Finite Satisfiability in Infinite-Valued Łukasiewicz Logic
SUM '09 Proceedings of the 3rd International Conference on Scalable Uncertainty Management
Minimum Cycle Bases of Weighted Outerplanar Graphs
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Minimum cycle bases of weighted outerplanar graphs
Information Processing Letters
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
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We consider the problem of computing a minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over $\mathbb{F}_{2}$generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Minimum cycle basis are useful in a number of contexts, e.g. the analysis of electrical networks and structural engineering. The previous best algorithm for computing a minimum cycle basis has running time O(m ω n), where ω is the best exponent of matrix multiplication. It is presently known that ωO(m 2 n+mn 2log n) algorithm. When the edge weights are integers, we have an O(m 2 n) algorithm. For unweighted graphs which are reasonably dense, our algorithm runs in O(m ω ) time. For any ε0, we also design an 1+ε approximation algorithm. The running time of this algorithm is O((m ω /ε)log (W/ε)) for reasonably dense graphs, where W is the largest edge weight.