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 Maximum difference about the size of optimal identifying codes in graphs differing by one vertex

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mikko Pelto
Keyword(s):  
Upper Bound ◽  
Undirected Graph ◽  
Minimum Size ◽  
Maximum Difference ◽  
Free Graph ◽  
Induced Subgraph ◽  
Identifying Code ◽  
Vertex Set ◽  
International Audience ◽  
Codes In Graphs

Graph Theory International audience Let G=(V,E) be a simple undirected graph. We call any subset C⊆V an identifying code if the sets I(v)={c∈C | {v,c}∈E or v=c } are distinct and non-empty for all vertices v∈V. A graph is called twin-free if there is an identifying code in the graph. The identifying code with minimum size in a twin-free graph G is called the optimal identifying code and the size of such a code is denoted by γ(G). Let GS denote the induced subgraph of G where the vertex set S⊂V is deleted. We provide a tight upper bound for γ(GS)-γ(G) when both graphs are twin-free and |V| is large enough with respect to |S|. Moreover, we prove tight upper bound when G is a bipartite graph and |S|=1.

scholarly journals Reducing the Computational Time for the Kemeny Method by Exploiting Condorcet Properties

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1380
Author(s):  
Noelia Rico ◽  
Camino R. Vela ◽  
Raúl Pérez-Fernández ◽  
Irene Díaz
Keyword(s):  
Choice Theory ◽  
State Of The Art ◽  
Social Choice Theory ◽  
Real Life ◽  
Search Space ◽  
Exact Algorithms ◽  
Computational Time ◽  
Preference Aggregation ◽  
Life Problems ◽  
Ranking Aggregation

Preference aggregation and in particular ranking aggregation are mainly studied by the field of social choice theory but extensively applied in a variety of contexts. Among the most prominent methods for ranking aggregation, the Kemeny method has been proved to be the only one that satisfies some desirable properties such as neutrality, consistency and the Condorcet condition at the same time. Unfortunately, the problem of finding a Kemeny ranking is NP-hard, which prevents practitioners from using it in real-life problems. The state of the art of exact algorithms for the computation of the Kemeny ranking experienced a major boost last year with the presentation of an algorithm that provides searching time guarantee up to 13 alternatives. In this work, we propose an enhanced version of this algorithm based on pruning the search space when some Condorcet properties hold. This enhanced version greatly improves the performance in terms of runtime consumption.

scholarly journals Speeding Up Incomplete GDL-based Algorithms for Multi-agent Optimization with Dense Local Utilities

2020 ◽  
Author(s):  
Yanchen Deng ◽  
Bo An
Keyword(s):  
Utility Function ◽  
Branch And Bound ◽  
State Of The Art ◽  
Search Space ◽  
Computational Effort ◽  
The State ◽  
Computational Overhead ◽  
Multi Agent

Incomplete GDL-based algorithms including Max-sum and its variants are important methods for multi-agent optimization. However, they face a significant scalability challenge as the computational overhead grows exponentially with respect to the arity of each utility function. Generic Domain Pruning (GDP) technique reduces the computational effort by performing a one-shot pruning to filter out suboptimal entries. Unfortunately, GDP could perform poorly when dealing with dense local utilities and ties which widely exist in many domains. In this paper, we present several novel sorting-based acceleration algorithms by alleviating the effect of densely distributed local utilities. Specifically, instead of one-shot pruning in GDP, we propose to integrate both search and pruning to iteratively reduce the search space. Besides, we cope with the utility ties by organizing the search space of tied utilities into AND/OR trees to enable branch-and-bound. Finally, we propose a discretization mechanism to offer a tradeoff between the reconstruction overhead and the pruning efficiency. We demonstrate the superiorities of our algorithms over the state-of-the-art from both theoretical and experimental perspectives.

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.

Automorphism group and fixing number of the orthogonality graph based on rank one upper triangular matrices

2020 ◽  
pp. 2250023
Author(s):  
Shikun Ou ◽  
Yanqi Fan ◽  
Fenglei Tian
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Canonical Form ◽  
Undirected Graph ◽  
Induced Subgraph ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisors ◽  
Rank One ◽  
Vertex Set

The orthogonality graph [Formula: see text] of a ring [Formula: see text] is the undirected graph with vertex set consisting of all nonzero two-sided zero divisors of [Formula: see text], in which for two vertices [Formula: see text] and [Formula: see text] (needless distinct), [Formula: see text] is an edge if and only if [Formula: see text]. Let [Formula: see text], [Formula: see text] be the set of all [Formula: see text] matrices over a finite field [Formula: see text], and [Formula: see text] the subset of [Formula: see text] consisting of all rank one upper triangular matrices. In this paper, we describe the full automorphism group, and using the technique of generalized equivalent canonical form of matrices, we compute the fixing number of [Formula: see text], the induced subgraph of [Formula: see text] with vertex set [Formula: see text].

scholarly journals On r-Noncommuting Graph of Finite Rings

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 233
Author(s):  
Rajat Kanti Nath ◽  
Monalisha Sharma ◽  
Parama Dutta ◽  
Yilun Shang
Keyword(s):  
Undirected Graph ◽  
Clique Number ◽  
Star Graph ◽  
Finite Rings ◽  
Induced Subgraph ◽  
Noncommutative Rings ◽  
Central Elements ◽  
Vertex Set ◽  
Vertex Degrees ◽  
Noncommuting Graph

