with the vertex-translation algorithm which
preserves the mesh layout. To accurately represent ceilings and floors, we
employ the mesh clipping algorithm which clips the ceiling and floor meshes
with respect to wall normals. In the case of indoor scenes, we apply a surface
reconstruction technique to enhance the fidelity. This paper focuses on the
intricate steps of the proposed scene simplification methodology, addressing
complex scenarios such as multi-story and slanted walls and ceilings. We also
conduct qualitative and quantitative performance comparisons against popular
surface reconstruction, shape approximation, and floorplan generation
approaches.</body></html>
2024-08-14 00:00:00 UTCarXiv: Computational GeometryDC3DO: Diffusion Classifier for 3D Objectshttp://arxiv.org/abs/2408.06693v1
http://arxiv.org/abs/2408.06693v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Nursena+Koprucu">Nursena Koprucu</a>, <a href="https://dblp.uni-trier.de/search?q=Meher+Shashwat+Nigam">Meher Shashwat Nigam</a>, <a href="https://dblp.uni-trier.de/search?q=Shicheng+Xu">Shicheng Xu</a>, <a href="https://dblp.uni-trier.de/search?q=Biruk+Abere">Biruk Abere</a>, <a href="https://dblp.uni-trier.de/search?q=Gabriele+Dominici">Gabriele Dominici</a>, <a href="https://dblp.uni-trier.de/search?q=Andrew+Rodriguez">Andrew Rodriguez</a>, <a href="https://dblp.uni-trier.de/search?q=Sharvaree+Vadgam">Sharvaree Vadgam</a>, <a href="https://dblp.uni-trier.de/search?q=Berfin+Inal">Berfin Inal</a>, <a href="https://dblp.uni-trier.de/search?q=Alberto+Tono">Alberto Tono</a></p>Inspired by Geoffrey Hinton emphasis on generative modeling, To recognize
shapes, first learn to generate them, we explore the use of 3D diffusion models
for object classification. Leveraging the density estimates from these models,
our approach, the Diffusion Classifier for 3D Objects (DC3DO), enables
zero-shot classification of 3D shapes without additional training. On average,
our method achieves a 12.5 percent improvement compared to its multiview
counterparts, demonstrating superior multimodal reasoning over discriminative
approaches. DC3DO employs a class-conditional diffusion model trained on
ShapeNet, and we run inferences on point clouds of chairs and cars. This work
highlights the potential of generative models in 3D object classification.</body></html>
2024-08-14 00:00:00 UTCarXiv: Computational GeometryStabilizer bootstrapping: A recipe for efficient agnostic tomography and
magic estimationhttp://arxiv.org/abs/2408.06967v1
http://arxiv.org/abs/2408.06967v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Sitan+Chen">Sitan Chen</a>, <a href="https://dblp.uni-trier.de/search?q=Weiyuan+Gong">Weiyuan Gong</a>, <a href="https://dblp.uni-trier.de/search?q=Qi+Ye">Qi Ye</a>, <a href="https://dblp.uni-trier.de/search?q=Zhihan+Zhang">Zhihan Zhang</a></p>We study the task of agnostic tomography: given copies of an unknown
$n$-qubit state $\rho$ which has fidelity $\tau$ with some state in a given
class $C$, find a state which has fidelity $\ge \tau - \epsilon$ with $\rho$.
We give a new framework, stabilizer bootstrapping, for designing
computationally efficient protocols for this task, and use this to get new
agnostic tomography protocols for the following classes:
Stabilizer states: We give a protocol that runs in time
$\mathrm{poly}(n,1/\epsilon)\cdot (1/\tau)^{O(\log(1/\tau))}$, answering an
open question posed by Grewal, Iyer, Kretschmer, Liang [40] and Anshu and
Arunachalam [6]. Previous protocols ran in time $\mathrm{exp}(\Theta(n))$ or
required $\tau>\cos^2(\pi/8)$.
States with stabilizer dimension $n - t$: We give a protocol that runs in
time $n^3\cdot(2^t/\tau)^{O(\log(1/\epsilon))}$, extending recent work on
learning quantum states prepared by circuits with few non-Clifford gates, which
only applied in the realizable setting where $\tau = 1$ [30, 37, 46, 61].
Discrete product states: If $C = K^{\otimes n}$ for some $\mu$-separated
discrete set $K$ of single-qubit states, we give a protocol that runs in time
$(n/\mu)^{O((1 + \log (1/\tau))/\mu)}/\epsilon^2$. This strictly generalizes a
prior guarantee which applied to stabilizer product states [39]. For stabilizer
product states, we give a further improved protocol that runs in time
$(n^2/\epsilon^2)\cdot (1/\tau)^{O(\log(1/\tau))}$.
As a corollary, we give the first protocol for estimating stabilizer
fidelity, a standard measure of magic for quantum states, to error $\epsilon$
in $n^3 \mathrm{quasipoly}(1/\epsilon)$ time.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsQuasi-Monte Carlo Beyond Hardy-Krausehttp://arxiv.org/abs/2408.06475v1
http://arxiv.org/abs/2408.06475v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Nikhil+Bansal">Nikhil Bansal</a>, <a href="https://dblp.uni-trier.de/search?q=Haotian+Jiang">Haotian Jiang</a></p>The classical approaches to numerically integrating a function $f$ are Monte
Carlo (MC) and quasi-Monte Carlo (QMC) methods. MC methods use random samples
to evaluate $f$ and have error $O(\sigma(f)/\sqrt{n})$, where $\sigma(f)$ is
the standard deviation of $f$. QMC methods are based on evaluating $f$ at
explicit point sets with low discrepancy, and as given by the classical
Koksma-Hlawka inequality, they have error
$\widetilde{O}(\sigma_{\mathsf{HK}}(f)/n)$, where $\sigma_{\mathsf{HK}}(f)$ is
the variation of $f$ in the sense of Hardy and Krause. These two methods have
distinctive advantages and shortcomings, and a fundamental question is to find
a method that combines the advantages of both.
In this work, we give a simple randomized algorithm that produces QMC point
sets with the following desirable features: (1) It achieves substantially
better error than given by the classical Koksma-Hlawka inequality. In
particular, it has error $\widetilde{O}(\sigma_{\mathsf{SO}}(f)/n)$, where
$\sigma_{\mathsf{SO}}(f)$ is a new measure of variation that we introduce,
which is substantially smaller than the Hardy-Krause variation. (2) The
algorithm only requires random samples from the underlying distribution, which
makes it as flexible as MC. (3) It automatically achieves the best of both MC
and QMC (and the above improvement over Hardy-Krause variation) in an optimal
way. (4) The algorithm is extremely efficient, with an amortized
$\widetilde{O}(1)$ runtime per sample.
Our method is based on the classical transference principle in geometric
discrepancy, combined with recent algorithmic innovations in combinatorial
discrepancy that besides producing low-discrepancy colorings, also guarantee
certain subgaussian properties. This allows us to bypass several limitations of
previous works in bridging the gap between MC and QMC methods and go beyond the
Hardy-Krause variation.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsImproved Counting under Continual Observation with Pure Differential
Privacyhttp://arxiv.org/abs/2408.07021v1
http://arxiv.org/abs/2408.07021v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Joel+Daniel+Andersson">Joel Daniel Andersson</a>, <a href="https://dblp.uni-trier.de/search?q=Rasmus+Pagh">Rasmus Pagh</a>, <a href="https://dblp.uni-trier.de/search?q=Sahel+Torkamani">Sahel Torkamani</a></p>Counting under continual observation is a well-studied problem in the area of
differential privacy. Given a stream of updates $x_1,x_2,\dots,x_T \in \{0,1\}$
the problem is to continuously release estimates of the prefix sums
$\sum_{i=1}^t x_i$ for $t=1,\dots,T$ while protecting each input $x_i$ in the
stream with differential privacy. Recently, significant leaps have been made in
our understanding of this problem under $\textit{approximate}$ differential
privacy, aka. $(\varepsilon,\delta)$$\textit{-differential privacy}$. However,
for the classical case of $\varepsilon$-differential privacy, we are not aware
of any improvement in mean squared error since the work of Honaker (TPDP 2015).
