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Intersection graphs of planar geometric objects such as intervals, disks,rectangles and pseudo-disks are well studied. Motivated by variousapplications, Butman et al. in SODA 2007 considered algorithmic questions inintersection graphs of $t$-intervals. A $t$-interval is a union of at most $t$distinct intervals (here $t$ is a parameter) -- these graphs are referred to asMultiple-Interval Graphs. Subsequent work by Kammer et al. in Approx 2010 alsoconsidered $t$-disks and other geometric shapes. In this paper we revisit someof these algorithmic questions via more recent developments in computationalgeometry. For the minimum weight dominating set problem, we give a simple $O(t\log t)$ approximation for Multiple-Interval Graphs, improving on thepreviously known bound of $t^2$ . We also show that it is NP-hard to obtain an$o(t)$-approximation in this case. In fact, our results hold for theintersection graph of a set of t-pseudo-disks which is a much larger class. Weobtain an ${\Omega}(1/t)$-approximation for the maximum weight independent setin the intersection graph of $t$-pseudo-disks. Our results are based on simplereductions to existing algorithms by appropriately bounding the unioncomplexity of the objects under consideration.

Algorithms for Intersection Graphs of Multiple Intervals and Pseudo Disks