Subexponential-Time Algorithms for Maximum Independent Set and Related Problems on Box Graphs

Abstract

A box graph is the intersection graph of orthogonal rectangles in the plane. We consider such basic combinatorial problems on box graphs as maximum independent set, minimum vertex cover and maximum induced subgraph with polynomial-time testable hereditary property Π. We show that they can be exactly solved in subexponential time, more precisely, in time \( 2^{O(\sqrt n \log n)} \), by applying Miller’s simple cycle planar separator theorem (in spite of the fact that the input box graph might be strongly non-planar).Furthermore we extend our idea to include the intersection graphs of orthogonal d-cubes of bounded aspect ratio and dimension. We present an algorithm that solves maximum independent set and the other aforementioned problems in time \( 2^{O(d2^d bn^{1 - 1/d} \log n)} \) on such box graphs in d-dimensions. We do this by applying a separator theorem by Smith and Wormald .Finally, we show that in general graph case substantially subexponential algorithms for maximum independent set and the maximum induced subgraph with polynomial-time testable hereditary property Π problems can yield non-trivial upper bounds on approximation factors achievable in polynomial time.