In this paper we present such an improvement, reducing the mean squared error
by a factor of about 4, asymptotically. The key technique is a new
generalization of the binary tree mechanism that uses a $k$-ary number system
with $\textit{negative digits}$ to improve the privacy-accuracy trade-off. Our
mechanism improves the mean squared error over all 'optimal'
$(\varepsilon,\delta)$-differentially private factorization mechanisms based on
Gaussian noise whenever $\delta$ is sufficiently small. Specifically, using
$k=19$ we get an asymptotic improvement over the bound given in the work by
Henzinger, Upadhyay and Upadhyay (SODA 2023) when $\delta = O(T^{-0.92})$.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsA $5/4$ Approximation for Two-Edge-Connectivityhttp://arxiv.org/abs/2408.07019v1
http://arxiv.org/abs/2408.07019v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Miguel+Bosch-Calvo">Miguel Bosch-Calvo</a>, <a href="https://dblp.uni-trier.de/search?q=Fabrizio+Grandoni">Fabrizio Grandoni</a>, <a href="https://dblp.uni-trier.de/search?q=Afrouz+Jabal+Ameli">Afrouz Jabal Ameli</a></p>The $2$-Edge Connected Spanning Subgraph problem (2ECSS) is among the most
basic survivable network design problems: given an undirected unweighted graph,
find a subgraph with the minimum number of edges which is 2-edge-connected
(i.e., it remains connected after the removal of any single edge). This NP-hard
problem is well-studied in terms of approximation algorithms. The current-best
approximation factor for 2ECSS is $1.3+\varepsilon$ for any constant
$\varepsilon >0$ [Garg, Grandoni, Jabal-Ameli'23; Kobayashi,Noguchi'23]. In
this paper we present a much simpler $9/7$ approximation algorithm, and a more
complex $5/4$ one. Our algorithms are also faster: their running time is
$n^{O(1)}$ instead of $n^{O(1/\varepsilon)}$.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsFaster Private Minimum Spanning Treeshttp://arxiv.org/abs/2408.06997v1
http://arxiv.org/abs/2408.06997v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Rasmus+Pagh">Rasmus Pagh</a>, <a href="https://dblp.uni-trier.de/search?q=Lukas+Retschmeier">Lukas Retschmeier</a></p>Motivated by applications in clustering and synthetic data generation, we
consider the problem of releasing a minimum spanning tree (MST) under
edge-weight differential privacy constraints where a graph topology $G=(V,E)$
with $n$ vertices and $m$ edges is public, the weight matrix $\vec{W}\in
\mathbb{R}^{n \times n}$ is private, and we wish to release an approximate MST
under $\rho$-zero-concentrated differential privacy. Weight matrices are
considered neighboring if they differ by at most $\Delta_\infty$ in each entry,
i.e., we consider an $\ell_\infty$ neighboring relationship. Existing private
MST algorithms either add noise to each entry in $\vec{W}$ and estimate the MST
by post-processing or add noise to weights in-place during the execution of a
specific MST algorithm. Using the post-processing approach with an efficient
MST algorithm takes $O(n^2)$ time on dense graphs but results in an additive
error on the weight of the MST of magnitude $O(n^2\log n)$. In-place algorithms
give asymptotically better utility, but the running time of existing in-place
algorithms is $O(n^3)$ for dense graphs. Our main result is a new
differentially private MST algorithm that matches the utility of existing
in-place methods while running in time $O(m + n^{3/2}\log n)$ for fixed privacy
parameter $\rho$. The technical core of our algorithm is an efficient sublinear
time simulation of Report-Noisy-Max that works by discretizing all edge weights
to a multiple of $\Delta_\infty$ and forming groups of edges with identical
weights. Specifically, we present a data structure that allows us to sample a
noisy minimum weight edge among at most $O(n^2)$ cut edges in $O(\sqrt{n} \log
n)$ time. Experimental evaluations support our claims that our algorithm
significantly improves previous algorithms either in utility or running time.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsEngineering Hypergraph $b$-Matching Algorithmshttp://arxiv.org/abs/2408.06924v1
http://arxiv.org/abs/2408.06924v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Ernestine+Gro%C3%9Fmann">Ernestine Großmann</a>, <a href="https://dblp.uni-trier.de/search?q=Felix+Joos">Felix Joos</a>, <a href="https://dblp.uni-trier.de/search?q=Henrik+Reinst%C3%A4dtler">Henrik Reinstädtler</a>, <a href="https://dblp.uni-trier.de/search?q=Christian+Schulz">Christian Schulz</a></p>Recently, researchers have extended the concept of matchings to the more
general problem of finding $b$-matchings in hypergraphs broadening the scope of
potential applications and challenges. The concept of $b$-matchings, where $b$
is a function that assigns positive integers to the vertices of the graph, is a
natural extension of matchings in graphs, where each vertex $v$ is allowed to
be matched to up to $b(v)$ edges, rather than just one. The weighted
$b$-matching problem then seeks to select a subset of the hyperedges that
fulfills the constraint and maximizes the weight.
In this work, we engineer novel algorithms for this generalized problem. More
precisely, we introduce exact data reductions for the problem as well as a
novel greedy initial solution and local search algorithms. These data
reductions allow us to significantly shrink the input size. This is done by
either determining if a hyperedge is guaranteed to be in an optimum
$b$-matching and thus can be added to our solution or if it can be safely
ignored. Our iterated local search algorithm provides a framework for finding
suitable improvement swaps of edges.
Experiments on a wide range of real-world hypergraphs show that our new set
of data reductions are highly practical, and our initial solutions are
competitive for graphs and hypergraphs as well.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsBetter Gaussian Mechanism using Correlated Noisehttp://arxiv.org/abs/2408.06853v1
http://arxiv.org/abs/2408.06853v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Christian+Janos+Lebeda">Christian Janos Lebeda</a></p>We present a simple variant of the Gaussian mechanism for answering
differentially private queries when the sensitivity space has a certain common
structure. Our motivating problem is the fundamental task of answering $d$
counting queries under the add/remove neighboring relation. The standard
Gaussian mechanism solves this task by adding noise distributed as a Gaussian
with variance scaled by $d$ independently to each count. We show that adding a
random variable distributed as a Gaussian with variance scaled by $(\sqrt{d} +
1)/4$ to all counts allows us to reduce the variance of the independent
Gaussian noise samples to scale only with $(d + \sqrt{d})/4$. The total noise
added to each counting query follows a Gaussian distribution with standard
deviation scaled by $(\sqrt{d} + 1)/2$ rather than $\sqrt{d}$. The central idea
of our mechanism is simple and the technique is flexible. We show that applying
our technique to another problem gives similar improvements over the standard
Gaussian mechanism.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsFaster Lattice Basis Computation -- The Generalization of the Euclidean
Algorithmhttp://arxiv.org/abs/2408.06685v1
http://arxiv.org/abs/2408.06685v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Kim-Manuel+Klein">Kim-Manuel Klein</a>, <a href="https://dblp.uni-trier.de/search?q=Janina+Reuter">Janina Reuter</a></p>The Euclidean algorithm the oldest algorithms known to mankind. Given two
integral numbers $a_1$ and $a_2$, it computes the greatest common divisor (gcd)
of $a_1$ and $a_2$ in a very elegant way. From a lattice perspective, it
computes a basis of the sum of two one-dimensional lattices $a_1 \mathbb{Z}$
and $a_2 \mathbb{Z}$ as $\gcd(a_1,a_2) \mathbb{Z} = a_1 \mathbb{Z} + a_2
\mathbb{Z}$. In this paper, we show that the classical Euclidean algorithm can
be adapted in a very natural way to compute a basis of a general lattice $L
(A_1, \ldots , A_n)$ given vectors $A_1, \ldots , A_n \in \mathbb{Z}^d$ with
$n> \mathrm{rank}(a_1, \ldots ,a_d)$. Similar to the Euclidean algorithm, our
algorithm is very easy to describe and implement and can be written within 12
lines of pseudocode.
