The American political science review

: This report briefly summarizes research on the following topics: game theory and energy; scheduling of large research and development programs; bimatrix games; cost/benefit analyses; measures of worth of weapons systems; hybrid primal algorithm; branch and round algorithm. A listing of papers prepared and published is included. (Author)

http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA049700&Location=U2&doc=GetTRDoc.pdf

Mathematics of Operations Research.

Definitions and basic properties of the Voronoi diagram generalizations of the Voronoi diagram algorithms for computing Voronoi diagrams poisson Voronoi diagrams spatial interpolation models of spatial processes point pattern analysis locational optimization through Voronoi diagrams.

Spatial tessellations. Concepts and Applications of Voronoi diagrams

We show a number of fine-grained hardness results for the Closest Vector Problem in the $\ell_p$ norm ($\mathrm{CVP}_p$), and its approximate and non-uniform variants. First, we show that $\mathrm{CVP}_p$ cannot be solved in $2^{(1-\varepsilon)n}$ time for all $p \notin 2\mathbb{Z}$ and $\varepsilon > 0$, assuming the Strong Exponential Time Hypothesis (SETH). Second, we extend this by showing that there is no $2^{(1-\varepsilon)n}$-time algorithm for approximating $\mathrm{CVP}_p$ to within a constant factor $\gamma$ for such $p$ assuming a "gap" version of SETH, with an explicit relationship between $\gamma$, $p$, and the arity $k = k(\varepsilon)$ of the underlying hard CSP. Third, we show the same hardness result for (exact) $\mathrm{CVP}_p$ with preprocessing (assuming non-uniform SETH). 
For exact "plain" $\mathrm{CVP}_p$, the same hardness result was shown in [Bennett, Golovnev, and Stephens-Davidowitz FOCS 2017] for all but finitely many $p \notin 2\mathbb{Z}$, where the set of exceptions depended on $\varepsilon$ and was not explicit. For the approximate and preprocessing problems, only very weak bounds were known prior to this work. 
We also show that the restriction to $p \notin 2\mathbb{Z}$ is in some sense inherent. In particular, we show that no "natural" reduction can rule out even a $2^{3n/4}$-time algorithm for $\mathrm{CVP}_2$ under SETH. For this, we prove that the possible sets of closest lattice vectors to a target in the $\ell_2$ norm have quite rigid structure, which essentially prevents them from being as expressive as $3$-CNFs. 
We prove these results using techniques from many different fields, including complex analysis, functional analysis, additive combinatorics, and discrete Fourier analysis. E.g., along the way, we give a new (and tighter) proof of Szemeredi's cube lemma for the boolean cube.

Fine-grained hardness of CVP(P) -- Everything that we can prove (and nothing else).

Given a number of obstacles in a plane, the problem of computing a geodesic (or the shortest path) between two points has been studied extensively. However, the case where the obstacles are circular discs has not been explored as much as it deserves. In this paper, we present an algorithm to compute a geodesic among a set of mutually disjoint discs, where the discs can have different radii. We devise two filters, an ellipse filter and a convex hull filter, which can significantly reduce the search space. After filtering, we apply Dijkstra’s algorithm to the remaining discs.

Shortest paths for disc obstacles

Let X = {f1,...,fn} be a set of scalar functions of the form fi : R2 → R which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered e-isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi-algebraic.We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results.

Planar minimization diagrams via subdivision with applications to anisotropic voronoi diagrams

Quadtrees are a well-known data structure for representing geometric data in the plane, and naturally generalize to higher dimensions. A basic operation is to expand the tree by splitting a given leaf. A quadtree is smooth if adjacent leaf boxes differ by at most one in height.

Amortized Analysis of Smooth Quadtrees in All Dimensions

Amortized analysis of smooth quadtrees in all dimensions

We study the complexity of lattice problems in a world where algorithms, reductions, and protocols can run in superpolynomial time. Specifically, we revisit four foundational results in this context—two protocols and two worst-case to average-case reductions. We show how to improve the approximation factor in each result by a factor of roughly √n/logn when running the protocol or reduction in 2є n time instead of polynomial time, and we show a novel protocol with no polynomial-time analog. Our results are as follows. (1) We show a worst-case to average-case reduction proving that secret-key cryptography (specifically, collision-resistant hash functions) exists if the (decision version of the) Shortest Vector Problem (SVP) cannot be approximated to within a factor of Õ(√n) in 2є n time. This extends to our setting Ajtai’s celebrated polynomial-time reduction for the Short Integer Solutions (SIS) problem (1996),which showed (after improvements by Micciancio and Regev (2004, 2007)) that secret-key cryptography exists if SVP cannot be approximated to within a factor of Õ(n) in polynomial time. (2) We show another worst-case to average-case reduction proving that public-key cryptography exists if SVP cannot be approximated to within a factor of Õ(n) in 2є n time. This extends Regev’s celebrated polynomial-time reduction for the Learning with Errors (LWE) problem (2005, 2009), which achieved an approximation factor of Õ(n1.5). In fact, Regev’s reduction is quantum, but we prove our result under a classical reduction, generalizing Peikert’s polynomial-time classical reduction (2009), which achieved an approximation factor of Õ(n2). (3) We show that the (decision version of the) Closest Vector Problem (CVP) with a constant approximation factor has a coAM protocol with a 2є n-time verifier. We prove this via a (very simple) generalization of the celebrated polynomial-time protocol due to Goldreich and Goldwasser (1998, 2000). It follows that the recent series of 2є n-time and even 2(1−є)n-time hardness results for CVP cannot be extended to large constant approximation factors γ unless AMETH is false. We also rule out 2(1−є)n-time lower bounds for any constant approximation factor γ > √2, under plausible complexity-theoretic assumptions. (These results also extend to arbitrary norms, with different constants.) (4) We show that O(√logn)-approximate SVP has a coNTIME protocol with a 2є n-time verifier. Here, the analogous (also celebrated!) polynomial-time result is due to Aharonov and Regev (2005), who showed a polynomial-time protocol achieving an approximation factor of √n (for both SVP and CVP, while we only achieve this result for CVP). This result implies similar barriers to hardness, with a larger approximation factor under a weaker complexity-theoretic conjectures (as does the next result). (5) Finally, we give a novel coMA protocol for constant-factor-approximate CVP with a 2є n-time verifier. Unlike our other results, this protocol has no known analog in the polynomial-time regime. All of the results described above are special cases of more general theorems that achieve time-approximation factor tradeoffs. In particular, the tradeoffs for the first four results smoothly interpolate from the polynomial-time results in prior work to our new results in the exponential-time world.

https://dl.acm.org/doi/pdf/10.1145/3564246.3585227

Lattice Problems beyond Polynomial Time

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Huck Bennett

Author Tools

Chat about Author