scholarly journals Computational topology and normal surfaces: Theoretical and experimental complexity bounds

Author(s):  
Benjamin A. Burton ◽  
João Paixão ◽  
Jonathan Spreer
2021 ◽  
Vol 15 (3) ◽  
pp. 1-35
Author(s):  
Muhammad Anis Uddin Nasir ◽  
Cigdem Aslay ◽  
Gianmarco De Francisci Morales ◽  
Matteo Riondato

“Perhaps he could dance first and think afterwards, if it isn’t too much to ask him.” S. Beckett, Waiting for Godot Given a labeled graph, the collection of -vertex induced connected subgraph patterns that appear in the graph more frequently than a user-specified minimum threshold provides a compact summary of the characteristics of the graph, and finds applications ranging from biology to network science. However, finding these patterns is challenging, even more so for dynamic graphs that evolve over time, due to the streaming nature of the input and the exponential time complexity of the problem. We study this task in both incremental and fully-dynamic streaming settings, where arbitrary edges can be added or removed from the graph. We present TipTap , a suite of algorithms to compute high-quality approximations of the frequent -vertex subgraphs w.r.t. a given threshold, at any time (i.e., point of the stream), with high probability. In contrast to existing state-of-the-art solutions that require iterating over the entire set of subgraphs in the vicinity of the updated edge, TipTap operates by efficiently maintaining a uniform sample of connected -vertex subgraphs, thanks to an optimized neighborhood-exploration procedure. We provide a theoretical analysis of the proposed algorithms in terms of their unbiasedness and of the sample size needed to obtain a desired approximation quality. Our analysis relies on sample-complexity bounds that use Vapnik–Chervonenkis dimension, a key concept from statistical learning theory, which allows us to derive a sufficient sample size that is independent from the size of the graph. The results of our empirical evaluation demonstrates that TipTap returns high-quality results more efficiently and accurately than existing baselines.


Author(s):  
Mareike Dressler ◽  
Adam Kurpisz ◽  
Timo de Wolff

AbstractVarious key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + d . Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.


2017 ◽  
Vol 53 (6) ◽  
pp. 1-4
Author(s):  
Pawel Dlotko ◽  
Bernard Kapidani ◽  
Ruben Specogna

2006 ◽  
Vol 365 (3) ◽  
pp. 184-198 ◽  
Author(s):  
K. Abe ◽  
J. Bisceglio ◽  
D.R. Ferguson ◽  
T.J. Peters ◽  
A.C. Russell ◽  
...  

2018 ◽  
Vol 19 (3) ◽  
pp. 591-621 ◽  
Author(s):  
Simon Abelard ◽  
Pierrick Gaudry ◽  
Pierre-Jean Spaenlehauer

2018 ◽  
Vol 115 (43) ◽  
pp. 10901-10907 ◽  
Author(s):  
Mark Bell ◽  
Joel Hass ◽  
Joachim Hyam Rubinstein ◽  
Stephan Tillmann

We describe an algorithm to compute trisections of orientable four-manifolds using arbitrary triangulations as input. This results in explicit complexity bounds for the trisection genus of a 4-manifold in terms of the number of pentachora (4-simplices) in a triangulation.


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