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A d-interval is the union of d disjoint intervals on the real line. A d-track interval is the union of d disjoint intervals on d disjoint parallel lines called tracks, one interval on each track. As generalizations of the ubiquitous interval graphs, d-interval graphs and d-track interval graphs have wide applications, traditionally to scheduling and resource allocation, and more recently to bioinformatics. In this paper, we prove that recognizing d-track interval graphs is NP-complete for any constant d≥2. This confirms a conjecture of Gyárfás and West in 1995. Previously only the complexity of the case d=2 was known. Our proof in fact implies that several restricted variants of this graph recognition problem, i.e., recognizing balanced d-track interval graphs, unit d-track interval graphs, and (2,...,2) d-track interval graphs, are all NP-complete. This partially answers another question recently raised by Gambette and Vialette. We also prove that recognizing depth-two 2-track interval graphs is NP-complete, even for the unit case. In sharp contrast, we present a simple linear-time algorithm for recognizing depth-two unit d-interval graphs. These and other results of ours give partial answers to a question of West and Shmoys in 1984 and a similar question of Gyárfás and West in 1995. Finally, we give the first bounds on the track number and the unit track number of a graph in terms of the number of vertices, the number of edges, and the maximum degree, and link the two numbers to the classical concepts of arboricity. [ABSTRACT FROM AUTHOR]

Given a set of points in the plane, and a sweep-line as a tool, what is best way to move the points to a target point using a sequence of sweeps? In a sweep, the sweep-line is placed at a start position somewhere in the plane, then moved orthogonally and continuously to another parallel end position, and then lifted from the plane. The cost of a sequence of sweeps is the total length of the sweeps. Another parameter of interest is the number of sweeps. Four variants are discussed, depending on whether the target is a hole or a pile, and whether the target is specified or freely selected by the algorithm. Here we present a ratio 4/ π≈1.27 approximation algorithm in the length measure, which performs at most four sweeps. We also prove that, for the two constrained variants, there are sets of n points for which any sequence of minimum cost requires 3 n/2− O(1) sweeps. [ABSTRACT FROM AUTHOR]

Abstract  We present a simple O(m+n 6/ε 12) time (1+ε)-approximation algorithm for finding a minimum-cost sequence of lines to cut a convex n-gon out of a convex m-gon. [ABSTRACT FROM AUTHOR]