In this paper, we address the following question: given a specific placement of wireless nodes in physical space and a specific traffic workload, what is the maximum throughput that can be supported by the resulting network? Unlike previous work that has focused on computing asymptotic performance bounds under assumptions of homogeneity or randomness in the network topology and/or workload, we work with any given network and workload specified as inputs.A key issue impacting performance is wireless interference between neighboring nodes. We model such interference using a conflict graph, and present methods for computing upper and lower bounds on the optimal throughput for the given network and workload. To compute these bounds, we assume that packet transmissions at the individual nodes can be finely controlled and carefully scheduled by an omniscient and omnipotent central entity, which is unrealistic. Nevertheless, using ns-2 simulations, we show that the routes derived from our analysis often yield noticeably better throughput than the default shortest path routes even in the presence of uncoordinated packet transmissions and MAC contention. This suggests that there is opportunity for achieving throughput gains by employing an interferenceaware routing protocol.

Abstract-The famous max-flow min-cut theorem states that a source node can send information through a network ( ) to a sink node at a rate determined by the min-cut separating and . Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.

We present approximation algorithms for the metric uncapacitated facility location problem and the metric k -median problem achieving guarantees of 3 and 6 respectively. The distinguishing feature of our algorithms is their low running time: O(m log m ) and O(m log m(L + log ( n ))) respectively, where n and m are the total number of vertices and edges in the underlying complete bipartite graph on cities and facilities. The main algorithmic ideas are a new extension of the primal-dual schema and the use of Lagrangian relaxation to derive approximation algorithms.

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In this paper, we will formalize the method of dual fitting and the idea of factor-revealing LP. This combination is used to design and analyze two greedy algorithms for the metric uncapacitated facility location problem. Their approximation factors are 1.861 and 1.61, with running times of O(m log m) and O(n 3 ), respectively, where n is the total number of vertices and m is the number of edges in the underlying complete bipartite graph between cities and facilities. The algorithms are used to improve recent results for several variants of the problem. . 1 This paper is based on the preliminary versions [31] and [21].1 clear so far was that the set cover problem did not require its full power. However, in retrospect, its salient features are best illustrated again in the simple setting of the set cover problem -we do this in Section 9. The method of dual fitting can be described as follows, assuming a minimization problem: The basic algorithm is combinatorial -in the case of set cover it is in fact a simple greedy algorithm. Using the linear programming relaxation of the problem and its dual, one first interprets the combinatorial algorithm as a primal-dual-type algorithm -an algorithm that is iteratively making primal and dual updates. Strictly speaking, this is not a primal-dual algorithm, since the dual solution computed is, in general, infeasible (see Section 9 for a discussion on this issue). However, one shows that the primal integral solution found by the algorithm is fully paid for by the dual computed. By fully paid for we mean that the objective function value of the primal solution is bounded by that of the dual. The main step in the analysis consists of dividing the dual by a suitable factor, say γ, and showing that the shrunk dual is feasible, i.e., it fits into the given instance. The shrunk dual is then a lower bound on OPT, and γ is the approximation guarantee of the algorithm.Clearly, we need to find the minimum γ that suffices. Equivalently, this amounts to finding the worst possible instance -one in which the dual solution needs to be shrunk the most in order to be rendered feasible. For each value of n c , the number of cities, we define a factor-revealing LP that encodes the problem of finding the worst possible instance with n c cities as a linear program. This gives a family of LP's, one for each value of n c . The supremum of the optimal solutions to these LP's is then the best value for γ. In our case, we do not know how to compute this supremum directly. Instead, we obtain a feasible solution to the dual of each of these LP's. An upper bound on the objective function values of these duals can be computed, and is an upper bound on the optimal γ. In our case, this upper bound is 1.861 for the first algorithm and 1.61 for the second one. In order to get a closely matching tight example, we numerically solve the factor-revealing LP for a large value of n c .The technique of factor-revealing LPs is similar to the idea of LP bounds in coding theory. LP bounds give the best known bounds on the...