TY - GEN

T1 - Choice is hard

AU - Arkin, Esther M.

AU - Banik, Aritra

AU - Carmi, Paz

AU - Citovsky, Gui

AU - Katz, Matthew J.

AU - Mitchell, Joseph S.B.

AU - Simakov, Marina

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2015.

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84952017861&partnerID=8YFLogxK

U2 - 10.1007/978-3-662-48971-0_28

DO - 10.1007/978-3-662-48971-0_28

M3 - Conference contribution

AN - SCOPUS:84952017861

SN - 9783662489703

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 318

EP - 328

BT - Algorithms and Computation - 26th International Symposium, ISAAC 2015, Proceedings

A2 - Elbassioni, Khaled

A2 - Makino, Kazuhisa

PB - Springer Verlag

T2 - 26th International Symposium on Algorithms and Computation, ISAAC 2015

Y2 - 9 December 2015 through 11 December 2015

ER -