Our generalized version of the Euclidean algorithm allows for several degrees
of freedom in the pivoting process. Hence, in a second step, we show that this
freedom can be exploited to make the algorithm perform more efficiently. As our
main result, we obtain an algorithm to compute a lattice basis for given
vectors $A_1, \ldots , A_n \in \mathbb{Z}^d$ in time (counting bit operations)
$LS + \tilde O ((n-d)d^2 \cdot \log(||A||)$, where $LS$ is the time required to
obtain the exact fractional solution of a certain system of linear equalities.
The analysis of the running time of our algorithms relies on fundamental
statements on the fractionality of solutions of linear systems of equations.
So far, the fastest algorithm for lattice basis computation was due to
Storjohann and Labhan [SL96] having a running time of $\tilde O (nd^\omega\log
||A||)$. For current upper bounds of $LS$, our algorithm has a running time
improvement of a factor of at least $d^{0.12}$ over [SL96]. Our algorithm is
therefore the first general algorithmic improvement to this classical problem
in nearly 30 years.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsPath Partitions of Phylogenetic Networkshttp://arxiv.org/abs/2408.06489v1
http://arxiv.org/abs/2408.06489v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Manuel+Lafond">Manuel Lafond</a>, <a href="https://dblp.uni-trier.de/search?q=Vincent+Moulton">Vincent Moulton</a></p>In phylogenetics, evolution is traditionally represented in a tree-like
manner. However, phylogenetic networks can be more appropriate for representing
evolutionary events such as hybridization, horizontal gene transfer, and
others. In particular, the class of forest-based networks was recently
introduced to represent introgression, in which genes are swapped between
between species. A network is forest-based if it can be obtained by adding arcs
to a collection of trees, so that the endpoints of the new arcs are in
different trees. This contrasts with so-called tree-based networks, which are
formed by adding arcs within a single tree.
We are interested in the computational complexity of recognizing forest-based
networks, which was recently left as an open problem by Huber et al.
Forest-based networks coincide with directed acyclic graphs that can be
partitioned into induced paths, each ending at a leaf of the original graph.
Several types of path partitions have been studied in the graph theory
literature, but to our knowledge this type of leaf induced path partition has
not been considered before. The study of forest-based networks in terms of
these partitions allows us to establish closer relationships between
phylogenetics and algorithmic graph theory, and to provide answers to problems
in both fields.
We show that deciding whether a network is forest-based is NP-complete, even
on input networks that are tree-based, binary, and have only three leaves. This
shows that partitioning a directed acyclic graph into three induced paths is
NP-complete, answering a recent question of Ferneau et al. We then show that
the problem is polynomial-time solvable on binary networks with two leaves and
on the class of orchards. Finally, for undirected graphs, we introduce unrooted
forest-based networks and provide hardness results for this class as well.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsMassively Parallel Minimum Spanning Tree in General Metric Spaceshttp://arxiv.org/abs/2408.06455v1
http://arxiv.org/abs/2408.06455v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Amir+Azarmehr">Amir Azarmehr</a>, <a href="https://dblp.uni-trier.de/search?q=Soheil+Behnezhad">Soheil Behnezhad</a>, <a href="https://dblp.uni-trier.de/search?q=Rajesh+Jayaram">Rajesh Jayaram</a>, <a href="https://dblp.uni-trier.de/search?q=Jakub+%C5%81%C4%85cki">Jakub Łącki</a>, <a href="https://dblp.uni-trier.de/search?q=Vahab+Mirrokni">Vahab Mirrokni</a>, <a href="https://dblp.uni-trier.de/search?q=Peilin+Zhong">Peilin Zhong</a></p>We study the minimum spanning tree (MST) problem in the massively parallel
computation (MPC) model. Our focus is particularly on the *strictly sublinear*
regime of MPC where the space per machine is $O(n^\delta)$. Here $n$ is the
number of vertices and constant $\delta \in (0, 1)$ can be made arbitrarily
small. The MST problem admits a simple and folklore $O(\log n)$-round algorithm
in the MPC model. When the weights can be arbitrary, this matches a conditional
lower bound of $\Omega(\log n)$ which follows from a well-known 1vs2-Cycle
conjecture. As such, much of the literature focuses on breaking the logarithmic
barrier in more structured variants of the problem, such as when the vertices
correspond to points in low- [ANOY14, STOC'14] or high-dimensional Euclidean
spaces [JMNZ, SODA'24].
In this work, we focus more generally on metric spaces. Namely, all pairwise
weights are provided and guaranteed to satisfy the triangle inequality, but are
otherwise unconstrained. We show that for any $\varepsilon > 0$, a
$(1+\varepsilon)$-approximate MST can be found in $O(\log \frac{1}{\varepsilon}
+ \log \log n)$ rounds, which is the first $o(\log n)$-round algorithm for
finding any constant approximation in this setting. Other than being applicable
to more general weight functions, our algorithm also slightly improves the
$O(\log \log n \cdot \log \log \log n)$ round-complexity of [JMNZ24, SODA'24]
and significantly improves its approximation from a large constant to
$1+\varepsilon$.
On the lower bound side, we prove that under the 1vs2-Cycle conjecture,
$\Omega(\log \frac{1}{\varepsilon})$ rounds are needed for finding a
$(1+\varepsilon)$-approximate MST in general metrics. It is worth noting that
while many existing lower bounds in the MPC model under the 1vs2-Cycle
conjecture only hold against "component stable" algorithms, our lower bound
applies to *all* algorithms.</body></html>
2024-08-14 00:00:00 UTCarXiv: Data Structures and AlgorithmsGuess Which Way?https://rjlipton.com/?p=24946
https://rjlipton.com/2024/08/13/guess-which-way/
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<p>Math problems get solved from time to time. Today I wonder if they are solved the way we guessed they were going to be solved? For example do most feel that P vs NP is likely to be equal or unequal? </p>
<blockquote><p>
William Gasarch has conducted three polls of researchers concerning this question. Confidence that P is not equal NP has been increasing: In 2002 61% believe unequal; in 2012 83% believe unequal; and in 2019 88% believed unequal. When restricted to experts, the 2019 answers became 99% believed unequal.
