We give a polynomial-time algorithm for synthesizing efficient regular strategies in adversarial patrolling games with general topology. Regular strategies use finite memory to gather some relevant information about the history of Defender's moves which results in substantially better protection of the targets. So far, the scope of automatic strategy synthesis was limited to positional strategies (which ignore the history) or to regular strategies where the underlying finite-memory observer had to be supplied manually. Furthermore, the existing methods do not give any information on how far are the constructed strategies from being optimal. In this paper, we try to overcome these limitations. We develop a novel gradient-based algorithm for synthesizing regular strategies where the underlying finite-memory observers are constructed algorithmically. The running time of our algorithm is polynomial which makes the algorithm applicable to instances of realistic size. Furthermore, we develop an algorithm for computing an upper bound on the best achievable protection, and compare the quality of the constructed strategies against this bound. Thus, we can effectively measure the "distance'' of the constructed strategies from optimal strategies, and our experiments show that this distance is often quite small.