Approximation algorithms for geometric problems
Approximation algorithms for NP-hard problems
Label placement by maximum independent set in rectangles
WADS '97 Selected papers presented at the international workshop on Algorithms and data structure
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Geometric Separator Theorems and Applications
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Which Problems Have Strongly Exponential Complexity?
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Finding small simple cycle separators for 2-connected planar graphs.
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
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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 II. We show that they can be exactly solved in subexponential time, more precisely, in time 2O(√n log n), by applying Miller's simple cycle planar separator theorem [6] (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 2O(d2dbn1-1/d log n) on such box graphs in d-dimensions. We do this by applying a separator theorem by Smith and Wormald [7]. 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 II problems can yield non-trivial upper bounds on approximation factors achievable in polynomial time.