The evolution of antibiotic resistance presents a practical and theoretical challenge: the design of strategies that limit the risk of evolved resistance while effectively treating current patients. Sequentially cycling antibiotics has been proposed as a way to slow the evolution of resistance by reducing the extent of adaptation to a given drug, and clinical trials have demonstrated its effectiveness in some settings. Empirical fitness landscapes in theory allow the sequence of drugs to be refined to maximize tradeoffs between drugs and thereby slow adaptation even further. Using the measured growth rates of 16 genotypes of Escherichia coli in the presence of β-lactam antibiotics, we test an adaptive strategy, based on a Markov chain transition matrix, to select drug sequences that continuously minimize resistance. Cycling is never selected over the long term. Instead, monotherapy with the antibiotic that permits the least growth in its landscape's absorbing state is rapidly selected from different starting conditions. Analysis of a synthetic fitness landscape shows that cycling drugs that induce sensitivity to one other could, in theory, outperform monotherapy. These results underscore the importance of considering the specific topologies of fitness landscape in determining whether to cycle drugs and suggest a general computational approach to identify high performing, practical strategies to manage resistance.