Capacitated Metric Labeling

• Authors: Matthew Andrews;Mohammad Taghi Hajiaghayi;Howard Karloff;Ankur Moitra
• Venue: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2011

We introduce Capacitated Metric Labeling. As in Metric Labeling, we are given a weighted graph G = (V, E), a label set L, a semimetric dl on this label set, and an assignment cost function &phis;: V x L → R+. The goal in Metric Labeling is to find an assignment f: V → L that minimizes a particular two-cost function. Here we add the additional restriction that each label ti receive at most li nodes, and we refer to this problem as Capacitated Metric Labeling. Allowing the problem to specify capacities on each label allows the problem to more faithfully represent the classification problems that Metric Labeling is intended to model. Our main positive result is a polynomial-time, O(log |V|)-approximation algorithm when the number of labels is fixed, which is the most natural parameter range for classification problems. We also prove that it is impossible to approximate the value of an instance of Capacitated Metric Labeling to within any finite factor, if P ≠ NP. Yet this does not address the more interesting question of how hard Capacitated Metric Labeling is to approximate when we are allowed to violate capacities. To study this question, we introduce the notion of the "congestion" of an instance of Capacitated Metric Labeling. We prove that (under certain complexity assumptions) there is no polynomial-time approximation algorithm that can approximate the congestion to within O((log|L|)1/2−∈) (for any ∈ 0) and this implies as a corollary that any polynomial-time approximation algorithm that achieves a finite approximation ratio must multiplicatively violate the label capacities by Ω((log |L|)1/2−∈). We also give a O(log |L|)-approximation algorithm for congestion.