scholarly journals Congruence Distributivity Implies Bounded Width

2010 ◽  
Vol 39 (4) ◽  
pp. 1531-1542 ◽  
Author(s):  
Libor Barto ◽  
Marcin Kozik
Keyword(s):  
Congruence Distributivity ◽  
Bounded Width

scholarly journals Twin-width I: Tractable FO Model Checking

2022 ◽  
Vol 69 (1) ◽  
pp. 1-46
Author(s):  
Édouard Bonnet ◽  
Eun Jung Kim ◽  
Stéphan Thomassé ◽  
Rémi Watrigant
Keyword(s):  
Model Checking ◽  
Elementary Function ◽  
First Order ◽  
Fpt Algorithm ◽  
Fixed Parameter ◽  
Dimension 2 ◽  
Bounded Size ◽  
Map Graphs ◽  
Dimensional Ball ◽  
Bounded Width

Inspired by a width invariant defined on permutations by Guillemot and Marx [SODA’14], we introduce the notion of twin-width on graphs and on matrices. Proper minor-closed classes, bounded rank-width graphs, map graphs, K t -free unit d -dimensional ball graphs, posets with antichains of bounded size, and proper subclasses of dimension-2 posets all have bounded twin-width. On all these classes (except map graphs without geometric embedding) we show how to compute in polynomial time a sequence of d -contractions , witness that the twin-width is at most d . We show that FO model checking, that is deciding if a given first-order formula ϕ evaluates to true for a given binary structure G on a domain D , is FPT in |ϕ| on classes of bounded twin-width, provided the witness is given. More precisely, being given a d -contraction sequence for G , our algorithm runs in time f ( d ,|ϕ |) · |D| where f is a computable but non-elementary function. We also prove that bounded twin-width is preserved under FO interpretations and transductions (allowing operations such as squaring or complementing a graph). This unifies and significantly extends the knowledge on fixed-parameter tractability of FO model checking on non-monotone classes, such as the FPT algorithm on bounded-width posets by Gajarský et al. [FOCS’15].

Metric Dimension of Bounded Width Graphs

2015 ◽  
pp. 115-126 ◽  
Author(s):  
Rémy Belmonte ◽  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
M. S. Ramanujan
Keyword(s):  
Metric Dimension ◽  
Bounded Width

On schemes for congruence distributivity

2004 ◽  
Vol 2 (3) ◽  
pp. 368-376
Author(s):  
I. Chajda ◽  
R. Halaš
Keyword(s):  
Congruence Distributivity

scholarly journals Universal Function Approximation by Deep Neural Nets with Bounded Width and ReLU Activations

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 992 ◽  
Author(s):  
Boris Hanin
Keyword(s):  
Convex Functions ◽  
Expressive Power ◽  
Scalar Function ◽  
Universal Function ◽  
Unit Cube ◽  
Neural Nets ◽  
Minimal Width ◽  
Continuous Convex Function ◽  
Dimensional Cube ◽  
Bounded Width

This article concerns the expressive power of depth in neural nets with ReLU activations and a bounded width. We are particularly interested in the following questions: What is the minimal width w min ( d ) so that ReLU nets of width w min ( d ) (and arbitrary depth) can approximate any continuous function on the unit cube [ 0 , 1 ] d arbitrarily well? For ReLU nets near this minimal width, what can one say about the depth necessary to approximate a given function? We obtain an essentially complete answer to these questions for convex functions. Our approach is based on the observation that, due to the convexity of the ReLU activation, ReLU nets are particularly well suited to represent convex functions. In particular, we prove that ReLU nets with width d + 1 can approximate any continuous convex function of d variables arbitrarily well. These results then give quantitative depth estimates for the rate of approximation of any continuous scalar function on the d-dimensional cube [ 0 , 1 ] d by ReLU nets with width d + 3 .

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