</p></blockquote>
<p><a href="https://rjlipton.com/2024/08/13/guess-which-way/wg/" rel="attachment wp-att-24948"><img fetchpriority="high" decoding="async" data-attachment-id="24948" data-permalink="https://rjlipton.com/2024/08/13/guess-which-way/wg/" data-orig-file="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/wg.jpeg?fit=225%2C225&ssl=1" data-orig-size="225,225" data-comments-opened="1" data-image-meta='{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}' data-image-title="wg" data-image-description="" data-image-caption="" data-medium-file="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/wg.jpeg?fit=225%2C225&ssl=1" data-large-file="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/wg.jpeg?fit=225%2C225&ssl=1" tabindex="0" role="button" src="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/wg.jpeg?resize=225%2C225&ssl=1" alt="" width="225" height="225" class="alignnone size-full wp-image-24948" srcset="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/wg.jpeg?w=225&ssl=1 225w, https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/wg.jpeg?resize=150%2C150&ssl=1 150w" sizes="(max-width: 225px) 100vw, 225px" data-recalc-dims="1"></a></p>
<p><strong>Many Are Guessed Right Before Being Solved</strong></p>
<p>Consider the famous: <a href="https://en.wikipedia.org/wiki/Four_color_theorem">4 color theorem</a>.It states that no more than four colors are required to color the regions of any planar map so that no two adjacent regions have the same color. Most seemed to believe that 4 colors was going to be enough. </p>
<p>False disproofs usually violated the assumptions of the theorem. Such as using a region that consists of multiple disconnected parts, or disallowing regions of the same color from touching at a point. They got the statement of the theorem incorrect. Martin Gardner (1975) played an April Fool’s joke by asserting that the McGregor map consisting of 110 regions required five colors and constitutes a counterexample to the four-color theorem.</p>
<p><strong>Some Guessed Wrong Before Being Solved</strong></p>
<p>The famous <a href="https://en.wikipedia.org/wiki/Fermat\%27s_Last_Theorem}">Fermat Last Theorem</a> was that there was no positive integer solution to:<br>
x^n + y^n = z^n<br>
with n >2 and xyz not zero. </p>
<p><a href="https://rjlipton.com/2024/08/13/guess-which-way/kr-3/" rel="attachment wp-att-24949"><img decoding="async" data-attachment-id="24949" data-permalink="https://rjlipton.com/2024/08/13/guess-which-way/kr-3/" data-orig-file="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/kr.jpg?fit=440%2C587&ssl=1" data-orig-size="440,587" data-comments-opened="1" data-image-meta='{"aperture":"0","credit":"","camera":"","caption":"","created_timestamp":"0","copyright":"","focal_length":"0","iso":"0","shutter_speed":"0","title":"","orientation":"0"}' data-image-title="kr" data-image-description="" data-image-caption="" data-medium-file="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/kr.jpg?fit=225%2C300&ssl=1" data-large-file="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/kr.jpg?fit=440%2C587&ssl=1" tabindex="0" role="button" src="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/kr.jpg?resize=225%2C300&ssl=1" alt="" width="225" height="300" class="alignnone size-medium wp-image-24949" srcset="https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/kr.jpg?resize=225%2C300&ssl=1 225w, https://i0.wp.com/rjlipton.com/wp-content/uploads/2024/08/kr.jpg?w=440&ssl=1 440w" sizes="(max-width: 225px) 100vw, 225px" data-recalc-dims="1"></a></p>
<p>The link occurred by contemplating the unthinkable—what would happen if Fermat’s Last Theorem was not true? This would mean that there existed a set of solutions to Fermat’s equation, and therefore this hypothetical combination of numbers could be used as the basis for constructing a hypothetical elliptic curve. Ribet demonstrated that this elliptic curve could not possibly be related to a modular form, and as such it would defy the Shimura-Taniyama conjecture.</p>
<p>Running the argument backwards, if somebody could prove the Shimura-Taniyama conjecture then every elliptic curve must be related to a modular form, hence any solution to Fermat’s equation is forbidden to exist, and hence Fermat’s Theorem must be true. If somebody could prove the Shimura-Taniyama conjecture, then this would immediately imply the proof of Fermat’s Last Theorem. By proving one of the most important conjectures of the twentieth century, mathematicians could solve a riddle from the seventeenth century.</p>
<p><strong>Open Problems</strong></p>
<p>Here are two open problems that are not clear which way we should guess. The first is the <a href="https://www.britannica.com/science/Riemann-hypothesis">Riemann Hypothesis</a> and the second is the <a href="https://www.livescience.com/prime-numbers-twin-proof.html#">twin prime conjecture</a>.</p>
<p>For the Riemann Hypothesis we could guess all the zeroes have real part 1/2. But it is not clear if all believe that this is likely to be the case. The twin prime could be the case that there are infinitely many primes p so that p+2 is also prime. But this might be false as some believe?</p>
<p class="authors">By rjlipton</p>
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2024-08-13 13:50:34 UTCRichard LiptonLuca Trevisan Memorial Session at RANDOM-APPROX 2024http://cstheory-events.org/2024/08/13/luca-trevisan-memorial-session-at-random-approx-2024/
https://cstheory-events.org/2024/08/13/luca-trevisan-memorial-session-at-random-approx-2024/
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<p>August 29, 2024 London School of Economics, London, UK https://approxrandom2024.site/ RANDOM-APPROX 2024 will include a session in memory of Luca Trevisan consisting of a few short talks highlighting Luca’s contributions to pseudorandomness and hardness of approximation, organized by Dieter van Melkebeek. The speakers will be Anupam Gupta (on unique games), Jelani Nelson (on approximation algorithms), … <a href="https://cstheory-events.org/2024/08/13/luca-trevisan-memorial-session-at-random-approx-2024/" class="more-link">Continue reading <span class="screen-reader-text">Luca Trevisan Memorial Session at RANDOM-APPROX 2024</span></a></p>
<p class="authors">By shacharlovett</p>
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2024-08-13 00:42:34 UTCCS Theory EventsComputability of Classification and Deep Learning: From Theoretical
Limits to Practical Feasibility through Quantizationhttp://arxiv.org/abs/2408.06212v1
http://arxiv.org/abs/2408.06212v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Holger+Boche">Holger Boche</a>, <a href="https://dblp.uni-trier.de/search?q=Vit+Fojtik">Vit Fojtik</a>, <a href="https://dblp.uni-trier.de/search?q=Adalbert+Fono">Adalbert Fono</a>, <a href="https://dblp.uni-trier.de/search?q=Gitta+Kutyniok">Gitta Kutyniok</a></p>The unwavering success of deep learning in the past decade led to the
increasing prevalence of deep learning methods in various application fields.
However, the downsides of deep learning, most prominently its lack of
trustworthiness, may not be compatible with safety-critical or
high-responsibility applications requiring stricter performance guarantees.
Recently, several instances of deep learning applications have been shown to be
subject to theoretical limitations of computability, undermining the
feasibility of performance guarantees when employed on real-world computers. We
extend the findings by studying computability in the deep learning framework
from two perspectives: From an application viewpoint in the context of
classification problems and a general limitation viewpoint in the context of
training neural networks. In particular, we show restrictions on the
algorithmic solvability of classification problems that also render the
algorithmic detection of failure in computations in a general setting
infeasible. Subsequently, we prove algorithmic limitations in training deep
neural networks even in cases where the underlying problem is well-behaved.
Finally, we end with a positive observation, showing that in quantized versions
of classification and deep network training, computability restrictions do not
arise or can be overcome to a certain degree.</body></html>
2024-08-13 00:00:00 UTCarXiv: Computational ComplexityQuantum Annealing-Based Algorithm for Efficient Coalition Formation
Among LEO Satelliteshttp://arxiv.org/abs/2408.06007v1
http://arxiv.org/abs/2408.06007v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Supreeth+Mysore+Venkatesh">Supreeth Mysore Venkatesh</a>, <a href="https://dblp.uni-trier.de/search?q=Antonio+Macaluso">Antonio Macaluso</a>, <a href="https://dblp.uni-trier.de/search?q=Marlon+Nuske">Marlon Nuske</a>, <a href="https://dblp.uni-trier.de/search?q=Matthias+Klusch">Matthias Klusch</a>, <a href="https://dblp.uni-trier.de/search?q=Andreas+Dengel">Andreas Dengel</a></p>The increasing number of Low Earth Orbit (LEO) satellites, driven by lower
manufacturing and launch costs, is proving invaluable for Earth observation
missions and low-latency internet connectivity. However, as the number of
satellites increases, the number of communication links to maintain also rises,
making the management of this vast network increasingly challenging and
highlighting the need for clustering satellites into efficient groups as a
promising solution. This paper formulates the clustering of LEO satellites as a
coalition structure generation (CSG) problem and leverages quantum annealing to
solve it. We represent the satellite network as a graph and obtain the optimal
partitions using a hybrid quantum-classical algorithm called GCS-Q. The
algorithm follows a top-down approach by iteratively splitting the graph at
each step using a quadratic unconstrained binary optimization (QUBO)
formulation. To evaluate our approach, we utilize real-world three-line element
set (TLE/3LE) data for Starlink satellites from Celestrak. Our experiments,
conducted using the D-Wave Advantage annealer and the state-of-the-art solver
Gurobi, demonstrate that the quantum annealer significantly outperforms
classical methods in terms of runtime while maintaining the solution quality.
The performance achieved with quantum annealers surpasses the capabilities of
classical computers, highlighting the transformative potential of quantum
computing in optimizing the management of large-scale satellite networks.</body></html>
2024-08-13 00:00:00 UTCarXiv: Computational ComplexityGromov's Approximating Tree and the All-Pairs Bottleneck Paths Problemhttp://arxiv.org/abs/2408.05338v1
http://arxiv.org/abs/2408.05338v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Anders+Cornect">Anders Cornect</a>, <a href="https://dblp.uni-trier.de/search?q=Eduardo+Mart%C3%ADnez-Pedroza">Eduardo Martínez-Pedroza</a></p>Given a pointed metric space $(X,\mathsf{dist}, w)$ on $n$ points, its
Gromov's approximating tree is a 0-hyperbolic pseudo-metric space
$(X,\mathsf{dist}_T)$ such that $\mathsf{dist}(x,w)=\mathsf{dist}_T(x,w)$ and
$\mathsf{dist}(x, y)-2 \delta \log_2n \leq \mathsf{dist}_T (x, y) \leq
\mathsf{dist}(x, y)$ for all $x, y \in X$ where $\delta$ is the Gromov
hyperbolicity of $X$. On the other hand, the all pairs bottleneck paths (APBP)
problem asks, given an undirected graph with some capacities on its edges, to
find the maximal path capacity between each pair of vertices. In this note, we
prove:
$\bullet$ Computing Gromov's approximating tree for a metric space with $n+1$
points from its matrix of distances reduces to solving the APBP problem on an
connected graph with $n$ vertices.
