Breaking the O(m^2 n) Barrier for Minimum Cycle Bases.

We give improved algorithms for constructing minimum directed and undirected cycle bases in graphs. For general graphs, the new algorithms are Monte Carlo and have running time O(m^ω ), where ω is the exponent of matrix multiplication. The previous best algorithm had running time O(m^2 n). For planar graphs, the new algorithm is deterministic and has running time O(n^2). The previous best algorithm had running time O(n^2 log n). A key ingredient to our improved running times is the insight that the search for minimum bases can be restricted to a set of candidate cycles of total length O(nm)