scholarly journals A Polynomial Time Algorithm for Graph Isomorphism and Automorphism

2021 ◽  
Author(s):  
Sardar Anisul Haque
Keyword(s):  
Data Structure ◽  
Polynomial Time ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Tree Data ◽  
Tree Data Structure

This paper describes a polynomial time algorithm for solving graph isomorphism and automorphism. We introduce a new tree data structure called Walk Length Tree. We show that such tree can be both constructed and compared with another in polynomial time. We prove that graph isomorphism and automorphism can be solved in polynomial time using Walk Length Trees.

Physically-motivated dynamical algorithms for the graph isomorphism problem

2005 ◽  
Vol 5 (6) ◽  
pp. 492-506
Author(s):  
S.-Y. Shiau ◽  
R. Joynt ◽  
S.N. Coppersmith
Keyword(s):  
Polynomial Time ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Strongly Regular Graphs ◽  
Regular Graphs ◽  
Original Graph ◽  
Graph Isomorphism Problem ◽  
Strongly Regular

The graph isomorphism problem (GI) plays a central role in the theory of computational complexity and has importance in physics and chemistry as well \cite{kobler93,fortin96}. No polynomial-time algorithm for solving GI is known. We investigate classical and quantum physics-based polynomial-time algorithms for solving the graph isomorphism problem in which the graph structure is reflected in the behavior of a dynamical system. We show that a classical dynamical algorithm proposed by Gudkov and Nussinov \cite{gudkov02} as well as its simplest quantum generalization fail to distinguish pairs of non-isomorphic strongly regular graphs. However, by combining the algorithm of Gudkov and Nussinov with a construction proposed by Rudolph \cite{rudolph02} in which one examines a graph describing the dynamics of two particles on the original graph, we find an algorithm that successfully distinguishes all pairs of non-isomorphic strongly regular graphs that we tested with up to 29 vertices.

scholarly journals A polynomial‐time algorithm for simple undirected graph isomorphism

2021 ◽  
Author(s):  
Jing He ◽  
Guangyan Huang ◽  
Jie Cao ◽  
Zhiwang Zhang ◽  
Hui Zheng ◽  
...  
Keyword(s):  
Polynomial Time ◽  
Undirected Graph ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm

On Some Equivalence Relations for Probabilistic Processes1

1992 ◽  
Vol 17 (3) ◽  
pp. 211-234
Author(s):  
Dung T. Huynh ◽  
Lu Tian
Keyword(s):  
Polynomial Time ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Equivalence Relations ◽  
Transition Systems ◽  
Bisimulation Equivalence ◽  
Finite Trace ◽  
Trace Equivalence

In this paper, we investigate several equivalence relations for probabilistic labeled transition systems: bisimulation equivalence, readiness equivalence, failure equivalence, trace equivalence, maximal trace equivalence and finite trace equivalence. We formally prove the inclusions (equalities) among these equivalences. We also show that readiness, failure, trace, maximum trace and finite trace equivalences for finite probabilistic labeled transition systems are decidable in polynomial time. This should be contrasted with the PSPACE completeness of the same equivalences for classical labeled transition systems. Moreover, we derive an efficient polynomial time algorithm for deciding bisimulation equivalence for finite probabilistic labeled transition systems. The special case of initiated probabilistic transition systems will be considered. We show that the isomorphism problem for finite initiated labeled probabilistic transition systems is NC(1) equivalent to graph isomorphism.

scholarly journals The graph isomorphism problem on geometric graphs

2014 ◽  
Vol Vol. 16 no. 2 (PRIMA 2013) ◽  
Author(s):  
Ryuhei Uehara
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Polynomial Hierarchy ◽  
Intersection Graphs ◽  
Graph Isomorphism Problem ◽  
Graph Classes

Special issue PRIMA 2013 International audience The graph isomorphism (GI) problem asks whether two given graphs are isomorphic or not. The GI problem is quite basic and simple, however, it\textquoterights time complexity is a long standing open problem. The GI problem is clearly in NP, no polynomial time algorithm is known, and the GI problem is not NP-complete unless the polynomial hierarchy collapses. In this paper, we survey the computational complexity of the problem on some graph classes that have geometric characterizations. Sometimes the GI problem becomes polynomial time solvable when we add some restrictions on some graph classes. The properties of these graph classes on the boundary indicate us the essence of difficulty of the GI problem. We also show that the GI problem is as hard as the problem on general graphs even for grid unit intersection graphs on a torus, that partially solves an open problem.

SPECTRUM BASED TECHNIQUES FOR GRAPH ISOMORPHISM

2009 ◽  
Vol 20 (03) ◽  
pp. 479-499
Author(s):  
SANGUTHEVAR RAJASEKARAN ◽  
VAMSI KUNDETI
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Graph Isomorphism ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Cospectral Graphs ◽  
Graph Isomorphism Problem ◽  
Spectra Of Graphs ◽  
New Construction

The graph isomorphism problem is to check if two given graphs are isomorphic. Graph isomorphism is a well studied problem and numerous algorithms are available for its solution. In this paper we present algorithms for graph isomorphism that employ the spectra of graphs. An open problem that has fascinated many a scientist is if there exists a polynomial time algorithm for graph isomorphism. Though we do not solve this problem in this paper, the algorithms we present take polynomial time. These algorithms have been tested on a good collection of instances. However, we have not been able to prove that our algorithms will work on all possible instances. In this paper, we also give a new construction for cospectral graphs.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

Sign in / Sign up

Export Citation Format

Share Document