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OV Graphs Are (Probably) Hard Instances

Josh Alman, Virginia Vassilevska Williams, Michael Wagner
2020 Innovations in Theoretical Computer Science  
We show that any problem which is NP-hard on constant-degree graphs is also NP-hard on OV graphs of dimension O(log n), and we give two problems which can be solved faster on OV graphs than in general:  ...  A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v1, . . . , vn ∈ {0, 1} d such that nodes i and j are adjacent in G if and only if vi, vj = 0 over Z.  ...  Graphs Are (Probably) Hard Instances Other Related Work OV graphs have been studied in a number of other settings in mathematics and computer science.  ... 
doi:10.4230/lipics.itcs.2020.83 dblp:conf/innovations/AlmanW20 fatcat:343pkhkgfncklhwq4eqrlibq7m

On the Representation of Probabilities over Structured Domains [chapter]

Marius Bozga, Oded Maler
1999 Lecture Notes in Computer Science  
We show h o w probabilistic vectors and matrices can be represented canonically and succinctly using probabilistic trees and graphs, and how s i m ulation of large-scale probabilistic systems can be performed  ...  Acknowledgements: W e are grateful to Moshe Tennenholz for raising the possibility of applying some veri cation techniques to AI problems of planning under uncertainty.  ...  0 1]) whose instances are written as x1 xn (y 1 y n ).  ... 
doi:10.1007/3-540-48683-6_24 fatcat:w3oypmtdc5g6rln6qfvw6ceufa

Nash inequalities for finite Markov chains

P. Diaconis, L. Saloff-Coste
1996 Journal of theoretical probability  
These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth.  ...  These are bounds on (,_-norms in terms of Dirichlet forms and l~-norms which yield decay rates for iterates of the kernel.  ...  Let X, d, n be a finite graph equipped with a probability measure n. Let o~q be a Dirichlet form and assume that ~r and Q are compatible.  ... 
doi:10.1007/bf02214660 fatcat:zemozlwxzjcxbjhphnjiy6gewm

Restricted Isometry Random Variables: Probability Distributions, RIC Prediction and Phase Transition Analysis for Gaussian Encoders [article]

Oliver James, Heung-No Lee
2015 arXiv   pre-print
Associated with a deterministic encoder there are two RICs, namely, the left and the right RIC.  ...  We derive the probability and the cumulative distribution functions of these RIVs for the most widely used i.i.d. Gaussian encoders.  ...  x R x ov ov R x R x R x                            are different, their conclusions are the same.  ... 
arXiv:1410.1956v4 fatcat:yv6fnrgtr5acfm4sunc3676yyi

Stochastic billiards on general tables

Steven N. Evans
2001 The Annals of Applied Probability  
Such c hains are closely related to those that appear in shake and bake s i m ulation algorithms.  ...  By analysing an associated non linear, rst order PDE, we i n vestigate which regions are such t h a t t h i s c hain is reversible with respect to surface measure on the boundary.  ...  It is not hard to see that the annulus example of Remark 1.2 and its higher dimensional analogues are reversible.  ... 
doi:10.1214/aoap/1015345298 fatcat:c4o7uzdktnhlhgrjggmxpl5mea

Asymptotics in the random assignment problem

David Aldous
1992 Probability theory and related fields  
We show that, in the usual probabilistic model for the random assignment problem, the optimal cost tends to a limit constant in probability and in expectation.  ...  Remarks The definition c*=inf~7(g) at (17) is hard to intuit because it is not clear how to produce any example of a function g~fr The existing proofs [,12, 4, 7] that lim sup EC(n)< ov proceed via the  ...  So x-a2(") restricted to D~ is a probability measure, and in the sequel computations with 2 are often done (or implicitly justified) by appeal to these probability measures.  ... 
doi:10.1007/bf01192719 fatcat:unazzpkrjjg3fkuyvnuu5clgdu

New Techniques for Proving Fine-Grained Average-Case Hardness [article]

Mina Dalirrooyfard, Andrea Lincoln, Virginia Vassilevska Williams
2020 arXiv   pre-print
This paper defines new versions of OV, kSUM and zero-k-clique that are both worst-case and average-case fine-grained hard assuming the core hypotheses of FGC.  ...  We show that factored-k-OV and factored kSUM are equivalent and are complete for a class of problems defined over Boolean functions.  ...  Thus there is some positive constant ε (for example ε = −2(1 − (1 + δ ) 1/2 )) such that the probability there are no orthogonal vectors in a #OV µ,d instance is at least 1 − 1/n ε .  ... 
arXiv:2008.06591v1 fatcat:wsec36opmfcdvoa3fehljcvbdm

Average-case fine-grained hardness

Marshall Ball, Alon Rosen, Manuel Sabin, Prashant Nalini Vasudevan
2017 Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing - STOC 2017  
We present functions that can be computed in some fixed polynomial time but are hard on average for any algorithm that runs in slightly smaller time, assuming widely-conjectured worst-case hardness for  ...  The conjectured hardness of OV and 3SUM then gives us functions that require n 2−o(1) time to compute on average, and that of APSP gives us a function that requires n 3−o(1) time.  ...  Are there samplable distributions over which well-studied problems like 3SUM, OV, etc. are hard on average?  ... 
doi:10.1145/3055399.3055466 dblp:conf/stoc/BallRSV17 fatcat:a7xcqudairelvngogbqi245ltu

