Online searching with turn cost

We consider the problem of searching for an object on a line at an unknown distance OPT from the original position of the searcher, in the presence of a cost of d for each time the searcher changes direction. This is a generalization of the well-studied linear-search problem. We describe a strategy that is guaranteed to find the object at a cost of at most 9 · OPT + 2 d, which has the optimal competitive ratio 9 with respect to OPT plus the minimum corresponding additive term. Our argument for upper and lower bound uses an infinite linear program, which we solve by experimental solution of an infinite series of approximating finite linear programs, estimating the limits, and solving the resulting recurrences for an explicit proof of optimality. We feel that this technique is interesting in its own right and should help solve other searching problems. In particular, we consider the star search or cow-path problem with turn cost, where the hidden object is placed on one of m rays emanating from the original position of the searcher. For this problem we give a tight bound of (1 + 2 m^m / (m - 1)^{m - 1}) OPT + m ((m / (m - 1))^{m - 1} - 1) d. We also discuss tradeoffs between the corresponding coefficients and we consider randomized strategies on the line.