Balanced Substructures in Bicolored Graphs

An edge-colored graph is said to be balanced if it has an equal number of edges of each color. Given a graph G whose edges are colored using two colors and a positive integer k, the objective in the Edge Balanced Connected Subgraph problem is to determine if G has a balanced connected subgraph containing at least k edges. We first show that this problem is $$\textsf{NP}$$ -complete and remains so even if the solution is required to be a tree or a path. Then, we focus on the parameterized complexity of Edge Balanced Connected Subgraph and its variants (where the balanced subgraph is required to be a path/tree) with respect to k as the parameter. Towards this, we show that if a graph has a balanced connected subgraph/tree/path of size at least k, then it has one of size at least k and at most f(k) where f is a linear function. We use this result combined with dynamic programming algorithms based on color coding and representative sets to show that Edge Balanced Connected Subgraph and its variants are $$\textsf{FPT}$$ . Further, using polynomial-time reductions to the Multilinear Monomial Detection problem, we give faster randomized $$\textsf{FPT}$$ algorithms for the problems. In order to describe these reductions, we define a combinatorial object called relaxed-subgraph. We define this object in such a way that balanced connected subgraphs, trees and paths are relaxed-subgraphs with certain properties. This object is defined in the spirit of branching walks known for the Steiner Tree problem and may be of independent interest.