Algorithms for shortest paths and d-cycle problems

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Let G be a weighted graph with n vertices and m edges. We address the d-cycle problem, i.e., the problem of finding a subgraph of minimum weight with given cyclomatic number d. Hartvigsen [Minimum path bases, J. Algorithms 15 (1993) 125–142] presented an algorithm with running time O( n 2 m) and O( n 2 d−1 m 2) for the cyclomatic numbers d=1 and d⩾2, respectively. Using a ( d+1)-shortest-paths algorithm, we develop a new more efficient algorithm for the d-cycle problem with running time O( n 2 d−1 + n 2 m+ n 3log n).