Bi-criteria and approximation algorithms for restricted matchings

In this work we study approximation algorithms for the Restricted matching problem which is defined as follows: given a graph in which each edge e   has a color cece and a profit pe∈Q+pe∈Q+, we want to compute a maximum (cardinality or profit) matching in which no more than wj∈Z+wj∈Z+ edges of color cjcj are present. This kind of problems, beside the theoretical interest on its own right, emerges in multi-fiber optical networking systems, where we interpret each unique wavelength that can travel through the fiber as a color class and we would like to establish communication between pairs of systems. We study approximation and bi-criteria algorithms for this problem which are based on linear programming techniques and, in particular, on polyhedral characterizations of the natural linear formulation of the problem. In our setting, we allow violations of the bounds wjwj and we can model our problem as a bi-criteria problem: we have two objectives that we want to optimize namely (a) to maximize the profit (maximum matching) while (b) minimizing the violation of the color bounds. We prove how we can “beat” the integrality gap of the natural linear programming formulation of the problem by allowing only a slight violation of the color bounds. In particular, our main result is constant approximation bounds for both criteria of the corresponding bi-criteria optimization problem.