Computational complexity of problems of combinatorics and graph theory

1981 ◽  
Vol 8 (16) ◽  
Author(s):  
M. M. Sysło
Keyword(s):  
Graph Theory ◽  
Computational Complexity

scholarly journals A Computational Perspective on Network Coding

2010 ◽  
Vol 2010 ◽  
pp. 1-11
Author(s):  
Qin Guo ◽  
Mingxing Luo ◽  
Lixiang Li ◽  
Yixian Yang
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bound ◽  
Network Coding ◽  
Upper Bound ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Multicast Networks ◽  
Computational Perspective ◽  
Source Rate

From the perspectives of graph theory and combinatorics theory we obtain some new upper bounds on the number of encoding nodes, which can characterize the coding complexity of the network coding, both in feasible acyclic and cyclic multicast networks. In contrast to previous work, during our analysis we first investigate the simple multicast network with source rateh=2, and thenh≥2. We find that for feasible acyclic multicast networks our upper bound is exactly the lower bound given by M. Langberg et al. in 2006. So the gap between their lower and upper bounds for feasible acyclic multicast networks does not exist. Based on the new upper bound, we improve the computational complexity given by M. Langberg et al. in 2009. Moreover, these results further support the feasibility of signatures for network coding.

scholarly journals A complete classification of the complexity of the vertex 3-colourability problem for quadruples of induced 5-vertex prohibitions

2020 ◽  
Vol 22 (1) ◽  
pp. 38-47
Author(s):  
Dmitry S. Malyshev
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Complete Classification ◽  
Induced Subgraphs ◽  
Simple Graphs ◽  
Forbidden Induced Subgraphs ◽  
The Third ◽  
Hereditary Class ◽  
Complexity Status

The vertex 3-colourability problem for a given graph is to check whether it is possible to split the set of its vertices into three subsets of pairwise non-adjacent vertices or not. A hereditary class of graphs is a set of simple graphs closed under isomorphism and deletion of vertices; the set of its forbidden induced subgraphs defines every such a class. For all but three the quadruples of 5-vertex forbidden induced subgraphs, we know the complexity status of the vertex 3-colourability problem. Additionally, two of these three cases are polynomially equivalent; they also polynomially reduce to the third one. In this paper, we prove that the computational complexity of the considered problem in all of the three mentioned classes is polynomial. This result contributes to the algorithmic graph theory.

Optimal networks for exact controllability

2020 ◽  
Vol 31 (10) ◽  
pp. 2050144
Author(s):  
Yunhua Liao ◽  
Mohamed Maama ◽  
M. A. Aziz-Alaoui
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Complex Networks ◽  
Open Problem ◽  
Topological Structure ◽  
Exact Controllability ◽  
Spectral Graph Theory ◽  
Algebraic Multiplicity ◽  
Spectral Graph ◽  
Optimal Networks

The exact controllability can be mapped to the problem of maximum algebraic multiplicity of all eigenvalues. In this paper, we focus on the exact controllability of deterministic complex networks. First, we explore the eigenvalues of two famous networks, i.e. the comb-of-comb network and the Farey graph. Due to their special structure, we find that the eigenvalues of each network are mutually distinct, showing that these two networks are optimal networks with respect to exact controllability. Second, we study how to optimize the exact controllability of a deterministic network. Based on the spectral graph theory, we find that reducing the order of duplicate sets or co-duplicate sets which are two special vertex subsets can decrease greatly the exact controllability. This result provides an answer to an open problem of Li et al. [X. F. Li, Z. M. Lu and H. Li, Int. J. Mod. Phys. C 26, 1550028 (2015)]. Finally, we discuss the relation between the topological structure and the multiplicity of two special eigenvalues and the computational complexity of our method.

scholarly journals An Algorithmic Approach to Wireless Sensor Networks Localization Using Rigid Graphs

