On a Special Co-cycle Basis of Graphs
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
An improved heuristic for computing short integral cycle bases
Journal of Experimental Algorithmics (JEA)
Minimum Cycle Bases and Their Applications
Algorithmics of Large and Complex Networks
Minimum cycle bases: Faster and simpler
ACM Transactions on Algorithms (TALG)
On the approximability of the minimum strictly fundamental cycle basis problem
Discrete Applied Mathematics
Properties of Gomory-Hu co-cycle bases
Theoretical Computer Science
Efficient deterministic algorithms for finding a minimum cycle basis in undirected graphs
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Survey: Cycle bases in graphs characterization, algorithms, complexity, and applications
Computer Science Review
Integral cycle bases for cyclic timetabling
Discrete Optimization
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In the last years, new variants of the minimum cycle basis (MCB) problem and new classes of cycle bases have been introduced, as motivated by several applications from disparate areas of scientific and technological inquiry. At present, the complexity status of the MCB problem is settled only for undirected, directed, and strictly fundamental cycle bases (SFCBs). Weakly fundamental cycle bases (WFCBs) form a natural superclass of SFCBs. A cycle basis of a graph G is a WFCB iff ½=0 or there exists an edge e of G and a circuit C i in such that is a WFCB of Ge. WFCBs still possess several of the nice properties offered by SFCBs. At the same time, several classes of graphs enjoying WFCBs of cost asymptotically inferior to the cost of the cheapest SFCBs have been found and exhibited in the literature. Considered also the computational difficulty of finding cheap SFCBs, these works advocated an in-depth study of WFCBs. In this paper, we settle the complexity status of the MCB problem for WFCBs (the MWFCB problem). The problem turns out to be -hard. However, in this paper, we also offer a simple and practical 2⌈log 2 n⌉-approximation algorithm for the MWFCB problem. In O(n ν) time, this algorithm actually returns a WFCB whose cost is at most 2⌈log 2 n⌉∑e∈E(G) w e, thus allowing a fast 2⌈log 2 n⌉-approximation also for the MCB problem. With this algorithm, we provide tight bounds on the cost of any MCB and MWFCB.