Choice is hard

Esther M. Arkin, Aritra Banik, Paz Carmi, Gui Citovsky, Matthew J. Katz, Joseph S.B. Mitchell, Marina Simakov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

18 Scopus citations


Let P = {C1, C2, . . . , Cn} be a set of color classes, where each color class Ci consists of a pair of objects. We focus on two problems in which the objects are points on the line. In the first problem (rainbow minmax gap), given P, one needs to select exactly one point from each color class, such that the maximum distance between a pair of consecutive selected points is minimized. This problem was studied by Consuegra and Narasimhan, who left the question of its complexity unresolved. We prove that it is NP-hard. For our proof we obtain the following auxiliary result. A 3-SAT formula is an LSAT formula if each clause (viewed as a set of literals) intersects at most one other clause, and, moreover, if two clauses intersect, then they have exactly one literal in common. We prove that the problem of deciding whether an LSAT formula is satisfiable or not is NP-complete. We present two additional applications of the LSAT result, namely, to rainbow piercing and rainbow covering. In the second problem (covering color classes with intervals), given P, one needs to find a minimum-cardinality set I of intervals, such that exactly one point from each color class is covered by an interval in I. Motivated by a problem in storage systems, this problem has received significant attention. Here, we settle the complexity question by proving that it is NP-hard.

Original languageEnglish
Title of host publicationAlgorithms and Computation - 26th International Symposium, ISAAC 2015, Proceedings
EditorsKhaled Elbassioni, Kazuhisa Makino
PublisherSpringer Verlag
Number of pages11
ISBN (Print)9783662489703
StatePublished - 1 Jan 2015
Event26th International Symposium on Algorithms and Computation, ISAAC 2015 - Nagoya, Japan
Duration: 9 Dec 201511 Dec 2015

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference26th International Symposium on Algorithms and Computation, ISAAC 2015

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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