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Shamantha Rai B ◽  
Shirshu Varma
Keyword(s):  
Wireless Sensor Networks ◽  
Graph Theory ◽  
Wireless Sensor Network ◽  
Sensor Networks ◽  
Computational Complexity ◽  
Path Loss ◽  
Wireless Sensor ◽  
Distance Information ◽  
Algorithmic Approach ◽  
Network Nodes

In this work estimating the position coordinates of Wireless Sensor Network nodes using the concept of rigid graphs is carried out in detail. The range based localization approaches use the distance information measured by the RSSI, which is prone to noise, due to effects of path loss, shadowing, and so forth. In this work, both the distance and the bearing information are used for localization using the trilateration technique. Rigid graph theory is employed to analyze the localizability, that is, whether the nodes of the WSN are uniquely localized. The WSN graph is divided into rigid patches by varying appropriately the communication power range of the WSN nodes and then localizing the patches by trilateration. The main advantage of localizing the network using rigid graph approach is that it overcomes the effect of noisy perturbed distance. Our approach gives a better performance compared to robust quads in terms of percentage of localizable nodes and computational complexity.

Treewidth-aware Reductions of Normal ASP to SAT - Is Normal ASP Harder than SAT after All?

2020 ◽  
Author(s):  
Markus Hecher
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Lower Bounds ◽  
Parameterized Complexity ◽  
No Reduction ◽  
Answer Set Programming ◽  
Propositional Satisfiability ◽  
Fine Grained ◽  
Problem Modeling ◽  
Structural Density

Answer Set Programming (ASP) is a paradigm and problem modeling/solving toolkit for KR that is often invoked. There are plenty of results dedicated to studying the hardness of (fragments of) ASP. So far, these studies resulted in characterizations in terms of computational complexity as well as in fine-grained insights presented in form of dichotomy-style results, lower bounds when translating to other formalisms like propositional satisfiability (SAT), and even detailed parameterized complexity landscapes. A quite generic and prominent parameter in parameterized complexity originating from graph theory is the so-called treewidth, which in a sense captures structural density of a program. Recently, there was an increase in the number of treewidth-based solvers related to SAT. While there exist several translations from (normal) ASP to SAT, yet there is no reduction preserving treewidth or at least being aware of the treewidth increase. This paper deals with a novel reduction from normal ASP to SAT that is aware of the treewidth, and guarantees that a slight increase of treewidth is indeed sufficient. Then, we also present a new result establishing that when considering treewidth, already the fragment of normal ASP is slightly harder than SAT (under reasonable assumptions in computational complexity). This also confirms that our reduction probably cannot be significantly improved and that the slight increase of treewidth is unavoidable.

An Optimization Algorithm for SDR Multi-Standard Systems Using Directed Hypergraphs

Frequenz ◽  
2012 ◽  
Vol 66 (9-10) ◽  
Author(s):  
Patricia Kaiser ◽  
Amine El Sahili ◽  
Yves Louët
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Software Defined Radio ◽  
Optimization Problem ◽  
Optimal Solution ◽  
Minimum Cost ◽  
Heuristic Methods ◽  
Standard Design ◽  
Directed Hypergraphs ◽  
The Cost

AbstractSoftware-Defined radio (SDR) is a future-proof solution for designing flexible and adaptable wireless networks and equipments. It replaces conventional radio hardware with reconfigurable, reprogrammable radios. A graphical approach for designing flexible SDR multi-standard systems is proposed, which provides all the possible alternatives of implementation capable of realizing the multi-standard design. However, a cost function which evaluates the cost of any one of these options is proposed in previous work. All these ideas are briefly mentioned in this paper but however, our goal is to help finding the option of implementation which has the minimum cost. Graph theory is adopted and particularly the study of directed hypergraphs, to present a new idea algorithm capable of solving this optimization problem. This algorithm provides an exact-optimal solution, unlike the previously applied heuristic methods which give a near-optimal solution. Furthermore in this work, we analyze the computational complexity of our algorithm.

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