Probable Trends in the Irradiation Treatment of Carcinoma of the Cervix Uteri with the Improved Expanding Type of Radium Applicator

Edwin C. Ernst
1949 Radiology  
The graph is similar in construction to that of Figure 8, but the exposures are 8,000 mg. hrs.  ...  I would like to say, in summary, that if we will work hard, we will get better results. We are having a lot of fun, we technicians, but the real heart of this problem is getting the patient in.  ... 
doi:10.1148/52.1.46 pmid:18107154 fatcat:2epvfeihlnf4phhs464dio2ptu

Settling SETH vs. Approximate Sparse Directed Unweighted Diameter (up to (NU)NSETH) [article]

Ray Li
2021 arXiv   pre-print
We prove additional hardness and non-reducibility results for undirected graphs.  ...  This settles the SETH-hardness of approximating the diameter of sparse directed unweighted graphs for deterministic reductions, up to NSETH.  ...  Recall that D /D-Diameter is a promise problem (Π Y ES , Π N O ) where the instances in Π Y ES are graphs of diameter at least D , and the instances in Π N O are graphs of diameter at most D.  ... 
arXiv:2008.05106v3 fatcat:lbxwv33pgne6haxecgt4iqenvm

Hardness for Triangle Problems under Even More Believable Hypotheses: Reductions from Real APSP, Real 3SUM, and OV [article]

Timothy M. Chan, Virginia Vassilevska Williams, Yinzhan Xu
2022 arXiv   pre-print
. * Real APSP and OV hardness for a variant of the Boolean Matrix Multiplication problem.  ...  The 3SUM hypothesis, the APSP hypothesis and SETH are the three main hypotheses in fine-grained complexity.  ...  Thus basing hardness on OV is arguably better than basing hardness on k-OV for k > 2, and basing hardness on any fixed k-OV is better than basing hardness on SETH, as all one needs to do to refute SETH  ... 
arXiv:2203.08356v2 fatcat:c2t2jzuf6ngbzfdpy7sonn5mzu

The Orthogonal Vectors Conjecture for Branching Programs and Formulas [article]

Daniel Kane, Ryan Williams
2017 arXiv   pre-print
> 0 such that for every n and d, OV instances where input bits are independently set to 1 with probability p (and 0 otherwise) can be solved with AC^0 formulas of size O(n^2-ϵ_p), on all but a o_n(1) fraction  ...  of instances.  ...  We are very grateful to Ramamohan Paturi for raising the question of whether the OV conjecture is true for AC0 circuits.  ... 
arXiv:1709.05294v1 fatcat:npnnoltelrfmpgvquj4i7z22om

Oblivious Stacking and MAX k-CUT for Circle Graphs [article]

Martin Olsen
2021 arXiv   pre-print
Our results can also be expressed within the context of the MAX k-CUT problem for circle graphs.  ...  We say that two items are conflicting if their storage time intervals overlap in which case one of the items needs to be rehandled if the items are stored at the same LIFO storage location.  ...  The conditional probability Pr(OV | C = xL) is computed as the area of the gray region divided by L 2 : Pr(OV | C = xL) = 1 − 2x 2 − (1 − 2x) 2 = 4x − 6x 2 , x ≤ 0.5 .  ... 
arXiv:2106.02113v1 fatcat:y67tm2knvvholer2236n5beoha

A Structural Investigation of the Approximability of Polynomial-Time Problems [article]

Karl Bringmann, Alejandro Cassis, Nick Fischer, Marvin Künnemann
2022 arXiv   pre-print
Assuming central hypotheses about clique detection in hypergraphs and MAX3SAT, we show that for any MaxSP_k problem definable by a quantifier-free m-edge graph formula ϕ, the best possible approximation  ...  OV graphs are (probably) hard instances. In Proceedings of the 11th Conference on Innovations in Theoretical Computer Science, volume 151 of ITCS '20, pages 83:1-83:18.  ...  An instance of k-OV on dimension d = c log n is reduced to multiple instances of k-MaxIP on dimension exp(c) log n. 11 Morally, just as SETH implies the hardness of low-dimensional OV (see Appendix B),  ... 
arXiv:2204.11681v1 fatcat:ttbagbimjvgvjp7law57zpblde

Time-Aware Probabilistic Knowledge Graphs

Melisachew Wudage Chekol, Heiner Stuckenschmidt, Michael Wagner
2019 International Symposium/Workshop on Temporal Representation and Reasoning  
The emergence of open information extraction as a tool for constructing and expanding knowledge graphs has aided the growth of temporal data, for instance, YAGO, NELL and Wikidata.  ...  Due to the imprecise nature of the extraction tools that are used to build and expand KG, such as NELL, the facts in the KG are weighted (a confidence value representing the correctness of a fact).  ...  The above definition shows that temporal scoping of instance assertions does not increase the complexity (since the ov function can be computed during ground which is before probability computation).  ... 
doi:10.4230/lipics.time.2019.8 dblp:conf/time/ChekolS19 fatcat:2mkqavw5pvfwjeb3juamg7fu5q
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