TY - GEN

T1 - Hardness and algorithms for rainbow connectivity

AU - Chakraborty, Sourav

AU - Fischer, Eldar

AU - Matsliah, Arie

AU - Yuster, Raphael

PY - 2009

Y1 - 2009

N2 - An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every ∈ > 0, a connected graph with minimum degree at least ∈n has bounded rainbow connectivity, where the bound depends only on ∈, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.

AB - An edge-colored graph G is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing rc(G) is NP-Hard. In fact, we prove that it is already NP-Complete to decide if rc(G) = 2, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every ∈ > 0, a connected graph with minimum degree at least ∈n has bounded rainbow connectivity, where the bound depends only on ∈, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.

UR - http://www.scopus.com/inward/record.url?scp=84880247437&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84880247437

SN - 9783939897095

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 243

EP - 254

BT - STACS 2009 - 26th International Symposium on Theoretical Aspects of Computer Science

T2 - 26th International Symposium on Theoretical Aspects of Computer Science, STACS 2009

Y2 - 26 February 2009 through 28 February 2009

ER -