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.

本文考虑了两个图覆盖问题，最小星座覆盖（CC）和最小k分裂星座覆盖（k-SCC）。这些问题的输入包括一个图\（G = \ left（V，E \ right）\）和G个星的一组\ {{\ mathcal {C}} \\}，并且输出是最小基数分集ç，使得任何两个不同的分ç是边缘不相交并的星的联合ç涵盖的所有边缘ģ。对于CC来说，集合C必须与G的边缘或\（{\ mathcal {C}} \），而对于k-SCC，给出整数k，并且C的元素必须是通过拆分\（{\ mathcal {C}} \）的星而获得的k星。这项工作证明，除非\（P = NP \），否则CC不允许多项式时间\（\ left | {\数学{C}} \ right | ^ {{\ mathcal {O}} \ left（1 \ right） } \） -近似算法和k-SCC不能\（\ left（\ left（1- \ epsilon \ right）\ ln \ left | E \ right | \ right）\） -对于任何\ （\ epsilon> 0 \）。另外，给出了最差的可行解的逼近比，对于k-SCC，提出了多项式时间逼近算法，实现了\（\ left（\ ln \ left | E \ right |-\ ln \ ln \ left | E \ right | + \ varTheta \ left（1 \ right）\ right）\）近似比率。此外，还针对包含边界度树和仙人掌的特殊类别的图提供了多项式时间算法。