scholarly journals Exact Algorithms for MRE Inference

2016 ◽  
Vol 55 ◽  
pp. 653-683 ◽  
Author(s):  
Xiaoyuan Zhu ◽  
Changhe Yuan
Keyword(s):  
Bayesian Networks ◽  
Branch And Bound ◽  
Upper Bound ◽  
Bayes Factor ◽  
Upper Bounds ◽  
Exact Algorithms ◽  
The Other ◽  
Generalized Bayes ◽  
Relevant Explanation ◽  
Inference Task

Most Relevant Explanation (MRE) is an inference task in Bayesian networks that finds the most relevant partial instantiation of target variables as an explanation for given evidence by maximizing the Generalized Bayes Factor (GBF). No exact MRE algorithm has been developed previously except exhaustive search. This paper fills the void by introducing two Breadth-First Branch-and-Bound (BFBnB) algorithms for solving MRE based on novel upper bounds of GBF. One upper bound is created by decomposing the computation of GBF using a target blanket decomposition of evidence variables. The other upper bound improves the first bound in two ways. One is to split the target blankets that are too large by converting auxiliary nodes into pseudo-targets so as to scale to large problems. The other is to perform summations instead of maximizations on some of the target variables in each target blanket. Our empirical evaluations show that the proposed BFBnB algorithms make exact MRE inference tractable in Bayesian networks that could not be solved previously.

scholarly journals A New Upper Bound Based on Vertex Partitioning for the Maximum K-plex Problem

2021 ◽  
Author(s):  
Hua Jiang ◽  
Dongming Zhu ◽  
Zhichao Xie ◽  
Shaowen Yao ◽  
Zhang-Hua Fu
Keyword(s):  
Branch And Bound ◽  
Upper Bound ◽  
Undirected Graph ◽  
State Of The Art ◽  
Search Space ◽  
Exact Algorithms ◽  
Induced Subgraph ◽  
Massive Graphs ◽  
Vertex Set ◽  
Number Of Branches

Given an undirected graph, the Maximum k-plex Problem (MKP) is to find a largest induced subgraph in which each vertex has at most k−1 non-adjacent vertices. The problem arises in social network analysis and has found applications in many important areas employing graph-based data mining. Existing exact algorithms usually implement a branch-and-bound approach that requires a tight upper bound to reduce the search space. In this paper, we propose a new upper bound for MKP, which is a partitioning of the candidate vertex set with respect to the constructing solution. We implement a new branch-and-bound algorithm that employs the upper bound to reduce the number of branches. Experimental results show that the upper bound is very effective in reducing the search space. The new algorithm outperforms the state-of-the-art algorithms significantly on real-world massive graphs, DIMACS graphs and random graphs.

scholarly journals Lengths for Which Fourth Degree PP Interleavers Lead to Weaker Performances Compared to Quadratic and Cubic PP Interleavers

Entropy ◽  
2020 ◽  
Vol 22 (1) ◽  
pp. 78
Author(s):  
Lucian Trifina ◽  
Daniela Tarniceriu ◽  
Jonghoon Ryu ◽  
Ana-Mirela Rotopanescu
Keyword(s):  
Upper Bound ◽  
Turbo Codes ◽  
Minimum Distance ◽  
Prime Numbers ◽  
Upper Bounds ◽  
The Other ◽  
Small Range ◽  
Generator Matrix ◽  
Fourth Degree ◽  
Minimum Distances

In this paper, we obtain upper bounds on the minimum distance for turbo codes using fourth degree permutation polynomial (4-PP) interleavers of a specific interleaver length and classical turbo codes of nominal 1/3 coding rate, with two recursive systematic convolutional component codes with generator matrix G = [ 1 , 15 / 13 ] . The interleaver lengths are of the form 16 Ψ or 48 Ψ , where Ψ is a product of different prime numbers greater than three. Some coefficient restrictions are applied when for a prime p i ∣ Ψ , condition 3 ∤ ( p i − 1 ) is fulfilled. Two upper bounds are obtained for different classes of 4-PP coefficients. For a 4-PP f 4 x 4 + f 3 x 3 + f 2 x 2 + f 1 x ( mod 16 k L Ψ ) , k L ∈ { 1 , 3 } , the upper bound of 28 is obtained when the coefficient f 3 of the equivalent 4-permutation polynomials (PPs) fulfills f 3 ∈ { 0 , 4 Ψ } or when f 3 ∈ { 2 Ψ , 6 Ψ } and f 2 ∈ { ( 4 k L − 1 ) · Ψ , ( 8 k L − 1 ) · Ψ } , k L ∈ { 1 , 3 } , for any values of the other coefficients. The upper bound of 36 is obtained when the coefficient f 3 of the equivalent 4-PPs fulfills f 3 ∈ { 2 Ψ , 6 Ψ } and f 2 ∈ { ( 2 k L − 1 ) · Ψ , ( 6 k L − 1 ) · Ψ } , k L ∈ { 1 , 3 } , for any values of the other coefficients. Thus, the task of finding out good 4-PP interleavers of the previous mentioned lengths is highly facilitated by this result because of the small range required for coefficients f 4 , f 3 and f 2 . It was also proven, by means of nonlinearity degree, that for the considered inteleaver lengths, cubic PPs and quadratic PPs with optimum minimum distances lead to better error rate performances compared to 4-PPs with optimum minimum distances.