$\bullet$ There is an explicit algorithm that computes Gromov's approximating
tree for a graph from its adjacency matrix in quadratic time.
$\bullet$ Solving the APBP problem on a weighted graph with $n$ vertices
reduces to finding Gromov's approximating tree for a metric space with $n+1$
points from its distance matrix.</body></html>
2024-08-13 00:00:00 UTCarXiv: Computational GeometryRapid Vector-based Any-angle Path Planning with Non-convex Obstacleshttp://arxiv.org/abs/2408.05806v1
http://arxiv.org/abs/2408.05806v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Yan+Kai+Lai">Yan Kai Lai</a></p>Vector-based algorithms are novel algorithms in optimal any-angle path
planning that are motivated by bug algorithms, bypassing free space by directly
conducting line-of-sight checks between two queried points, and searching along
obstacle contours if a check collides with an obstacle. The algorithms
outperform conventional free-space planners such as A* especially when the
queried points are far apart. The thesis presents novel search methods to speed
up vector-based algorithms in non-convex obstacles by delaying line-of-sight
checks. The "best hull" is a notable method that allows for monotonically
increasing path cost estimates even without verifying line-of-sight, utilizing
"phantom points" placed on non-convex corners to mimic future turning points.
Building upon the methods, the algorithms R2 and R2+ are formulated, which
outperform other vector-based algorithms when the optimal path solution is
expected to have few turning points. Other novel methods include a novel and
versatile multi-dimensional ray tracer for occupancy grids, and a description
of the three-dimensional angular sector for future works.</body></html>
2024-08-13 00:00:00 UTCarXiv: Computational GeometryInvariants of almost embeddings of graphs in the plane: results and
problemshttp://arxiv.org/abs/2408.06392v1
http://arxiv.org/abs/2408.06392v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=E.+Alkin">E. Alkin</a>, <a href="https://dblp.uni-trier.de/search?q=E.+Bordacheva">E. Bordacheva</a>, <a href="https://dblp.uni-trier.de/search?q=A.+Miroshnikov">A. Miroshnikov</a>, <a href="https://dblp.uni-trier.de/search?q=O.+Nikitenko">O. Nikitenko</a>, <a href="https://dblp.uni-trier.de/search?q=A.+Skopenkov">A. Skopenkov</a></p>A graph drawing in the plane is called an almost embedding if images of any
two non-adjacent simplices (i.e. vertices or edges) are disjoint. We introduce
integer invariants of almost embeddings: winding number, cyclic and triodic Wu
numbers. We construct almost embeddings realizing some values of these
invariants. We prove some relations between the invariants. We study values
realizable as invariants of some almost embedding, but not of any embedding.
This paper is expository and is accessible to mathematicians not specialized
in the area (and to students). However elementary, this paper is motivated by
frontline of research.</body></html>
2024-08-13 00:00:00 UTCarXiv: Computational GeometryTolerant testing stabilizer stateshttp://arxiv.org/abs/2408.06289v1
http://arxiv.org/abs/2408.06289v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Srinivasan+Arunachalam">Srinivasan Arunachalam</a>, <a href="https://dblp.uni-trier.de/search?q=Arkopal+Dutt">Arkopal Dutt</a></p>We consider the following task: suppose an algorithm is given copies of an
unknown $n$-qubit quantum state $|\psi\rangle$ promised $(i)$ $|\psi\rangle$ is
$\varepsilon_1$-close to a stabilizer state in fidelity or $(ii)$
$|\psi\rangle$ is $\varepsilon_2$-far from all stabilizer states, decide which
is the case. We give a $\textsf{poly}(1/\varepsilon_1)$-sample and $n\cdot
\textsf{poly}(1/\varepsilon_1)$-time algorithm for this task for every
$\varepsilon_1>0$ and $\varepsilon_2\leq 2^{-\textsf{poly}(1/\varepsilon_1)}$.
Our proof includes a new definition of Gowers norm for quantum states, an
inverse theorem for the Gowers-$3$ norm of states and new bounds on stabilizer
covering for structured subsets of Paulis using results in additive
combinatorics.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsSequential non-determinism in tile self-assembly: a general framework
and an application to efficient temperature-1 self-assembly of squareshttp://arxiv.org/abs/2408.06241v1
http://arxiv.org/abs/2408.06241v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=David+Furcy">David Furcy</a>, <a href="https://dblp.uni-trier.de/search?q=Scott+M.+Summers">Scott M. Summers</a></p>In this paper, we work in a 2D version of the probabilistic variant of
Winfree's abstract Tile Assembly Model defined by Chandran, Gopalkrishnan and
Reif (SICOMP 2012) in which attaching tiles are sampled uniformly with
replacement. First, we develop a framework called ``sequential
non-determinism'' for analyzing the probabilistic correctness of a
non-deterministic, temperature-1 tile assembly system (TAS) in which most (but
not all) tile attachments are deterministic and the non-deterministic
attachments always occur in a specific order. Our main sequential
non-determinism result equates the probabilistic correctness of such a TAS to a
finite product of probabilities, each of which 1) corresponds to the
probability of the correct type of tile attaching at a point where it is
possible for two different types to attach, and 2) ignores all other tile
attachments that do not affect the non-deterministic attachment. We then show
that sequential non-determinism allows for efficient and geometrically
expressive self-assembly. To that end, we constructively prove that for any
positive integer $N$ and any real $\delta \in (0,1)$, there exists a TAS that
self-assembles into an $N \times N$ square with probability at least $1 -
\delta$ using only $O\left( \log N + \log \frac{1}{\delta} \right)$ types of
tiles. Our bound improves upon the previous state-of-the-art bound for this
problem by Cook, Fu and Schweller (SODA 2011).</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsDynamic Traffic Assignment for Public Transport with Vehicle Capacitieshttp://arxiv.org/abs/2408.06308v1
http://arxiv.org/abs/2408.06308v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Julian+Patzner">Julian Patzner</a>, <a href="https://dblp.uni-trier.de/search?q=Matthias+M%C3%BCller-Hannemann">Matthias Müller-Hannemann</a></p>Traffic assignment is a core component of many urban transport planning
tools. It is used to determine how traffic is distributed over a transportation
network. We study the task of computing traffic assignments for public
transport: Given a public transit network, a timetable, vehicle capacities and
a demand (i.e. a list of passengers, each with an associated origin,
destination, and departure time), the goal is to predict the resulting
passenger flow and the corresponding load of each vehicle. Microscopic
stochastic simulation of individual passengers is a standard, but
computationally expensive approach. Briem et al. (2017) have shown that a
clever adaptation of the Connection Scan Algorithm (CSA) can lead to highly
efficient traffic assignment algorithms, but ignores vehicle capacities,
resulting in overcrowded vehicles. Taking their work as a starting point, we
here propose a new and extended model that guarantees capacity-feasible
assignments and incorporates dynamic network congestion effects such as crowded
vehicles, denied boarding, and dwell time delays. Moreover, we also incorporate
learning and adaptation of individual passengers based on their experience with
the network. Applications include studying the evolution of perceived travel
times as a result of adaptation, the impact of an increase in capacity, or
network effects due to changes in the timetable such as the addition or the
removal of a service or a whole line. The proposed framework has been
experimentally evaluated with public transport networks of G\"ottingen and
Stuttgart (Germany). The simulation proves to be highly efficient. On a
standard PC the computation of a traffic assignment takes just a few seconds
per simulation day.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsBatched Ranged Random Integer Generationhttp://arxiv.org/abs/2408.06213v1
http://arxiv.org/abs/2408.06213v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Nevin+Brackett-Rozinsky">Nevin Brackett-Rozinsky</a>, <a href="https://dblp.uni-trier.de/search?q=Daniel+Lemire">Daniel Lemire</a></p>Pseudorandom values are often generated as 64-bit binary words. These random
