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Dense Subset Sum may be the hardest
[article]

2015
*
arXiv
*
pre-print

*The*

*Subset*

*Sum*problem asks whether a given set of n positive integers contains a

*subset*of elements that

*sum*up to a given target t. ... Continuing an earlier work [STACS 2015], we study

*Subset*

*Sum*parameterized by

*the*maximum bin size β, defined as

*the*largest number of

*subsets*of

*the*n input integers that yield

*the*same

*sum*. ... An optimist

*may*interpret this as an indication that truly faster algorithms indeed exist, while a pessimist

*may*conclude

*the*remaining instances must

*be*

*the*(strictly)

*hardest*instances. ...

##
###
Dense Subset Sum May Be the Hardest

2016
*
Symposium on Theoretical Aspects of Computer Science
*

*The*

*Subset*

*Sum*problem asks whether a given set of n positive integers contains a

*subset*of elements that

*sum*up to a given target t. ... Continuing an earlier work [STACS 2015], we study

*Subset*

*Sum*parameterized by

*the*maximum bin size β, defined as

*the*largest number of

*subsets*of

*the*n input integers that yield

*the*same

*sum*. ... σ 2 ± o γ (1) , |z −1 (1)| n = 1 − σ 2 + (1 − τ )σ 2 ± o γ (1) , |z −1 (2)| n = (1 − τ )σ 2 ± o γ (1) , 13:12

*Dense*

*Subset*

*Sum*

*May*

*Be*

*the*

*Hardest*Further Discussion Our original ambition was to resolve ...

##
###
Dense Subset Sum May Be the Hardest

unpublished

*The*

*Subset*

*Sum*problem asks whether a given set of n positive integers contains a

*subset*of elements that

*sum*up to a given target t. ... Continuing an earlier work [STACS 2015], we study

*Subset*

*Sum*parameterized by

*the*maximum bin size β, defined as

*the*largest number of

*subsets*of

*the*n input integers that yield

*the*same

*sum*. ... An optimist

*may*interpret this as an indication that truly faster algorithms indeed exist, while a pessimist

*may*conclude

*the*remaining instances must

*be*

*the*(strictly)

*hardest*instances. ...

##
###
From Approximate to Optimal Solutions: Constructing Pruning and Propagation Rules

1997
*
International Joint Conference on Artificial Intelligence
*

We show that

dblp:conf/ijcai/GentW97
fatcat:nte4xt7ibzh4lkwwmyeb36zvtq
*the*pruning rule adds essentially a constant factor overhead to runtime, whilst reducing search significantly. On*the**hardest*problems, runtime can*be*reduced by an order of magnitude. ...*be*effective on other NP-complete problems. ...*Subset**sum*Given a set is there a*subset*whose weights have a target*sum**Subset**sum*is problem SP13 in [Garey and Johnson, 1979] . ...##
###
Sharp estimate on the supremum of a class of partial sums of small i.i.d. random variables
[article]

2014
*
arXiv
*
pre-print

In a subsequent paper [2] we shall give a sharp bound for

arXiv:1407.1224v1
fatcat:palzma2dfnbmpjlklvm3ee5tk4
*the*supremum of normalized*sums*of i.i.d. random variables in a more general case. ... We take an L_1-*dense*class of functions F on a measurable space (X, X) together with a sequence of independent, identically distributed X-space valued random variables ξ_1,... ... . , x N }*be*a finite set of N elements, and let X*be**the*σ-algebra consisting of all*subsets*of X. ...##
###
Non-injective knapsack public-key cryptosystems

2001
*
Theoretical Computer Science
*

Decryption is fairly slow and

doi:10.1016/s0304-3975(99)00297-2
fatcat:obysbpyseve4dluzllwmrg6bpu
*may*produce more than one decipherment, but all alternative decipherments can*be*found. Disambiguating protocols are needed to determine*the*correct decipherment. ... Two public-key 0-1 knapsack cryptosystems are proposed, that have so high a density and use so weak a modular multiplication as a trapdoor, that known attacks can*be*avoided. ...*The*initial knapsack B 0 is to*be**dense*, injective and small enough for complete enumeration of its*subset**sums*. Decipherment can thus*be*carried out by table lookup. ...##
###
Vector lattice approach to the Riesz and Stone Representation theorems
[article]

2021
*
arXiv
*
pre-print

Let h ∈ F (A)

arXiv:2104.01576v1
fatcat:4r3c6iobubesrhrxqehvi7m6eu
*be*such that f, −f ≤ h. Since Qg does not depend on*the*disjoint representation, we*may*assume that h has*the*same one as f , i.e. there is w = γ i a i , such that Qw = h. ... In this article we focus on*the*"*hardest*part", and in our proof most of*the*heavy lifting is done by*the*vector lattice theory. ...##
###
A Hybrid-Similarity Exemplar Model for Predicting Distinctiveness Effects in Perceptual Old-New Recognition

2003
*
Journal of Experimental Psychology. Learning, Memory and Cognition
*

In this case, large old-item distinctive effects were observed, with

doi:10.1037/0278-7393.29.6.1194
pmid:14622055
fatcat:mg4xnhtgcvcdvhdmswkyydg5de
*the*nature of*the*effects*being*modulated by*the*structure of*the*test lists. ... In this case, any beneficial effects of distinctiveness were absent or small, regardless of*the*structure of*the*test list used to assess recognition memory. ... One concern that might*be*raised is that*the*proportion of old versus new items varied across*the*three testing conditions (complete set, fixed*subset*, and correlated*subset*). ...##
###
Sharp estimate on the supremum of a class of sums of small i.i.d. random variables

