Parallel Connectivity in Edge-Colored Complete Graphs: Complexity Results

Given an edge-colored graph \(G_c\), a set of p pairs of vertices \((a_i,b_i)\) together with p numbers \(k_1,k_2, \ldots k_p\) associated with the pairs, can we find a set of alternating paths linking the pairs \((a_1,b_1)\), \((a_2,b_2), \ldots \), in their respective numbers \(k_1,k_2,\ldots k_p\)? Such is the question addressed in this paper. The problem being highly intractable, we consider a restricted version of it to edge-colored complete graphs. Even so restricted, the problem remains intractable if the paths/trails must be edge-disjoint, but it ceases to be so if the paths/trails are to be vertex-disjoint, as is proved in this paper. An approximation algorithm is presented in the end, with a performance ratio asymptotically close to 3/4 for a restricted version of the problem.