words need to be converted into ranged values without statistical bias. We
present an efficient algorithm to generate multiple independent
uniformly-random bounded integers from a single uniformly-random binary word,
without any bias. In the common case, our method uses one multiplication and no
division operations per value produced. In practice, our algorithm can more
than double the speed of unbiased random shuffling for small to moderately
large arrays.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsSpectral Sparsification by Deterministic Discrepancy Walkhttp://arxiv.org/abs/2408.06146v1
http://arxiv.org/abs/2408.06146v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Lap+Chi+Lau">Lap Chi Lau</a>, <a href="https://dblp.uni-trier.de/search?q=Robert+Wang">Robert Wang</a>, <a href="https://dblp.uni-trier.de/search?q=Hong+Zhou">Hong Zhou</a></p>Spectral sparsification and discrepancy minimization are two well-studied
areas that are closely related. Building on recent connections between these
two areas, we generalize the "deterministic discrepancy walk" framework by
Pesenti and Vladu [SODA~23] for vector discrepancy to matrix discrepancy, and
use it to give a simpler proof of the matrix partial coloring theorem of Reis
and Rothvoss [SODA~20]. Moreover, we show that this matrix discrepancy
framework provides a unified approach for various spectral sparsification
problems, from stronger notions including unit-circle approximation and
singular-value approximation to weaker notions including graphical spectral
sketching and effective resistance sparsification. In all of these
applications, our framework produces improved results with a simpler and
deterministic analysis.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsHigh Probability Low Latency Sequential Change Detection over an Unknown
Finite Horizonhttp://arxiv.org/abs/2408.05817v1
http://arxiv.org/abs/2408.05817v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Yu-Han+Huang">Yu-Han Huang</a>, <a href="https://dblp.uni-trier.de/search?q=Venugopal+V.+Veeravalli">Venugopal V. Veeravalli</a></p>A finite horizon variant of the quickest change detection problem is studied,
in which the goal is to minimize a delay threshold (latency), under constraints
on the probability of false alarm and the probability that the latency is
exceeded. In addition, the horizon is not known to the change detector. A
variant of the cumulative sum (CuSum) test with a threshold that increasing
logarithmically with time is proposed as a candidate solution to the problem.
An information-theoretic lower bound on the minimum value of the latency under
the constraints is then developed. This lower bound is used to establish
certain asymptotic optimality properties of the proposed test in terms of the
horizon and the false alarm probability. Some experimental results are given to
illustrate the performance of the test.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsSeparate Generation and Evaluation for Parallel Greedy Best-First Searchhttp://arxiv.org/abs/2408.05682v1
http://arxiv.org/abs/2408.05682v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Takumi+Shimoda">Takumi Shimoda</a>, <a href="https://dblp.uni-trier.de/search?q=Alex+Fukunaga">Alex Fukunaga</a></p>Parallelization of Greedy Best First Search (GBFS) has been difficult because
straightforward parallelization can result in search behavior which differs
significantly from sequential GBFS, exploring states which would not be
explored by sequential GBFS with any tie-breaking strategy. Recent work has
proposed a class of parallel GBFS algorithms which constrains search to
exploration of the Bench Transition System (BTS), which is the set of states
that can be expanded by GBFS under some tie-breaking policy. However, enforcing
this constraint is costly, as such BTS-constrained algorithms are forced to
spend much of the time waiting so that only states which are guaranteed to be
in the BTS are expanded. We propose an improvement to parallel search which
decouples state generation and state evaluation and significantly improves
state evaluation rate, resulting in better search performance.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsMixing on Generalized Associahedrahttp://arxiv.org/abs/2408.05611v1
http://arxiv.org/abs/2408.05611v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=William+Chang">William Chang</a>, <a href="https://dblp.uni-trier.de/search?q=Colin+Defant">Colin Defant</a>, <a href="https://dblp.uni-trier.de/search?q=Daniel+Frishberg">Daniel Frishberg</a></p>Eppstein and Frishberg recently proved that the mixing time for the simple
random walk on the $1$-skeleton of the associahedron is $O(n^3\log^3 n)$. We
obtain similar rapid mixing results for the simple random walks on the
$1$-skeleta of the type-$B$ and type-$D$ associahedra. We adapt Eppstein and
Frishberg's technique to obtain the same bound of $O(n^3\log^3 n)$ in type $B$
and a bound of $O(n^{13} \log^2 n)$ in type $D$; in the process, we establish
an expansion bound that is tight up to logarithmic factors in type $B$.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsConvergence Analysis for Deep Sparse Coding via Convolutional Neural
Networkshttp://arxiv.org/abs/2408.05540v1
http://arxiv.org/abs/2408.05540v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Jianfei+Li">Jianfei Li</a>, <a href="https://dblp.uni-trier.de/search?q=Han+Feng">Han Feng</a>, <a href="https://dblp.uni-trier.de/search?q=Ding-Xuan+Zhou">Ding-Xuan Zhou</a></p>In this work, we explore the intersection of sparse coding theory and deep
learning to enhance our understanding of feature extraction capabilities in
advanced neural network architectures. We begin by introducing a novel class of
Deep Sparse Coding (DSC) models and establish a thorough theoretical analysis
of their uniqueness and stability properties. By applying iterative algorithms
to these DSC models, we derive convergence rates for convolutional neural
networks (CNNs) in their ability to extract sparse features. This provides a
strong theoretical foundation for the use of CNNs in sparse feature learning
tasks. We additionally extend this convergence analysis to more general neural
network architectures, including those with diverse activation functions, as
well as self-attention and transformer-based models. This broadens the
applicability of our findings to a wide range of deep learning methods for deep
sparse feature extraction. Inspired by the strong connection between sparse
coding and CNNs, we also explore training strategies to encourage neural
networks to learn more sparse features. Through numerical experiments, we
demonstrate the effectiveness of these approaches, providing valuable insights
for the design of efficient and interpretable deep learning models.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsImproved Approximation Guarantees for Joint Replenishment in Continuous
Timehttp://arxiv.org/abs/2408.05443v1
http://arxiv.org/abs/2408.05443v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Danny+Segev">Danny Segev</a></p>The primary objective of this work is to revisit and revitalize one of the
most fundamental models in deterministic inventory management, the
continuous-time joint replenishment problem. Our main contribution consists of
resolving several long-standing open questions in this context. For most of
these questions, we obtain the first quantitative improvement over power-of-$2$
policies and their nearby derivatives, which have been state-of-the-art in
terms of provable performance guarantees since the mid-80's.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsSimple and Nearly-Optimal Sampling for Rank-1 Tensor Completion via
Gauss-Jordanhttp://arxiv.org/abs/2408.05431v1
http://arxiv.org/abs/2408.05431v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Alejandro+Gomez-Leos">Alejandro Gomez-Leos</a>, <a href="https://dblp.uni-trier.de/search?q=Oscar+L%C3%B3pez">Oscar López</a></p>We revisit the sample and computational complexity of completing a rank-1
tensor in $\otimes_{i=1}^{N} \mathbb{R}^{d}$, given a uniformly sampled subset
of its entries. We present a characterization of the problem (i.e. nonzero
entries) which admits an algorithm amounting to Gauss-Jordan on a pair of
random linear systems. For example, when $N = \Theta(1)$, we prove it uses no
more than $m = O(d^2 \log d)$ samples and runs in $O(md^2)$ time. Moreover, we
show any algorithm requires $\Omega(d\log d)$ samples.