An upper bound for the breadth of the Jones polynomial

1988 ◽  
Vol 103 (3) ◽  
pp. 451-456 ◽  
Author(s):  
Morwen B. Thistlethwaite
Keyword(s):  
Simple Proof ◽  
Upper Bound ◽  
Recent Article ◽  
Jones Polynomial ◽  
Upper Bounds ◽  
The Other ◽  
Link Diagram ◽  
Kauffman Bracket ◽  
Bracket Polynomial ◽  
Connected Diagram

In the recent article [2], a kind of connected link diagram called adequate was investigated, and it was shown that the Jones polynomial is never trivial for such a diagram. Here, on the other hand, upper bounds are considered for the breadth of the Jones polynomial of an arbitrary connected diagram, thus extending some of the results of [1,4,5]. Also, in Theorem 2 below, a characterization is given of those connected, prime diagrams for which the breadth of the Jones polynomial is one less than the number of crossings; recall from [1,4,5] that the breadth equals the number of crossings if and only if that diagram is reduced alternating. The article is concluded with a simple proof, using the Jones polynomial, of W. Menasco's theorem [3] that a connected, alternating diagram cannot represent a split link. We shall work with the Kauffman bracket polynomial 〈D〉 ∈ Z[A, A−1 of a link diagram D.

An Exact Algorithm for Minimum Vertex Cover Problem

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 603
Author(s):  
Luzhi Wang ◽  
Shuli Hu ◽  
Mingyang Li ◽  
Junping Zhou
Keyword(s):  
Lower Bound ◽  
Branch And Bound ◽  
Heuristic Algorithms ◽  
Vertex Cover ◽  
Exact Algorithm ◽  
Search Space ◽  
Exact Algorithms ◽  
The Other ◽  
Branch And Bound Algorithm ◽  
Cover Problem

In this paper, we propose a branch-and-bound algorithm to solve exactly the minimum vertex cover (MVC) problem. Since a tight lower bound for MVC has a significant influence on the efficiency of a branch-and-bound algorithm, we define two novel lower bounds to help prune the search space. One is based on the degree of vertices, and the other is based on MaxSAT reasoning. The experiment confirms that our algorithm is faster than previous exact algorithms and can find better results than heuristic algorithms.

scholarly journals Isosceles Sets

10.37236/230 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Yury J. Ionin
Keyword(s):  
Euclidean Space ◽  
Upper Bound ◽  
Isosceles Triangle ◽  
Upper Bounds ◽  
The Other ◽  
The Third ◽  
Exact Answer ◽  
Hamming Space ◽  
Erdős Problem ◽  
Set Of Points

In 1946, Paul Erdős posed a problem of determining the largest possible cardinality of an isosceles set, i.e., a set of points in plane or in space, any three of which form an isosceles triangle. Such a question can be asked for any metric space, and an upper bound ${n+2\choose 2}$ for the Euclidean space ${\Bbb E}^{n}$ was found by Blokhuis. This upper bound is known to be sharp for $n=1$, 2, 6, and 8. We will consider Erdős' question for the binary Hamming space $H_{n}$ and obtain the following upper bounds on the cardinality of an isosceles subset $S$ of $H_{n}$: if there are at most two distinct nonzero distances between points of $S$, then $|S|\leq{n+1\choose 2}+1$; if, furthermore, $n\geq 4$, $n\ne 6$, and, as a set of vertices of the $n$-cube, $S$ is contained in a hyperplane, then $|S|\leq{n\choose 2}$; if there are more than two distinct nonzero distances between points of $S$, then $|S|\leq{n\choose 2}+1$. The first bound is sharp if and only if $n=2$ or $n=5$; the other two bounds are sharp for all relevant values of $n$, except the third bound for $n=6$, when the sharp upper bound is 12. We also give the exact answer to the Erdős problem for ${\Bbb E}^{n}$ with $n\leq 7$ and describe all isosceles sets of the largest cardinality in these dimensions.

