REDUCING THE OPTIMALITY GAP OF STRICTLY FUNDAMENTAL CYCLE BASES IN PLANAR GRIDS

The Minimum Cycle Basis (MCB) Problem is a classical problem in combinatorial optimization. AnO(mn + mn log n)-algorithm for this problem is known. Much faster heuristics have been examined in the context of sever al practical applications. These heuristics restrain the solution space to strictly fundame ntal cycle bases, hereby facing a significant loss in quality. We complement these experiment al studies by giving theoretical evidencewhystrictly fundamental cycle bases (SFCB) in general must be m uch worse than general MCB. Alon et al. (1995) provide the first non-trivial lower bound f or the minimum SFCB problem, which in general is NP-hard. For unweighted planar squa re grid graphs they achieve a lower bound of ln 2 2048 n log2 n − O(n), where ln 2 2048 ≈ 1 2955 . Using a new recursive approach, we are able to establish a sub stantially better lower bound. Our explicit method yields a lower bound of only 1 12 n log2 n−O(n). In addition, we provide an exact way of counting a short SFCB that was presented b y Alon et al. In particular, we improve their upper bound from 2n log2 n+ o(n log n) to only 4 3 n log2 n−Θ(n). We thus reduce the optimality gap for the MSFCB problem on plana r square grids to a factor of 16—compared to about 5900 being the former state-of-the-art. As a consequence, we conclude that for unweighted planar squ e grid graphs the ratio of the length of a minimum SFCB over a general MCB is Θ(log n).