By contrast, existing upper bounds on the sample complexity are at least as
large as $d^{1.5} \mu^{\Omega(1)} \log^{\Omega(1)} d$, where $\mu$ can be
$\Theta(d)$ in the worst case. Prior work obtained these looser guarantees in
higher rank versions of our problem, and tend to involve more complicated
algorithms.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsCompetitive Capacitated Online Recoloringhttp://arxiv.org/abs/2408.05370v1
http://arxiv.org/abs/2408.05370v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Rajmohan+Rajaraman">Rajmohan Rajaraman</a>, <a href="https://dblp.uni-trier.de/search?q=Omer+Wasim">Omer Wasim</a></p>In this paper, we revisit the online recoloring problem introduced recently
by Azar et al. In online recoloring, there is a fixed set $V$ of $n$ vertices
and an initial coloring $c_0: V\rightarrow [k]$ for some $k\in
\mathbb{Z}^{>0}$. Under an online sequence $\sigma$ of requests where each
request is an edge $(u_t,v_t)$, a proper vertex coloring $c$ of the graph $G_t$
induced by requests until time $t$ needs to be maintained for all $t$; i.e.,
for any $(u,v)\in G_t$, $c(u)\neq c(v)$. The objective is to minimize the total
weight of vertices recolored for the sequence $\sigma$.
We obtain the first competitive algorithms for capacitated online recoloring
and fully dynamic recoloring. Our first set of results is for $2$-recoloring
using algorithms that are $(1+\varepsilon)$-resource augmented where
$\varepsilon\in (0,1)$ is an arbitrarily small constant. Our main result is an
$O(\log n)$-competitive deterministic algorithm for weighted bipartite graphs,
which is asymptotically optimal in light of an $\Omega(\log n)$ lower bound
that holds for an unbounded amount of augmentation. We also present an $O(n\log
n)$-competitive deterministic algorithm for fully dynamic recoloring, which is
optimal within an $O(\log n)$ factor in light of a $\Omega(n)$ lower bound that
holds for an unbounded amount of augmentation.
Our second set of results is for $\Delta$-recoloring in an
$(1+\varepsilon)$-overprovisioned setting where the maximum degree of $G_t$ is
bounded by $(1-\varepsilon)\Delta$ for all $t$, and each color assigned to at
most $(1+\varepsilon)\frac{n}{\Delta}$ vertices, for an arbitrary $\varepsilon
> 0$. Our main result is an $O(1)$-competitive randomized algorithm for $\Delta
= O(\sqrt{n/\log n})$. We also present an $O(\Delta)$-competitive deterministic
algorithm for $\Delta \le \varepsilon n/2$. Both results are asymptotically
optimal.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsTwo-Edge Connectivity via Pac-Man Gluinghttp://arxiv.org/abs/2408.05282v1
http://arxiv.org/abs/2408.05282v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Mohit+Garg">Mohit Garg</a>, <a href="https://dblp.uni-trier.de/search?q=Felix+Hommelsheim">Felix Hommelsheim</a>, <a href="https://dblp.uni-trier.de/search?q=Alexander+Lindermayr">Alexander Lindermayr</a></p>We study the 2-edge-connected spanning subgraph (2-ECSS) problem: Given a
graph $G$, compute a connected subgraph $H$ of $G$ with the minimum number of
edges such that $H$ is spanning, i.e., $V(H) = V(G)$, and $H$ is
2-edge-connected, i.e., $H$ remains connected upon the deletion of any single
edge, if such an $H$ exists. The $2$-ECSS problem is known to be NP-hard. In
this work, we provide a polynomial-time $(\frac 5 4 +
\varepsilon)$-approximation for the problem for an arbitrarily small
$\varepsilon>0$, improving the previous best approximation ratio of
$\frac{13}{10}+\varepsilon$.
Our improvement is based on two main innovations: First, we reduce solving
the problem on general graphs to solving it on structured graphs with high
vertex connectivity. This high vertex connectivity ensures the existence of a
4-matching across any bipartition of the vertex set with at least 10 vertices
in each part. Second, we exploit this property in a later gluing step, where
isolated 2-edge-connected components need to be merged without adding too many
edges. Using the 4-matching property, we can repeatedly glue a huge component
(containing at least 10 vertices) to other components. This step is reminiscent
of the Pac-Man game, where a Pac-Man (a huge component) consumes all the dots
(other components) as it moves through a maze. These two innovations lead to a
significantly simpler algorithm and analysis for the gluing step compared to
the previous best approximation algorithm, which required a long and tedious
case analysis.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsFast John Ellipsoid Computation with Differential Privacy Optimizationhttp://arxiv.org/abs/2408.06395v1
http://arxiv.org/abs/2408.06395v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Jiuxiang+Gu">Jiuxiang Gu</a>, <a href="https://dblp.uni-trier.de/search?q=Xiaoyu+Li">Xiaoyu Li</a>, <a href="https://dblp.uni-trier.de/search?q=Yingyu+Liang">Yingyu Liang</a>, <a href="https://dblp.uni-trier.de/search?q=Zhenmei+Shi">Zhenmei Shi</a>, <a href="https://dblp.uni-trier.de/search?q=Zhao+Song">Zhao Song</a>, <a href="https://dblp.uni-trier.de/search?q=Junwei+Yu">Junwei Yu</a></p>Determining the John ellipsoid - the largest volume ellipsoid contained
within a convex polytope - is a fundamental problem with applications in
machine learning, optimization, and data analytics. Recent work has developed
fast algorithms for approximating the John ellipsoid using sketching and
leverage score sampling techniques. However, these algorithms do not provide
privacy guarantees for sensitive input data. In this paper, we present the
first differentially private algorithm for fast John ellipsoid computation. Our
method integrates noise perturbation with sketching and leverage score sampling
to achieve both efficiency and privacy. We prove that (1) our algorithm
provides $(\epsilon,\delta)$-differential privacy, and the privacy guarantee
holds for neighboring datasets that are $\epsilon_0$-close, allowing
flexibility in the privacy definition; (2) our algorithm still converges to a
$(1+\xi)$-approximation of the optimal John ellipsoid in
$O(\xi^{-2}(\log(n/\delta_0) + (L\epsilon_0)^{-2}))$ iterations where $n$ is
the number of data point, $L$ is the Lipschitz constant, $\delta_0$ is the
failure probability, and $\epsilon_0$ is the closeness of neighboring input
datasets. Our theoretical analysis demonstrates the algorithm's convergence and
privacy properties, providing a robust approach for balancing utility and
privacy in John ellipsoid computation. This is the first differentially private
algorithm for fast John ellipsoid computation, opening avenues for future
research in privacy-preserving optimization techniques.</body></html>
2024-08-13 00:00:00 UTCarXiv: Data Structures and AlgorithmsTight Time Complexities in Parallel Stochastic Optimization with
Arbitrary Computation Dynamicshttp://arxiv.org/abs/2408.04929v1
http://arxiv.org/abs/2408.04929v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Alexander+Tyurin">Alexander Tyurin</a></p>In distributed stochastic optimization, where parallel and asynchronous
methods are employed, we establish optimal time complexities under virtually
any computation behavior of workers/devices/CPUs/GPUs, capturing potential
disconnections due to hardware and network delays, time-varying computation
powers, and any possible fluctuations and trends of computation speeds. These
real-world scenarios are formalized by our new universal computation model.
