Finding a Shortest Odd Hole

An odd hole in a graph is an induced cycle with odd length greater than 3. In an earlier paper (with Sophie Spirkl), solving a longstanding open problem, we gave a polynomial-time algorithm to test if a graph has an odd hole. We subsequently showed that, for every t , there is a polynomial-time algorithm to test whether a graph contains an odd hole of length at least t . In this article, we give an algorithm that finds a shortest odd hole, if one exists.

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The graph isomorphism problem on geometric graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2076 ◽

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.

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SPECTRUM BASED TECHNIQUES FOR GRAPH ISOMORPHISM

International Journal of Foundations of Computer Science ◽

10.1142/s0129054109006693 ◽

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.

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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.

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