This work is devoted to the study of the Balanced Connected Subgraph Problem (BCS) from a complexity, inapproximability and approximation point of view. The input is a graph $G=(V,E)$, with each vertex having been colored, “red” or “blue”; the goal is to find a maximum connected subgraph $G^{\prime }=(V^{\prime },E^{\prime })$ from G that is color-balanced (having exactly $|V^{\prime }|/2$ red vertices and $|V^{\prime }|/2$ blue vertices). This problem is known to be $\mathcal{NP}$-complete in general but polynomial in paths and trees. We propose a polynomial-time algorithm for block graph. We propose some complexity results for bounded-degree or bounded-diameter graphs, and also for bipartite graphs. We also propose inapproximability results for some graph classes, including chordal, planar, or subcubic graphs.

这项工作致力于从复杂性、不可逼近性和逼近性的角度研究平衡连接子图问题（BCS）。输入是一个图形$G=(伏,乙)$，每个顶点都被着色，“红色”或“蓝色”；目标是找到最大连通子图$G^{\prime }=({伏}^{\prime },{乙}^{\prime })$来自颜色平衡的G（恰好具有$|{伏}^{\prime }|/2$ 红色顶点和 $|{伏}^{\prime }|/2$蓝色顶点）。这个问题已知是$\mathcal{NP}$- 一般而言是完整的，但路径和树中的多项式。我们提出了一种用于块图的多项式时间算法。我们为有界度或有界直径图以及二部图提出了一些复杂性结果。我们还提出了一些图类的不可近似性结果，包括弦图、平面图或亚立方图。