Minimum constellation covers: hardness, approximability and polynomial cases

Abstract

This paper considers two graph covering problems, the Minimum Constellation Cover (CC) and the Minimum k-Split Constellation Cover (k-SCC). The input of these problems consists on a graph $G=\left( V,E\right)$ and a set ${\mathcal {C}}$ of stars of G, and the output is a minimum cardinality set of stars C, such that any two different stars of C are edge-disjoint and the union of the stars of C covers all edges of G. For CC, the set C must be compound by edges of G or stars of ${\mathcal {C}}$ while, for k-SCC, an integer k is given and the elements of C must be k-stars obtained by splitting stars of ${\mathcal {C}}$. This work proves that unless $P=NP$, CC does not admit polynomial time $\left| {\mathcal {C}}\right| ^{{\mathcal {O}}\left( 1\right) }$-approximation algorithms and k-SCC cannot be $\left( \left( 1-\epsilon \right) \ln \left| E\right| \right)$-approximated in polynomial time, for any $\epsilon >0$. Additionally, approximation ratios are given for the worst feasible solutions of the problems and, for k-SCC, polynomial time approximation algorithms are proposed, achieving a $\left( \ln \left| E\right| -\ln \ln \left| E\right| +\varTheta \left( 1\right) \right)$ approximation ratio. Also, polynomial time algorithms are presented for special classes of graphs that include bounded degree trees and cacti.