scholarly journals Complexity bounds for approximately solving discounted MDPs by value iterations

2020 ◽  
Vol 48 (5) ◽  
pp. 543-548
Author(s):  
Eugene A. Feinberg ◽  
Gaojin He
Keyword(s):  
Complexity Bounds ◽  
Value Iterations

Approximate Mining of Frequent -Subgraph Patterns in Evolving Graphs

2021 ◽  
Vol 15 (3) ◽  
pp. 1-35
Author(s):  
Muhammad Anis Uddin Nasir ◽  
Cigdem Aslay ◽  
Gianmarco De Francisci Morales ◽  
Matteo Riondato
Keyword(s):  
Sample Size ◽  
State Of The Art ◽  
Empirical Evaluation ◽  
Connected Subgraph ◽  
Dynamic Graphs ◽  
High Quality ◽  
Complexity Bounds ◽  
Approximation Quality ◽  
Uniform Sample ◽  
Waiting For Godot

“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.

scholarly journals Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

2021 ◽  
Author(s):  
Mareike Dressler ◽  
Adam Kurpisz ◽  
Timo de Wolff
Keyword(s):  
Computer Science ◽  
Affine Transformation ◽  
Optimization Problems ◽  
Polynomial Optimization ◽  
Theoretical Computer Science ◽  
Sums Of Squares ◽  
Theoretical Computer ◽  
Complexity Bounds ◽  
Polynomial Optimization Problems ◽  
Transformation Of Variables

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.

scholarly journals Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

2018 ◽  
Vol 19 (3) ◽  
pp. 591-621 ◽  
Author(s):  
Simon Abelard ◽  
Pierrick Gaudry ◽  
Pierre-Jean Spaenlehauer
Keyword(s):  
Hyperelliptic Curves ◽  
Complexity Bounds ◽  
Counting Points

scholarly journals Computing trisections of 4-manifolds

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

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.

scholarly journals Bisimulations on Data Graphs

2018 ◽  
Vol 61 ◽  
pp. 171-213 ◽  
Author(s):  
Sergio Abriola ◽  
Pablo Barceló ◽  
Diego Figueira ◽  
Santiago Figueira
Keyword(s):  
Constraint Satisfaction ◽  
Transitive Closure ◽  
Modal Logics ◽  
Closure Operators ◽  
External Observer ◽  
Complexity Bounds ◽  
Fundamental Notion ◽  
Labeled Graphs ◽  
Non Locality

Bisimulation provides structural conditions to characterize indistinguishability from an external observer between nodes on labeled graphs. It is a fundamental notion used in many areas, such as verification, graph-structured databases, and constraint satisfaction. However, several current applications use graphs where nodes also contain data (the so called "data graphs"), and where observers can test for equality or inequality of data values (e.g., asking the attribute 'name' of a node to be different from that of all its neighbors). The present work constitutes a first investigation of "data aware" bisimulations on data graphs. We study the problem of computing such bisimulations, based on the observational indistinguishability for XPath ---a language that extends modal logics like PDL with tests for data equality--- with and without transitive closure operators. We show that in general the problem is PSpace-complete, but identify several restrictions that yield better complexity bounds (coNP, PTime) by controlling suitable parameters of the problem, namely the amount of non-locality allowed, and the class of models considered (graphs, DAGs, trees). In particular, this analysis yields a hierarchy of tractable fragments.

scholarly journals Lower complexity bounds for lifted inference

2014 ◽  
Vol 15 (2) ◽  
pp. 246-263 ◽  
Author(s):  
MANFRED JAEGER
Keyword(s):  
Knowledge Bases ◽  
Modeling Languages ◽  
Complexity Bounds ◽  
Lifted Inference ◽  
Probabilistic Relational Models ◽  
Lower Complexity ◽  
Logical Modeling ◽  
Inference Techniques ◽  
And Function ◽  
Lower Complexity Bounds

AbstractOne of the big challenges in the development of probabilistic relational (or probabilistic logical) modeling and learning frameworks is the design of inference techniques that operate on the level of the abstract model representation language, rather than on the level of ground, propositional instances of the model. Numerous approaches for such “lifted inference” techniques have been proposed. While it has been demonstrated that these techniques will lead to significantly more efficient inference on some specific models, there are only very recent and still quite restricted results that show the feasibility of lifted inference on certain syntactically defined classes of models. Lower complexity bounds that imply some limitations for the feasibility of lifted inference on more expressive model classes were established earlier in Jaeger (2000; Jaeger, M. 2000. On the complexity of inference about probabilistic relational models. Artificial Intelligence 117, 297–308). However, it is not immediate that these results also apply to the type of modeling languages that currently receive the most attention, i.e., weighted, quantifier-free formulas. In this paper we extend these earlier results, and show that under the assumption that NETIME≠ETIME, there is no polynomial lifted inference algorithm for knowledge bases of weighted, quantifier-, and function-free formulas. Further strengthening earlier results, this is also shown to hold for approximate inference and for knowledge bases not containing the equality predicate.

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