This paper considers jointly optimal design of crosslayer congestion control, routing and scheduling for ad hoc wireless networks. We first formulate the rate constraint and scheduling constraint using multicommodity flow variables, and formulate resource allocation in networks with fixed wireless channels (or single-rate wireless devices that can mask channel variations) as a utility maximization problem with these constraints. By dual decomposition, the resource allocation problem naturally decomposes into three subproblems: congestion control, routing and scheduling that interact through congestion price. The global convergence property of this algorithm is proved. We next extend the dual algorithm to handle networks with timevarying channels and adaptive multi-rate devices. The stability of the resulting system is established, and its performance is characterized with respect to an ideal reference system which has the best feasible rate region at link layer. We then generalize the aforementioned results to a general model of queueing network served by a set of interdependent parallel servers with time-varying service capabilities, which models many design problems in communication networks. We show that for a general convex optimization problem where a subset of variables lie in a polytope and the rest in a convex set, the dual-based algorithm remains stable and optimal when the constraint set is modulated by an irreducible finite-state Markov chain. This paper thus presents a step toward a systematic way to carry out cross-layer design in the framework of “layering as optimization decomposition” for time-varying channel models.

We present a linear time approximation algorithm with a performance ratio of 1/2 for nding a maximum weight matching in an arbitrary graph. Such a result is already known and is due to Preis [7].

Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that has maximum weight. The first polynomial time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O(nm+n 2 log n). Many real world problems require graphs of such large size that this running time is too costly. Therefore there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/3

Approximation algorithms have so far mainly been studied for problems that are not known to have polynomial time algorithms for solving them exactly. Here we propose an approximation algorithm for the weighted matching problem in graphs which can be solved in polynomial time. The weighted matching problem is to find a matching in an edge weighted graph that has maximum weight. The first polynomial-time algorithm for this problem was given by Edmonds in 1965. The fastest known algorithm for the weighted matching problem has a running time of O(nm + n² log n). Many real world problems require graphs of such large size that this running time is too costly. Therefore, there is considerable need for faster approximation algorithms for the weighted matching problem. We present a linear-time approximation algorithm for the weighted matching problem with a performance ratio arbitrarily close to 2/1. This improves the previously best performance ratio of 3/2. Our algorithm is not only of theoretical interest, but because it is easy to implement and the constants involved are quite small it is also useful in practice.

It has been an important research topic since 1992 to maximize stability region in constrained queueing systems, which includes the study of scheduling over wireless ad hoc networks. In this paper, we propose a framework to study a wide range of existing and future scheduling algorithms and characterize the achieved tradeoffs in stability, delay, and complexity. These characterizations reveal interesting properties hidden in the study of any one or two dimensions in isolation. For example, decreasing complexity from exponential to polynomial, while keeping stability region the same, generally comes at the expense of exponential growth of delays. Investigating trade-offs in the 3-dimensional space allows a designer to fix one dimension and vary the other two jointly. For example, incentives for using scheduling algorithms with only partial throughput-guarantee can be quantified with regards to delay and complexity. Tradeoff analysis is then extended to systems with congestion control through utility maximization for non-stabilizable arrival inputs, where the complexity-utility-delay trade-off is shown to be different from the complexity-stability-delay tradeoff. Finally, we analyze more practical models with bounded message size, and consider “effective throughput” which reflects resource occupied by control messages. We show that effective throughput may degrade significantly in certain scheduling algorithms, and suggest a mechanism to avoid this problem in light of the 3D tradeoff framework.

We reduce the best known approximation ratio for finding a weighted matching of a graph using a one-pass semi-streaming algorithm from 5.828 to 5.585. The semi-streaming model forbids random access to the input and restricts the memory to O(n · polylog n) bits. It was introduced by Muthukrishnan in 2003 and is appropriate when dealing with massive graphs.

The objective of this paper is to characterize classes of problems for which a greedy algorithm finds solutions provably close to optimum. To that end, we introduce the notion of k-extendible systems, a natural generalization of matroids, and show that a greedy algorithm is a 1-factor approximation for these systems. Many seemly unrelated k problems fit in our framework, e.g.: b-matching, maximum profit scheduling and maximum asymmetric TSP. In the second half of the paper we focus on the maximum weight b-matching problem. The problem forms a 2-extendible system, so greedy gives us a 1-factor solution which runs in 2 O(m log n) time. We improve this by providing two linear time approximation algorithms for the problem: a 1 2-factor algorithm that runs in O(bm) time, and a `2 3 − ǫ ´-factor algorithm which runs in expected O ` bm log 1 ´ time.

Wattenhofer et al. [WW04] derive a complicated distributed algorithm to compute a weighted matching of an arbitrary weighted graph, that is at most a factor 5 away from the maximum weighted matching of that graph. We show that a variant of the obvious sequential greedy algorithm [Pre99], that computes a weighted matching at most a factor 2 away from the maximum, is easily distributed. This yields the best known distributed approximation algorithm for this problem so far. 1

Motivated by the dynamics of the ever-popular online movie rental business, we study a range of assignment problems in rental markets. The assignment problems associated with rental markets possess a rich mathematical structure and are closely related to many well-studied one-sided matching problems. We formalize and characterize the assignment problems in rental markets in terms of one-sided matching problems, and consider several solution concepts for these problems. In order to evaluate and compare these solution concepts (and the corresponding algorithms), we define some “value ” functions to capture our objectives, which include fairness, efficiency and social welfare. Then, we bound the value of the output of these algorithms in terms of the chosen value functions. We also consider models of rental markets corresponding to static, online, and dynamic customer valuations. We provide several constant-factor approximation algorithms for the assignment problem, as well as hardness of approximation results for the different models. Finally, we describe some experiments with a discrete event simulator compare the various algorithms in a practical setting, and present some interesting experimental results. 1

Abstract. In this paper, we study distributed algorithms to compute a weighted matching that have constant (or at least sub-logarithmic) running time and that achieve approximation ratio 2 + ɛ or better. In fact we present two such synchronous algorithms, that work on arbitrary weighted trees. The first algorithm is a randomised distributed algorithm that computes a weighted matching of an arbitrary weighted tree, that approximates the maximum weighted matching by a factor 2 + ɛ. The running time is O(1). The second algorithm is deterministic, and approximates the maximum weighted matching by a factor 2 + ɛ, but has running time O(log ∗ |V |). Our algorithms can also be used to compute maximum unweighted matchings on regular and almost regular graphs within a constant approximation. 1