Parameterized Complexity of Geometric Problems

This paper surveys parameterized complexity results for NP-hard geometric problems. Geometric problems arise frequently in application domains as diverse as computer graphics [19], computer vision [4, 35, 43], VLSI design [64], geographic information systems [73, 30], graph drawing [72], and robotics [65, 37], and typically involve (sets of) geometric objects, such as, points, line segments, balls, or polytopes, which usually lie in some metric space. Designing efficient algorithms and data structures for such problems is the main focus in the field of computational geometry [60, 57, 16]. Many geometric problems can be formulated as combinatorial optimization problems where the objective is to maximize or minimize a function subject to constraints induced by a given collection of geometric objects; by exploiting the geometric nature of such problems, one can obtain faster and simpler algorithms. Geometric optimization has been a topic of extensive research, and many general techniques have been developed that yield efficient polynomial-time algorithms for a wide range of problems; see the excellent survey by Agarwal and Sharir [1].