In this paper, we consider Steiner forest and its generalizations, prize-collecting Steiner forest and k-Steiner forest, when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we present a simpler analysis of the polynomial-time approximation scheme (PTAS) of Borradaile et al. [12] for the Euclidean Steiner forest problem. This is done by proving a new structural property and modifying the dynamic programming by adding a new piece of information to each dynamic programming state. Next we develop a PTAS for a well-motivated case, i.e., the multiplicative case, of prize-collecting and budgeted Steiner forest. The ideas used in the algorithm may have applications in design of a broad class of bicriteria PTASs. At the end, we demonstrate why PTASs for these problems can be hard in the general Euclidean case (and thus for PTASs we cannot go beyond the multiplicative case).

This thesis studies the problem of determining achievable rates in heterogeneous wireless networks. We analyze the impact of location, traffic, and service heterogeneity. Consider a wireless network with n nodes located in a square area of size n communicating with each other over Gaussian fading channels. Location heterogeneity is modeled by allowing the nodes in the wireless network to be deployed in an arbitrary manner on the square area instead of the usual random uniform node placement. For traffic heterogeneity, we analyze the n × n dimensional unicast capacity region. For service heterogeneity, we consider the impact of multicasting and caching. This gives rise to the n × 2 n dimensional multicast capacity region and the 2 n × n dimensional caching capacity region. In each of these cases, we obtain an explicit informationtheoretic characterization of the scaling of achievable rates by providing a converse and a matching (in the scaling sense) communication architecture.