Leveraging this model and new proof techniques, we discover tight lower bounds
that apply to virtually all synchronous and asynchronous methods, including
Minibatch SGD, Asynchronous SGD (Recht et al., 2011), and Picky SGD (Cohen et
al., 2021). We show that these lower bounds, up to constant factors, are
matched by the optimal Rennala SGD and Malenia SGD methods (Tyurin &
Richt\'arik, 2023).</body></html>
2024-08-12 00:00:00 UTCarXiv: Computational ComplexitySearching in Euclidean Spaces with Predictionshttp://arxiv.org/abs/2408.04964v1
http://arxiv.org/abs/2408.04964v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Sergio+Cabello">Sergio Cabello</a>, <a href="https://dblp.uni-trier.de/search?q=Panos+Giannopoulos">Panos Giannopoulos</a></p>We study the problem of searching for a target at some unknown location in
$\mathbb{R}^d$ when additional information regarding the position of the target
is available in the form of predictions. In our setting, predictions come as
approximate distances to the target: for each point $p\in \mathbb{R}^d$ that
the searcher visits, we obtain a value $\lambda(p)$ such that $|p\mathbf{t}|\le
\lambda(p) \le c\cdot |p\mathbf{t}|$, where $c\ge 1$ is a fixed constant,
$\mathbf{t}$ is the position of the target, and $|p\mathbf{t}|$ is the
Euclidean distance of $p$ to $\mathbf{t}$. The cost of the search is the length
of the path followed by the searcher. Our main positive result is a strategy
that achieves $(12c)^{d+1}$-competitive ratio, even when the constant $c$ is
unknown. We also give a lower bound of roughly $(c/16)^{d-1}$ on the
competitive ratio of any search strategy in $\mathbb{R}^d$.</body></html>
2024-08-12 00:00:00 UTCarXiv: Computational GeometryDistributed Augmentation, Hypersweeps, and Branch Decomposition of
Contour Trees for Scientific Explorationhttp://arxiv.org/abs/2408.04836v1
http://arxiv.org/abs/2408.04836v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Mingzhe+Li">Mingzhe Li</a>, <a href="https://dblp.uni-trier.de/search?q=Hamish+Carr">Hamish Carr</a>, <a href="https://dblp.uni-trier.de/search?q=Oliver+R%C3%BCbel">Oliver Rübel</a>, <a href="https://dblp.uni-trier.de/search?q=Bei+Wang">Bei Wang</a>, <a href="https://dblp.uni-trier.de/search?q=Gunther+H.+Weber">Gunther H. Weber</a></p>Contour trees describe the topology of level sets in scalar fields and are
widely used in topological data analysis and visualization. A main challenge of
utilizing contour trees for large-scale scientific data is their computation at
scale using high-performance computing. To address this challenge, recent work
has introduced distributed hierarchical contour trees for distributed
computation and storage of contour trees. However, effective use of these
distributed structures in analysis and visualization requires subsequent
computation of geometric properties and branch decomposition to support contour
extraction and exploration. In this work, we introduce distributed algorithms
for augmentation, hypersweeps, and branch decomposition that enable parallel
computation of geometric properties, and support the use of distributed contour
trees as query structures for scientific exploration. We evaluate the parallel
performance of these algorithms and apply them to identify and extract
important contours for scientific visualization.</body></html>
2024-08-12 00:00:00 UTCarXiv: Computational GeometryLocalized Evaluation for Constructing Discrete Vector Fieldshttp://arxiv.org/abs/2408.04769v1
http://arxiv.org/abs/2408.04769v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Tanner+Finken">Tanner Finken</a>, <a href="https://dblp.uni-trier.de/search?q=Julien+Tierny">Julien Tierny</a>, <a href="https://dblp.uni-trier.de/search?q=Joshua+A+Levine">Joshua A Levine</a></p>Topological abstractions offer a method to summarize the behavior of vector
fields but computing them robustly can be challenging due to numerical
precision issues. One alternative is to represent the vector field using a
discrete approach, which constructs a collection of pairs of simplices in the
input mesh that satisfies criteria introduced by Forman's discrete Morse
theory. While numerous approaches exist to compute pairs in the restricted case
of the gradient of a scalar field, state-of-the-art algorithms for the general
case of vector fields require expensive optimization procedures. This paper
introduces a fast, novel approach for pairing simplices of two-dimensional,
triangulated vector fields that do not vary in time. The key insight of our
approach is that we can employ a local evaluation, inspired by the approach
used to construct a discrete gradient field, where every simplex in a mesh is
considered by no more than one of its vertices. Specifically, we observe that
for any edge in the input mesh, we can uniquely assign an outward direction of
flow. We can further expand this consistent notion of outward flow at each
vertex, which corresponds to the concept of a downhill flow in the case of
scalar fields. Working with outward flow enables a linear-time algorithm that
processes the (outward) neighborhoods of each vertex one-by-one, similar to the
approach used for scalar fields. We couple our approach to constructing
discrete vector fields with a method to extract, simplify, and visualize
topological features. Empirical results on analytic and simulation data
demonstrate drastic improvements in running time, produce features similar to
the current state-of-the-art, and show the application of simplification to
large, complex flows.</body></html>
2024-08-12 00:00:00 UTCarXiv: Computational GeometryOn the Number of Non-equivalent Parameterized Squares in a Stringhttp://arxiv.org/abs/2408.04920v1
http://arxiv.org/abs/2408.04920v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Rikuya+Hamai">Rikuya Hamai</a>, <a href="https://dblp.uni-trier.de/search?q=Kazushi+Taketsugu">Kazushi Taketsugu</a>, <a href="https://dblp.uni-trier.de/search?q=Yuto+Nakashima">Yuto Nakashima</a>, <a href="https://dblp.uni-trier.de/search?q=Shunsuke+Inenaga">Shunsuke Inenaga</a>, <a href="https://dblp.uni-trier.de/search?q=Hideo+Bannai">Hideo Bannai</a></p>A string $s$ is called a parameterized square when $s = xy$ for strings $x$,
$y$ and $x$ and $y$ are parameterized equivalent. Kociumaka et al. showed the
number of parameterized squares, which are non-equivalent in parameterized
equivalence, in a string of length $n$ that contains $\sigma$ distinct
characters is at most $2 \sigma! n$ [TCS 2016]. In this paper, we show that the
maximum number of non-equivalent parameterized squares is less than $\sigma n$,
which significantly improves the best-known upper bound by Kociumaka et al.</body></html>
2024-08-12 00:00:00 UTCarXiv: Data Structures and AlgorithmsLocally Private Histograms in All Privacy Regimeshttp://arxiv.org/abs/2408.04888v1
http://arxiv.org/abs/2408.04888v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Cl%C3%A9ment+L.+Canonne">Clément L. Canonne</a>, <a href="https://dblp.uni-trier.de/search?q=Abigail+Gentle">Abigail Gentle</a></p>Frequency estimation, a.k.a. histograms, is a workhorse of data analysis, and
as such has been thoroughly studied under differentially privacy. In
particular, computing histograms in the local model of privacy has been the
focus of a fruitful recent line of work, and various algorithms have been
proposed, achieving the order-optimal $\ell_\infty$ error in the high-privacy
(small $\varepsilon$) regime while balancing other considerations such as time-
and communication-efficiency. However, to the best of our knowledge, the
picture is much less clear when it comes to the medium- or low-privacy regime
(large $\varepsilon$), despite its increased relevance in practice. In this
paper, we investigate locally private histograms, and the very related
distribution learning task, in this medium-to-low privacy regime, and establish
near-tight (and somewhat unexpected) bounds on the $\ell_\infty$ error
achievable. Our theoretical findings emerge from a novel analysis, which
appears to improve bounds across the board for the locally private histogram
problem. We back our theoretical findings by an empirical comparison of
existing algorithms in all privacy regimes, to assess their typical performance
and behaviour beyond the worst-case setting.</body></html>
2024-08-12 00:00:00 UTCarXiv: Data Structures and AlgorithmsA simple quadratic kernel for Token Jumping on surfaceshttp://arxiv.org/abs/2408.04743v1
http://arxiv.org/abs/2408.04743v1
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<p class="arxiv-authors"><b>Authors:</b> <a href="https://dblp.uni-trier.de/search?q=Daniel+W.+Cranston">Daniel W. Cranston</a>, <a href="https://dblp.uni-trier.de/search?q=Moritz+M%C3%BChlenthaler">Moritz Mühlenthaler</a>, <a href="https://dblp.uni-trier.de/search?q=Benjamin+Peyrille">Benjamin Peyrille</a></p>The problem \textsc{Token Jumping} asks whether, given a graph $G$ and two
independent sets of \emph{tokens} $I$ and $J$ of $G$, we can transform $I$ into
$J$ by changing the position of a single token in each step and having an
independent set of tokens throughout. We show that there is a polynomial-time
algorithm that, given an instance of \textsc{Token Jumping}, computes an
equivalent instance of size $O(g^2 + gk + k^2)$, where $g$ is the genus of the
input graph and $k$ is the size of the independent sets.</body></html>
2024-08-12 00:00:00 UTCarXiv: Data Structures and Algorithms