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 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.

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 Isomorphism, canonization, and definability for graphs of bounded rank width

2021 ◽  
Vol 64 (5) ◽  
pp. 98-105
Author(s):  
Martin Grohe ◽  
Daniel Neuen
Keyword(s):  
Fixed Point ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Graph Isomorphism ◽  
Isomorphism Problem ◽  
Descriptive Complexity ◽  
Graph Isomorphism Problem ◽  
Structural Graph ◽  
Graph Classes ◽  
Bounded Rank

We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. We prove that the combinatorial Weisfeiler-Leman algorithm of dimension (3 k + 4) is a complete isomorphism test for the class of all graphs of rank width at most k. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width. Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.

scholarly journals Marking Vertices to Find Graph Isomorphism Mapping Based on Continuous-Time Quantum Walk

Entropy ◽  
2018 ◽  
Vol 20 (8) ◽  
pp. 586 ◽  
Author(s):  
Xin Wang ◽  
Yi Zhang ◽  
Kai Lu ◽  
Xiaoping Wang ◽  
Kai Liu
Keyword(s):  
Polynomial Time ◽  
Continuous Time ◽  
Graph Isomorphism ◽  
Quantum Walk ◽  
Isomorphism Problem ◽  
Quantum Walks ◽  
Regular Graphs ◽  
Structure Preserving ◽  
Topological Structures ◽  
Time Quantum

The isomorphism problem involves judging whether two graphs are topologically the same and producing structure-preserving isomorphism mapping. It is widely used in various areas. Diverse algorithms have been proposed to solve this problem in polynomial time, with the help of quantum walks. Some of these algorithms, however, fail to find the isomorphism mapping. Moreover, most algorithms have very limited performance on regular graphs which are generally difficult to deal with due to their symmetry. We propose IsoMarking to discover an isomorphism mapping effectively, based on the quantum walk which is sensitive to topological structures. Firstly, IsoMarking marks vertices so that it can reduce the harmful influence of symmetry. Secondly, IsoMarking can ascertain whether the current candidate bijection is consistent with existing bijections and eventually obtains qualified mapping. Thirdly, our experiments on 1585 pairs of graphs demonstrate that our algorithm performs significantly better on both ordinary graphs and regular graphs.

scholarly journals Oriented Coloring on Recursively Defined Digraphs

Algorithms ◽  
2019 ◽  
Vol 12 (4) ◽  
pp. 87 ◽  
Author(s):  
Frank Gurski ◽  
Dominique Komander ◽  
Carolin Rehs
Keyword(s):  
Chromatic Number ◽  
Linear Time ◽  
Directed Graphs ◽  
Graph Isomorphism ◽  
Oriented Graph ◽  
Time Algorithm ◽  
Isomorphism Problem ◽  
Np Hard ◽  
Graph Isomorphism Problem ◽  
Acyclic Digraph

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G = ( V , A ) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows that finding the chromatic number of an oriented graph is an NP-hard problem, too. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.

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