The Rainbow Cycle Cover Problem

Abstract

We model and solve the Rainbow Cycle Cover Problem (RCCP). Given a connected and undirected graph G = ( V , E , L ) and a coloring function ℓ that assigns a color to each edge of G from the finite color set L , a cycle whose edges have all different colors is called a rainbow cycle. The RCCP consists of finding the minimum number of disjoint rainbow cycles covering G . The RCCP on general graphs is known to be NP-complete. We model the RCCP as an integer linear program, we derive valid inequalities and we solve it by branch-and-cut. Computational results are reported on randomly generated instances

A branch‐and‐price algorithm for the rainbow cycle cover problems

“…We also report the ratio of the objective function value of an optimal LP relaxation of the RMP to that of an IP optimal solution to the MP as to analyze how the gap changes with respect to the number of vertices, the number of edges and the number of colors. In order to be consistent with Silvestri et al , we set w c = 2⌊ n /3⌋ for all trivial cycles as ⌊ n /3⌋ provides an upper bound on the number of non‐trivial rainbow cycles that can be used in any rainbow cycle cover. In the branch‐and‐bound tree, as the branching strategy, we select a variable with the value closest to 0.5 and branch on this variable.…”

“…However, the average computation time is much lower with the branch‐and‐price, particularly for settings with a high number of vertices. The highest average computation time required by the branch‐and‐price algorithm is less than 5 minutes (259.28 seconds) whereas Silvestri et al and Moreno et al report seven and six settings, respectively, with an average computation time exceeding 1 hour. In addition, Silvestri et al and Moreno et al fail to solve at least one instance for eight and seven settings respectively in 3 hours.…”

“…The TC‐RCCP has two objectives: minimize the number of vertices that are covered by trivial cycles and minimize the number of rainbow cycles. In this respect, a trivial cycle is associated with an artificially large cost (set as 2⌊ n /3⌋ by Silvestri et al , where n is the number of vertices) while all nontrivial cycles have unit cost. w c denotes the cost of using cycle c in the rainbow cycle cover.…”

“…However, in terms of the RCCP both solutions are optimal and cover the graph using two cycles. In the subsequent parts of the paper, we first focus on the TC‐RCCP, studied by Silvestri et al earlier under the name of the RCCP.…”

“…In Section 2, we present a mathematical formulation for the TC‐RCCP and the corresponding branch‐and‐price algorithm. In Section 3, we discuss the computational results on a set of instances in Silvestri et al . We present our concluding remarks in Section 4.…”