On finding a cycle basis with a shortest maximal cycle

doi:10.1016/0020-0190(94)00231-M

Information Processing Letters

Volume 54, Issue 1, 14 April 1995, Pages 55-58

https://doi.org/10.1016/0020-0190(94)00231-M Get rights and content

Abstract

The Shortest Maximal Cycle Basis (SMCB) problem is that of finding a cycle basis B of a given graph G such that the length of the longest cycle included in B is the smallest among all bases of G. We show that any cycle basis B′ of G such that the sum of the lengths of the cycles included in B′ is the smallest among all cycle bases of G constitutes a solution to the SMCB problem. Finding a basis with the latter property requires at most O(m³n) steps using Horton's algorithm where m is the number of edges and n is the number of vertices.

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An undirected biconnected graph G with nonnegative integer lengths on the edges is given. The problem we consider is that of finding a cycle basis B of G such that the length of the longest cycle included in B is the smallest among all cycle bases of G. We first observe that Horton's algorithm [SIAM J. Comput. 16 (2) (1987) 358–366] provides a fast solution of the problem that extends the one given by Chickering et al. [Inform. Process. Lett. 54 (1995) 55–58] for uniform graphs. On the other hand we show that, if the basis is required to be fundamental, then the problem is NP-hard and cannot be approximated within 2−ϵ, ∀ϵ>0, even with uniform lengths, unless P=NP. This problem remains NP-hard even restricted to the class of complete graphs; in this case it cannot be approximated within 13/11−ϵ, ∀ϵ>0, unless P=NP; it is instead approximable within 2 in general, and within 3/2 if the triangle inequality holds.

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^∗: Author's primary affiliation: Computer Science Department, Technion, Haifa 32000, Israel.

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On finding a cycle basis with a shortest maximal cycle

Abstract

On the inherent intractability of certain coding problems

IEEE Trans. Inform. Theory

On cycle bases of a graph

Minimum length fundamental cycle set

IEEE Trans. Circuits and Systems

Algorithms for generating fundamental cycles in a graph

ACM Trans. Math. Software

A polynomial-time algorithm to find the shortest cycle basis of a graph

SIAM J. Comput.

Minimal bases of cycles of a graph