Shortest Color-Spanning Intervals

Abstract

Given a set of n points on a line, where each point has one of k colors, and given an integer s _i  ≥ 1 for each color i, 1 ≤ i ≤ k, the problem Shortest Color-Spanning t Intervals (SCSI-t) aims at finding t intervals to cover at least s _i points of each color i, such that the maximum length of the intervals is minimized. Chen and Misiolek introduced the problem SCSI-1, and presented an algorithm running in O(n) time if the input points are sorted. Khanteimouri et al. gave an O(n ²logn) time algorithm for the special case of SCSI-2 with s _i  = 1 for all colors i. In this paper, we present an improved algorithm with running time of O(n ²) for SCSI-2 with arbitrary s _i  ≥ 1. We also obtain some interesting results for the general problem SCSI-t. From the negative direction, we show that approximating SCSI-t within any ratio is NP-hard when t is part of the input, is W[2]-hard when t is the parameter, and is W[1]-hard with both t and k as parameters. Moreover, the NP-hardness and the W[2]-hardness with parameter t hold even if s _i  = 1 for all i. From the positive direction, we show that SCSI-t with s _i  = 1 for all i is fixed-parameter tractable with k as the parameter, and admits an exact algorithm running in O(2^k n· max {k,logn}) time.