On the approximability of the minimum strictly fundamental cycle basis problem

Authors:
Giulia Galbiati;Romeo Rizzi;Edoardo Amaldi
Affiliations:
Dipartimento di Informatica e Sistemistica, Universití degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy;Dipartimento di Matematica e Informatica, Universití degli Studi di Udine, Via delle Scienze 208, 33100 Udine, Italy;Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
Venue:
Discrete Applied Mathematics
Year:
2011

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Abstract

We consider the problem of finding a strictly fundamental cycle basis of minimum weight in the cycle space associated with an undirected connected graph G, where a nonnegative weight is assigned to each edge of G and the total weight of a basis is defined as the sum of the weights of all the cycles in the basis. Several heuristics have been proposed to tackle this NP-hard problem, which has some interesting applications. In this paper we show that this problem is APX-hard, even when restricted to unweighted graphs, and hence does not admit a polynomial-time approximation scheme, unless P=NP. Using a recent result on the approximability of lower-stretch spanning trees (Elkin et al. (2005) [7]), we obtain that the problem is approximable within O(log^2nloglogn) for arbitrary graphs. We obtain tighter approximability bounds for dense graphs. In particular, the problem restricted to complete graphs admits a polynomial-time approximation scheme.