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OV Graphs Are (Probably) Hard Instances

2020
*
Innovations in Theoretical Computer Science
*

We show that any problem which is NP-

doi:10.4230/lipics.itcs.2020.83
dblp:conf/innovations/AlmanW20
fatcat:343pkhkgfncklhwq4eqrlibq7m
*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. ...##
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On the Representation of Probabilities over Structured Domains
[chapter]

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

doi:10.1007/3-540-48683-6_24
fatcat:w3oypmtdc5g6rln6qfvw6ceufa
*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 ). ...##
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Nash inequalities for finite Markov chains

1996
*
Journal of theoretical probability
*

These

doi:10.1007/bf02214660
fatcat:zemozlwxzjcxbjhphnjiy6gewm
*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. ...##
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Restricted Isometry Random Variables: Probability Distributions, RIC Prediction and Phase Transition Analysis for Gaussian Encoders
[article]

2015
*
arXiv
*
pre-print

Associated with a deterministic encoder there

arXiv:1410.1956v4
fatcat:yv6fnrgtr5acfm4sunc3676yyi
*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. ...##
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Stochastic billiards on general tables

2001
*
The Annals of Applied Probability
*

Such c hains

doi:10.1214/aoap/1015345298
fatcat:c4o7uzdktnhlhgrjggmxpl5mea
*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. ...##
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Asymptotics in the random assignment problem

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

doi:10.1007/bf01192719
fatcat:unazzpkrjjg3fkuyvnuu5clgdu
*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. ...##
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New Techniques for Proving Fine-Grained Average-Case Hardness
[article]

2020
*
arXiv
*
pre-print

This paper defines new versions of

arXiv:2008.06591v1
fatcat:wsec36opmfcdvoa3fehljcvbdm
*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 . ...##
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Average-case fine-grained hardness

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

doi:10.1145/3055399.3055466
dblp:conf/stoc/BallRSV17
fatcat:a7xcqudairelvngogbqi245ltu
*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? ...##
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Probable Trends in the Irradiation Treatment of Carcinoma of the Cervix Uteri with the Improved Expanding Type of Radium Applicator

1949
*
Radiology
*

The

doi:10.1148/52.1.46
pmid:18107154
fatcat:2epvfeihlnf4phhs464dio2ptu
*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. ...##
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Settling SETH vs. Approximate Sparse Directed Unweighted Diameter (up to (NU)NSETH)
[article]

2021
*
arXiv
*
pre-print

We prove additional

arXiv:2008.05106v3
fatcat:lbxwv33pgne6haxecgt4iqenvm
*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. ...##
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Hardness for Triangle Problems under Even More Believable Hypotheses: Reductions from Real APSP, Real 3SUM, and OV
[article]

2022
*
arXiv
*
pre-print

. * Real APSP and

arXiv:2203.08356v2
fatcat:c2t2jzuf6ngbzfdpy7sonn5mzu
*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 ...##
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The Orthogonal Vectors Conjecture for Branching Programs and Formulas
[article]

2017
*
arXiv
*
pre-print

> 0 such that for every n and d,

arXiv:1709.05294v1
fatcat:npnnoltelrfmpgvquj4i7z22om
*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. ...##
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Oblivious Stacking and MAX k-CUT for Circle Graphs
[article]

2021
*
arXiv
*
pre-print

Our results can also be expressed within the context of the MAX k-CUT problem for circle

arXiv:2106.02113v1
fatcat:y67tm2knvvholer2236n5beoha
*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 . ...##
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A Structural Investigation of the Approximability of Polynomial-Time Problems
[article]

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

arXiv:2204.11681v1
fatcat:ttbagbimjvgvjp7law57zpblde
*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), ...##
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Time-Aware Probabilistic Knowledge Graphs

2019
*
International Symposium/Workshop on Temporal Representation and Reasoning
*

The emergence of open information extraction as a tool for constructing and expanding knowledge

doi:10.4230/lipics.time.2019.8
dblp:conf/time/ChekolS19
fatcat:2mkqavw5pvfwjeb3juamg7fu5q
*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). ...
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