2016
*
Stochastic Processes and their Applications
*

Because of

doi:10.1016/j.spa.2015.07.016
fatcat:jmpkodqdpbbk5pkuyi6iq54wum
*the*special form of*the*functionsf j if a set of functionsF ε,ν ⊂F is an ε-*dense**subset*ofF in*the*L 1 (ν) norm, then it is also a ε-*dense**subset*ofF in*the*L 1 (ν) norm. ... elements which is an ε-*dense*set in F with respect to*the*L 1 (ν) norm with a probability measure ν, and f j ∈ F, but these functions*may**be*not contained in F ′ . ...##
###
Coarser connected topologies

2004
*
Topology and its Applications
*

In

doi:10.1016/j.topol.2003.10.007
fatcat:dyzpy3iycjcrhhcv3fafat3vf4
*the*previous sentence,*the*separation property collectionwise normal can*be*replaced by other separation properties-for example, Hausdorff, Urysohn, regular, metrizable. ... We present a nowhere locally compact Hausdorff space X with no coarser connected Hausdorff topology, yet X is*dense*in a connected Hausdorff space Y . ... Let ξ n*be**the*order type of X n . We*may*assume that {ξ n : n ∈ ω} is cofinal in β. (Let {β n : n ∈ ω}*be*cofinal in β, and make β n a*subset*of X n .) ...##
###
Asymptotic variation of L functions of one-variable exponential sums
[article]

2005
*
arXiv
*
pre-print

We prove that there is a Zariski

arXiv:math/0206284v3
fatcat:4rbt4vjz4ve3rmmjw7ybfvxadu
*dense*open*subset*U defined over Q in A^d such that for every geometric point f(x) in U(Qbar) we have lim_p-->oo NP(f mod P) = HP(A^d), where P is any prime ideal in*the*... Let d>2 and let p*be*a prime coprime to d. Let Z_pbar*be**the*ring of integers of Q_pbar. Suppose f(x) is a degree-d polynomial over Qbar and Z_pbar. ... Suppose there is such a Zariski*dense*open*subset*U defined over Q in A d . ...##
###
Convolutional neural networks for structured omics: OmicsCNN and the OmicsConv layer
[article]

2017
*
arXiv
*
pre-print

In particular, for metagenomics data, a metric can

arXiv:1710.05918v1
fatcat:lf2qhbkehzcfvmw2vkewoy4oli
*be*derived from*the*patristic distance on*the*phylogenetic tree. ... ., they can*be*endowed with a metric structure, enabling to adopt a convolutional based deep learning framework, e.g., for prediction. ... As a consequence,*the*direct application of deep networks to microarray, genetic or other omics can*be*disappointing, as found by Ditzler and coauthors [6] that claimed that deep learning "*may*not*be*...##
###
Asymptotic variation of L functions of one-variable exponential sums

2004
*
Journal für die Reine und Angewandte Mathematik
*

We prove that there is a Zariski

doi:10.1515/crll.2004.051
fatcat:kcnbf36yaza7xhlp6akz7tsgiy
*dense*open*subset*U defined over Q in A d such that for every geometric point f (x) in U(Q) and p large enough (depending only on f ) one has NP(f mod P) = GNP(A d ; Fp ... Suppose f ∈ A d (Q ∩ Zp), let NP(f (x) mod P) denote*the*q-adic Newton polygon of*the*L function L(f (x) mod P; T ) of exponential*sums*of f mod P. ... Suppose there is such a Zariski*dense*open*subset*U defined over Q in A d . ...##
###
Parallel modularity computation for directed weighted graphs with overlapping communities

Параллельное вычисление модулярности для направленных взвешенных графов с пересекающимися сообществами

2016
*
Proceedings of the Institute for System Programming of RAS
*

Параллельное вычисление модулярности для направленных взвешенных графов с пересекающимися сообществами

*The*paper presents new versions of modularity measure for directed weighted graphs with overlapping communities. We consider several approaches to computing modularity and try to extend them. ...

*The*only problem here is that

*the*biggest community

*may*have size square much more than

*sum*of size squares of

*the*rest ones, i.e.

*the*chunk which gets this community will

*be*overloaded. ... To achieve this we used a greedy algorithm, which iterates over communities in descending order and assigns each of them to a

*subset*that has

*the*smallest

*sum*of size squares. ...

##
###
IMFD IMPRESEE at TRECVID 2019: Ad-Hoc Video Search and Video To Text

2019
*
TREC Video Retrieval Evaluation
*

Second, for

dblp:conf/trecvid/HernandezPBBB19
fatcat:uc57auanvzagfd5rbjfue7vm7q
*the*VTT Matching and Ranking Task, we develop another deep learning model based on Word2VisualVec++, extracting temporal information of*the*video by using*Dense*Trajectories [16] and a clustering ... We participated in*the*task Ad-hoc Video Search (AVS) and*the*subtasks Description Generation and Matching and Ranking of Video to Text (VTT) task. ... Acknowledgements*The*authors of this paper would like to thank NIST and all*the*coordinators of TRECVID for organizing this event. ...
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