Let R be a finite ring and r∈R. The r-noncommuting graph of R, denoted by ΓRr, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x,y]≠r and [x,y]≠−r. In this paper, we obtain expressions for vertex degrees and show that ΓRr is neither a regular graph nor a lollipop graph if R is noncommutative. We characterize finite noncommutative rings such that ΓRr is a tree, in particular a star graph. It is also shown that ΓR1r and ΓR2ψ(r) are isomorphic if R1 and R2 are two isoclinic rings with isoclinism (ϕ,ψ). Further, we consider the induced subgraph ΔRr of ΓRr (induced by the non-central elements of R) and obtain results on clique number and diameter of ΔRr along with certain characterizations of finite noncommutative rings such that ΔRr is n-regular for some positive integer n. As applications of our results, we characterize certain finite noncommutative rings such that their noncommuting graphs are n-regular for n≤6.

scholarly journals Dot product graphs and domination number

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
Dina Saleh ◽  
Nefertiti Megahed
Keyword(s):  
Commutative Ring ◽  
Upper Bound ◽  
Undirected Graph ◽  
Multiplicative Group ◽  
Domination Number ◽  
Product Graph ◽  
Group Of Units ◽  
Product Graphs ◽  
Vertex Set ◽  
Dot Product

Abstract Let A be a commutative ring with 1≠0 and R=A×A. The unit dot product graph of R is defined to be the undirected graph UD(R) with the multiplicative group of units in R, denoted by U(R), as its vertex set. Two distinct vertices x and y are adjacent if and only if x·y=0∈A, where x·y denotes the normal dot product of x and y. In 2016, Abdulla studied this graph when $A=\mathbb {Z}_{n}$ A = ℤ n , $n \in \mathbb {N}$ n ∈ ℕ , n≥2. Inspired by this idea, we study this graph when A has a finite multiplicative group of units. We define the congruence unit dot product graph of R to be the undirected graph CUD(R) with the congruent classes of the relation $\thicksim $ ∽ defined on R as its vertices. Also, we study the domination number of the total dot product graph of the ring $R=\mathbb {Z}_{n}\times... \times \mathbb {Z}_{n}$ R = ℤ n × ... × ℤ n , k times and k<∞, where all elements of the ring are vertices and adjacency of two distinct vertices is the same as in UD(R). We find an upper bound of the domination number of this graph improving that found by Abdulla.

scholarly journals An Exact Algorithm Based on MaxSAT Reasoning for the Maximum Weight Clique Problem

2016 ◽  
Vol 55 ◽  
pp. 799-833 ◽  
Author(s):  
Zhiwen Fang ◽  
Chu-Min Li ◽  
Ke Xu
Keyword(s):  
Upper Bound ◽  
Optimal Solution ◽  
Maximum Clique ◽  
Exact Algorithm ◽  
Search Space ◽  
Exact Algorithms ◽  
Maximum Weight ◽  
Winner Determination ◽  
Sound Transformation ◽  
Maximum Weight Clique

Recently, MaxSAT reasoning is shown very effective in computing a tight upper bound for a Maximum Clique (MC) of a (unweighted) graph. In this paper, we apply MaxSAT reasoning to compute a tight upper bound for a Maximum Weight Clique (MWC) of a wighted graph. We first study three usual encodings of MWC into weighted partial MaxSAT dealing with hard clauses, which must be satisfied in all solutions, and soft clauses, which are weighted and can be falsified. The drawbacks of these encodings motivate us to propose an encoding of MWC into a special weighted partial MaxSAT formalism, called LW (Literal-Weighted) encoding and dedicated for upper bounding an MWC, in which both soft clauses and literals in soft clauses are weighted. An optimal solution of the LW MaxSAT instance gives an upper bound for an MWC, instead of an optimal solution for MWC. We then introduce two notions called the Top-k literal failed clause and the Top-k empty clause to extend classical MaxSAT reasoning techniques, as well as two sound transformation rules to transform an LW MaxSAT instance. Successive transformations of an LW MaxSAT instance driven by MaxSAT reasoning give a tight upper bound for the encoded MWC. The approach is implemented in a branch-and-bound algorithm called MWCLQ. Experimental evaluations on the broadly used DIMACS benchmark, BHOSLIB benchmark, random graphs and the benchmark from the winner determination problem show that our approach allows MWCLQ to reduce the search space significantly and to solve MWC instances effectively. Consequently, MWCLQ outperforms state-of-the-art exact algorithms on the vast majority of instances. Moreover, it is surprisingly effective in solving hard and dense instances.

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 An Iterated Tabu Search Approach for the Clique Partitioning Problem

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Gintaras Palubeckis ◽  
Armantas Ostreika ◽  
Arūnas Tomkevičius
Keyword(s):  
Tabu Search ◽  
Undirected Graph ◽  
State Of The Art ◽  
Computational Results ◽  
Clique Partitioning ◽  
Vertex Set ◽  
Clique Partitioning Problem ◽  
Partitioning Problem ◽  
Search Approach ◽  
Iterated Tabu Search

Given an edge-weighted undirected graph with weights specifying dissimilarities between pairs of objects, represented by the vertices of the graph, the clique partitioning problem (CPP) is to partition the vertex set of the graph into mutually disjoint subsets such that the sum of the edge weights over all cliques induced by the subsets is as small as possible. We develop an iterated tabu search (ITS) algorithm for solving this problem. The proposed algorithm incorporates tabu search, local search, and solution perturbation procedures. We report computational results on CPP instances of size up to 2000 vertices. Performance comparisons of ITS against state-of-the-art methods from the literature demonstrate the competitiveness of our approach.

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