Complexity of Paths, Trails and Circuits in Arc-Colored Digraphs

Abstract

We deal with different algorithmic questions regarding properly arc-colored s-t paths, trails and circuits in arc-colored digraphs. Given an arc-colored digraph D c with c ≥ 2 colors, we show that the problem of maximizing the number of arc disjoint properly arc-colored s-t trails can be solved in polynomial time. Surprisingly, we prove that the determination of one properly arc-colored s-t path is NP-complete even for planar digraphs containing no properly arc-colored circuits and c = Ω(n), where n denotes the number of vertices in D c . If the digraph is an arc-colored tournament, we show that deciding whether it contains a properly arc-colored circuit passing through a given vertex x (resp., properly arc-colored Hamiltonian s-t path) is NP-complete, even if c = 2. As a consequence, we solve a weak version of an open problem posed in Gutin et. al. [17].

The Rainbow Cycle Cover Problem

“…Note that, thanks to constraints (5), constraints (6) are redundant in an integer solution, but we add them as valid cuts. We also add constraints (17), (18), (20), (21), (22), and (23) to the initial subproblem. From this point, in line 12 a search for violated constraints (7) is performed on the integer solutions, to prevent solutions with more than one cycle associated to each variable γ c .…”

“…These authors proved that the problem is NP-hard and since then it has been studied by numerous researchers [3,6,9,11,12,24,31]. Many other colored problems have been studied in the literature, like the Colorful Traveling Salesman Problem [5,14,23,28,32], the Minimum Labeling Steiner Problem [7,10,13], the Labeled Maximum Matching Problem [4], the Minimum Reload Cost Cycle Cover [18], general colored problems [20], rainbow graph structures problems [25], label optimization problems [21,22,27], reload optimization problems [19,15], and some other cycle cover problems [1,2,17]. Li and Zhang [26] investigated the complexity of the rainbow tree, cycle and path partition problems and proved that identifying a RCC with the minimum number of cycles is NP-hard.…”

The Complexity of Connectivity Problems in Forbidden-Transition Graphs and Edge-Colored Graphs

“…Similarly, many known results about forbidden-transition graphs are proofs of NP-completeness of problems that are polynomially solvable on standard graphs (e.g. [1], [7], [15], [21], [22], [28], [29], [39]).…”

The Euler and Chinese Postman Problems on 2-Arc-Colored Digraphs

“…In comparison, the problem of deciding the existence of PC cycle in a c-edge-colored undirected graph is polynomial-time solvable for every c ≥ 2 [33]. In [13], the authors proved that it is NP-hard to decide whether there is a PC path between two given vertices even in a c-edge-colored planar digraph which contains no PC cycle for c = Ω(|V (G)|). In the same paper, it was proved that deciding the existence of a PC cycle through a given vertex in a c-edge-colored tournament T is NP-hard.…”

“…In addition, deciding whether T has a PC s-t path or a PC Hamilton s-t path is NP-complete. However, there were a couple of positive results proved in [13], in particular, it was proved to be polynomial-time solvable to decide whether an edge-colored digraph contains a PC closed trail and to compute the maximum number of edge disjoint PC trails between any two vertices.…”