We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance trade-off as the best known approximation schemes for planar graphs. We also define the concept of -precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance trade-off than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for -precision unit disk graphs, many more graph problems have efficient approximation schemes. Our NC approximation schemes can also be extended to obtain efficient NC approximation schemes for several PSPACE-hard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann and Widmayer. The approximation schemes for hierarchically specified un...

We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number k of nodes are required to be connected in the solution. A prototypical example is the kMST problem in which we require a tree of minimum weight spanning at least k nodes in an edge-weighted graph. We show that the kMST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio 2 p k for the general edge-weighted case and O(k 1=4 ) for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of treewidth-bounded graphs which includes trees, series-parallel graphs, and bounded bandwidth graphs, and for points on the boundary of a convex region in the Euclidean plane. We also investigate the problem of finding short trees, and more generally, that of finding networks with minimum diameter. A simple technique is used to prov...

A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.

We study a general class of bicriteria network design problems. A generic problem in this class is as follows: Given an undirected graph and two minimization objectives (under different cost functions), with a budget specified on the first, find a subgraph from a given subgraph-class that minimizes the second objective subject to the budget on the first. We consider three different criteria - the total edge cost, the diameter and the maximum degree of the network. Here, we present the first polynomial-time approximation algorithms for a large class of bicriteria network design problems for the above mentioned criteria. The following general types of results are presented. First, we develop a framework for bicriteria problems and their approximations. Second, when the two criteria are the same we present a "black box" parametric search technique. This black box takes in as input an (approximation) algorithm for the unicriterion situation and generates an approximation algorit...

We consider the frequency assignment (broadcast scheduling) problem for packet radio networks. Such networks are naturally modeled by graphs with a certain geometric structure. The problem of broadcast scheduling can be cast as a variant of the vertex coloring problem (called the distance-2 coloring problem) on the graph that models a given packet radio network. We present efficient approximation algorithms for the distance-2 coloring problem for various geometric graphs including those that naturally model a large class of packet radio networks. The class of graphs considered include (r, s)-civilized graphs, planar graphs, graphs with bounded genus, etc.

We consider the channel assignment (broadcast scheduling) problem for packet radio networks. Such networks are naturally modeled by graphs with certain geometric structure. The channel assignment problem can be cast as a variant of the vertex coloring problem (called the distance-2 coloring problem) on the graph that models a given packet radio network. We present efficient approximation algorithms for the distance-2 coloring problem for several classes of graphs including a class of geometric graphs that naturally model a large class of packet radio networks. The classes of graphs considered include (r, s)-civilized graphs, planar graphs, graphs with bounded genus, etc. Many of the approximation results presented here are the first such results in the literature.

Given an alphabet Sigma, a (directed) graph G whose edges are weighted and Sigma-labeled, and a formal language L , the Formal Language Constrained Shortest/Simple Path problem consists of finding a shortest (simple) path p in G complying with the additional constraint that l(p) 2 L. Here l(p) denotes the unique word given by concatenating the Sigma-labels of the edges along the path p. The main contributions of this paper include the following: 1. We show that the formal language constrained shortest path problem is solvable efficiently in polynomial time when L is restricted to be a context free language. When L is specified as a regular language we provide algorithms with improved space and time bounds...

Abstract. Given an alphabet Σ, a (directed) graph G whose edges are weighted and Σ-labeled, and a formal language L ⊆ Σ ∗ , the formal-language-constrained shortest/simple path problem consists of finding a shortest (simple) path p in G complying with the additional constraint that l(p) ∈ L. Here l(p) denotes the unique word obtained by concatenating the Σ-labels of the edges along the path p. The main contributions of this paper include the following: (1) We show that the formal-language-constrained shortest path problem is solvable efficiently in polynomial time when L is restricted to be a context-free language (CFL). When L is specified as a regular language we provide algorithms with improved space and time bounds. (2) In contrast, we show that the problem of finding a simple path between a source and a given destination is NP-hard, even when L is restricted to fixed simple regular languages and to very simple classes of graphs (e.g., complete grids). (3) For the class of treewidth-bounded graphs, we show that (i) the problem of finding a regular-language-constrained simple path between source and destination is solvable in polynomial time and (ii) the extension to finding CFL-constrained simple paths is NP-complete.

This paper considers problems of the following type: We are given an edge weighted graph G = (V, E). It is assumed that each edge e of the given network has an associated function c_e that specifies the cost of shortening the edge by a given amount and that there is a budget B on the total reduction cost. The goal is to develop a reduction strategy satisfying the budget constraint so that the total length of a minimum spanning tree in the modified network is the smallest possible over all reduction strategies that obey the budget constraint. We show that in general the problem of computing an optimal reduction strategy for modifying the network as above is NP-hard even for simple classes of graphs and linear functions c_e. We present the first polynomial time approximation algorithms for the problem, where the cost functions c_e are allowed to be taken from a broad class of functions. We also present improved approximation algorithms for the class of treewidth-bounded graphs when the cost functions are linear...

Abstract—In third generation (3G) wireless data networks, multicast throughput decreases with the increase in multicast group size, since a conservative strategy for the base station is to use the lowest data rate of all the receivers so that the receiver with the worst downlink channel condition can decode the transmission correctly. This paper proposes ICAM, Integrated Cellular and Ad hoc Multicast, to increase 3G multicast throughput through opportunistic use of ad hoc relays. In ICAM, a 3G base station delivers packets to proxy mobile devices with better 3G channel quality. The proxy then forwards the packets to the receivers through an IEEE 802.11based ad hoc network. In this paper, we first propose a localized greedy algorithm that discovers for each multicast receiver the proxy with the highest 3G downlink channel rate. We discover that due to capacity limitations and interference of the ad hoc relay network, maximizing the 3G downlink data rate of each multicast receiver’s proxy does not lead to maximum throughput for the multicast group. We then show that the optimal ICAM problem is NP-hard, and derive a polynomial-time 4-approximation algorithm for the construction of the multicast forest. This bound holds when the underlying wireless MAC supports broadcast or unicast, single rate or multiple rates (4ð1 þ Þ approximation scheme for the latter), and even when there are multiple simultaneous multicast sessions. Through both analysis and simulations, we show that our algorithms achieve throughput gains up to 840 percent for 3G downlink multicast with modest overhead on the 3G uplink. Index Terms—Ad hoc networks, cellular networks, multicast, network architecture, routing. 1