scholarly journals Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint

2021 ◽  
Vol 5 (1) ◽  
pp. 1-22
Author(s):  
Jing Tang ◽  
Xueyan Tang ◽  
Andrew Lim ◽  
Kai Han ◽  
Chongshou Li ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Greedy Algorithm ◽  
Branch And Bound ◽  
Upper Bound ◽  
Optimization Problem ◽  
Approximation Factor ◽  
Real World Application ◽  
Efficiency Of Algorithms ◽  
Knapsack Constraint ◽  
Submodular Maximization

Monotone submodular maximization with a knapsack constraint is NP-hard. Various approximation algorithms have been devised to address this optimization problem. In this paper, we revisit the widely known modified greedy algorithm. First, we show that this algorithm can achieve an approximation factor of 0.405, which significantly improves the known factors of 0.357 given by Wolsey and (1-1/e)/2\approx 0.316 given by Khuller et al. More importantly, our analysis closes a gap in Khuller et al.'s proof for the extensively mentioned approximation factor of (1-1/\sqrte )\approx 0.393 in the literature to clarify a long-standing misconception on this issue. Second, we enhance the modified greedy algorithm to derive a data-dependent upper bound on the optimum. We empirically demonstrate the tightness of our upper bound with a real-world application. The bound enables us to obtain a data-dependent ratio typically much higher than 0.405 between the solution value of the modified greedy algorithm and the optimum. It can also be used to significantly improve the efficiency of algorithms such as branch and bound.

scholarly journals A BOUND FOR THE INDEX OF A QUADRATIC FORM AFTER SCALAR EXTENSION TO THE FUNCTION FIELD OF A QUADRIC

2018 ◽  
Vol 19 (2) ◽  
pp. 421-450 ◽  
Author(s):  
Stephen Scully
Keyword(s):  
Quadratic Form ◽  
Upper Bound ◽  
Function Field ◽  
The Other ◽  
Direct Generalization ◽  
Other Hand ◽  
General Field ◽  
The One ◽  
Anisotropic Quadratic Form

Let $q$ be an anisotropic quadratic form defined over a general field $F$. In this article, we formulate a new upper bound for the isotropy index of $q$ after scalar extension to the function field of an arbitrary quadric. On the one hand, this bound offers a refinement of an important bound established in earlier work of Karpenko–Merkurjev and Totaro; on the other hand, it is a direct generalization of Karpenko’s theorem on the possible values of the first higher isotropy index. We prove its validity in two key cases: (i) the case where $\text{char}(F)\neq 2$, and (ii) the case where $\text{char}(F)=2$ and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated separately using completely different approaches, the first being algebraic–geometric, and the second being purely algebraic.

scholarly journals On connectedness of power graphs of finite groups

2018 ◽  
Vol 17 (10) ◽  
pp. 1850184 ◽  
Author(s):  
Ramesh Prasad Panda ◽  
K. V. Krishna
Keyword(s):  
Finite Groups ◽  
Upper Bound ◽  
The Other ◽  
Number Of Components ◽  
Cyclic Groups ◽  
Power Graph ◽  
Vertex Set ◽  
Proper Power

The power graph of a group [Formula: see text] is the graph whose vertex set is [Formula: see text] and two distinct vertices are adjacent if one is a power of the other. This paper investigates the minimal separating sets of power graphs of finite groups. For power graphs of finite cyclic groups, certain minimal separating sets are obtained. Consequently, a sharp upper bound for their connectivity is supplied. Further, the components of proper power graphs of [Formula: see text]-groups are studied. In particular, the number of components of that of abelian [Formula: see text]-groups are determined.

A Yield Criterion of Porous Materials With Anisotropy Caused by Geometry or Distribution of Pores

1991 ◽  
Vol 113 (4) ◽  
pp. 425-429 ◽  
Author(s):  
T. Hisatsune ◽  
T. Tabata ◽  
S. Masaki
Keyword(s):  
Velocity Field ◽  
Porous Materials ◽  
Upper Bound ◽  
Volume Fraction ◽  
Yield Criterion ◽  
Longitudinal Direction ◽  
Axisymmetric Deformation ◽  
The Other ◽  
The Matrix ◽  
Spherical Cells

Axisymmetric deformation of anisotropic porous materials caused by geometry of pores or by distribution of pores is analyzed. Two models of the materials are proposed: one consists of spherical cells each of which has a concentric ellipsoidal pore; and the other consists of ellipsoidal cells each of which has a concentric spherical pore. The velocity field in the matrix is assumed and the upper bound approach is attempted. Yield criteria are expressed as ellipses on the σm σ3 plane which are longer in longitudinal direction with increasing anisotropy and smaller with increasing volume fraction of the pore. Furthermore, the axes rotate about the origin at an angle α from the σm-axis, while the axis for isotropic porous materials is on the σm-axis.

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