On variations of geometric covering-packing

Abstract

This thesis aims at designing exact and approximation algorithms and showing NP-completeness results for variations and special cases of covering, hitting, and packing problems naturally arising in the field of computationalgeometry. A well known variation of set cover is the class cover problem. In spite of its application to data classification, it has some limitations. Keeping these limitations in mind, we investigate a generalization of class cover problem with two variations: multiplecoverageandsinglecoverage. Theobjectsarerestrictedtoaxis-parallelstrips andhalf-strips. Weprovethatsinglecoveragewithstripsandbothsingleandmultiple coveragewithhalf-stripsinmorethanonedirectionareNP-complete. Fortheseproblems, constant factor approximation algorithms are proposed. Further, we prove that multiplecoveragewithstripsandbothsingleandmultiplecoveragewithhalf-stripsin exactly one direction are in P. Two interesting results are the NP-completeness of single and multiple coverage variations of generalized class cover problem with half-strips in two opposite directions. To prove these results, we first prove the NP-completeness of two intermediate problems: hitting set and set cover with half-strips in two opposite directions. These intermediate problems are interesting themselves. Hitting set has application in visibility preserving terrain simplification whereas set cover has military application or can be used to design computer games. An intriguing special case occurs when the type as well as the location of given objects are restricted. For example, practical applications like restricted wireless coverage may require that the objects intersect an inclined line. We consider covering, hitting, piercing, and packing problems with axis-parallel unit squares, squares, unitheight rectangles, and rectangles intersecting an inclined line. We prove that both set cover and hitting set with unit squares or unit-height rectangles and piercing set with rectanglesareNP-complete. Polynomialtimealgorithmsaredesignedforpiercingset withunitsquaresandforindependentsetwithunit-heightrectangles. Finally,anexact algorithm for independent set with squares is presented.