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This paper classifies common mobile robot on-line motion planning problems according to their competitive complexity. The competitiveness of an on-line algorithm measures its worst case performance relative to the optimal off-line solution to the problem. Competitiveness usually means constant relative performance. This paper generalizes competitiveness to any functional relationship between on-line performance and optimal off-line solution. The constants in the functional relationship must be scalable and may depend only upon on-line information. Given an on-line task, its competitive complexity class is a pair of lower and upper bounds on the competitive performance of all on-line algorithms for the task, such that the two bounds satisfy the same functional relationship. The paper classifies the following on-line motion planning problems into competitive classes: area coverage, navigation to a target, and on-line search for an optimal path. In particular, it is shown that navigation to a target whose position is either apriori known or recognized upon arrival belongs to a quadratic competitive complexity class. The hardest on-line problem involves navigation in unknown variable traversibility environments. Under certain restriction on traversibility, this last problem belongs to an exponential competitive complexity class.

We explore an online problem where a group of robots has to find a target whose position is unknown in an unknown planar environment whose geometry is acquired by the robots during task execution. The critical parameter in such a problem is the physical motion time, which, under the assumption of uniform velocity of all the robots, corresponds to length or cost of the path traveled by the robot which finds the target. The Competitiveness of an online algorithm measures its performance relative to the optimal offline solution to the problem. While competitiveness usually means constant relative performance, this paper uses generalized competitiveness, i.e. any functional relationship between online performance and optimal offline solution. Given an online task, its Competitive Complexity Class is a pair of lower and upper bounds on the competitive performance of all online algorithms for the task, such that the two bounds satisfy the same functional relationship. We classify a common online motion planning problem into competitive class. In particular, it is shown that group of robots navigation to a target whose position is recognized only upon arrival belongs to a quadratic competitive class. This paper describes a new online navigation algorithm, called MRSAM (short for Multi-Robot Search Area Multiplication), which requires linear memory and has a quadratic competitive performance. Moreover, it is shown that in general any online navigation algorithm must have at least a quadratic competitive performance. The MRSAM algorithm achieves the quadratic lower bound and thus has optimal competitiveness. The algorithm's performance is illustrated in an office-like environments.