We consider the channel assignment problem in a multi-radio wireless mesh network that involves assigning channels to radio interfaces for achieving efficient channel utilization. We propose the notion of a traffic-independent base channel assignment to ease coordination and enable dynamic, efficient and flexible channel assignment. We present a novel formulation of the base channel assignment as a topology control problem, and show that the resulting optimization problem is NP-complete. We then develop a new greedy heuristic channel assignment algorithm (termed CLICA) for finding connected, low interference topologies by utilizing multiple channels. Our extensive simulation studies show that the proposed CLICA algorithm can provide large reduction in interference (even with a small number of radios per node), which in turn leads to significant gains in both link layer and multihop performance in 802.11-based multi-radio mesh networks.

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A topology control approach for utilizing multiple channels in multi-radio wireless mesh networks

We study the question of what optimization problems can be optimally or approximately solved by "greedy-like" algorithms. For definiteness, we will limit the present discussion to some well-studied scheduling problems although the underlying issues apply in a much more general setting. Of course, the main benefit of greedy algorithms lies in both their conceptual simplicity and their computational efficiency. Based on the experience from online competitive analysis, it seems plausible that we should be able to derive approximation bounds for "greedy-like" algorithms exploiting only the conceptual simplicity of these algorithms. To this end, we will provide a precise definition of what we mean by greedy and greedy-like. A full version of this paper is available at http://www.cs.toronto.edu/~ bor/priority.ps.

Incremental) priority algorithms

Probability and Computing: Chernoff Bounds

Borodin et al. (Algorithmica 37(4):295–326, 2003) gave a model of greedy-like algorithms for scheduling problems and Angelopoulos and Borodin (Algorithmica 40(4):271–291, 2004) extended their work to facility location and set cover problems. We generalize their model to include other optimization problems, and apply the generalized framework to graph problems. Our goal is to define an abstract model that captures the intrinsic power and limitations of greedy algorithms for various graph optimization problems, as Borodin et al. (Algorithmica 37(4):295–326, 2003) did for scheduling. We prove bounds on the approximation ratio achievable by such algorithms for basic graph problems such as shortest path, weighted vertex cover, Steiner tree, and independent set. For example, we show that, for the shortest path problem, no algorithm in the FIXED priority model can achieve any approximation ratio (even one dependent on the graph size), but the well-known Dijkstra’s algorithm is an optimal ADAPTIVE priority algorithm. We also prove that the approximation ratio for weighted vertex cover achievable by ADAPTIVE priority algorithms is exactly 2. Here, a new lower bound matches the known upper bounds (Johnson in J. Comput. Syst. Sci. 9(3):256–278, 1974). We give a number of other lower bounds for priority algorithms, as well as a new approximation algorithm for minimum Steiner tree problem with weights in the interval [1,2].

Models of Greedy Algorithms for Graph Problems

In this dissertation we study a broad class of stochastic scheduling problems characterized by the presence of hard deadline constraints. The input to such a problem is a set of jobs, each with an associated value, processing time, and deadline. We would like to schedule these jobs on a set of machines over time. In our stochastic setting, the processing time of each job is random, known in advance only as a probability distribution (and we make no assumptions about the structure of this distribution). Only after a job completes do we know its actual “instantiated” processing time with certainty. Each machine can process only a singe job at a time, and each job must be assigned to only one machine for processing. After a job starts processing we require that it must be allowed to complete—it cannot be canceled or “preempted” (put on hold and resumed later). Our goal is to devise a scheduling policy that maximizes the expected value of jobs that are scheduled by their deadlines. 
A scheduling policy observes the state of our machines over time, and any time a machine becomes available for use, it selects a new job to execute on that machine. Scheduling policies can be classified as adaptive or non-adaptive based on whether or not they utilize information learned from the instantiation of processing times of previously-completed jobs in their future scheduling decisions. A novel aspect of our work lies in studying the benefit one can obtain through adaptivity, as we show that for all of our stochastic scheduling problems, adaptivity can only allow us to improve the expected value obtained by an optimal policy by at most a small constant factor. 
All of the problems we consider are at least NP-hard since they contain the deterministic 0/1 knapsack problem as a special case. We therefore seek to develop approximation algorithms: algorithms that run in polynomial time and compute a policy whose expected value is provably close to that of an optimal adaptive policy. For all the problems we consider, we can approximate the expected value obtained by an optimal adaptive policy to within a small constant factor (which depends on the problem under consideration, but is always less than 10). A small handful of our results are pseudo-approximation algorithms, delivering an approximately optimal policy that is feasible with respect to a slightly expanded set of deadlines. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.) (Abstract shortened by UMI.)

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Approximation algorithms for stochastic scheduling problems

Borodin, Nielsen, and Rackoff ([5]) gave a model of greedy-like algorithms for scheduling problems and [1] extended their work to facility location and set cover problems. We generalize their notion to include other optimization problems, and apply the generalized framework to graph problems. Our goal is to define an abstract model that captures the intrinsic power and limitations of greedy algorithms for various graph optimization problems. We prove bounds on the approximation ratio achievable by such algorithms for basic graph problems such as shortest path, vertex cover, and others. Shortest path is an example of a problem where no algorithm in the FIXED priority model can achieve any approximation ratio (even one dependent on the graph size), but for which the well-known Dijkstra's algorithm shows that an ADAPTIVE priority algorithm can be optimal. We also prove that the approximation ratio for vertex cover achievable by ADAPTIVE priority algorithms is exactly 2. Here, a new lower bound matches the known upper bounds ([8]).

Models of greedy algorithms for graph problems

We define a formal model of dynamic programming algorithms which we call Prioritized Branching Programs (pBP). Our model is a generalization of the BT model of Alekhnovich et al. (IEEE Conference on Computational Complexity, pp. 308–322, 2005), which is in turn a generalization of the priority algorithms model of Borodin, Nielson and Rackoff. One of the distinguishing features of these models is that they not only capture large classes of algorithms generally considered to be greedy, backtracking or dynamic programming algorithms, but they also allow characterizations of their limitations. Hence they give meaning to the statement that a given problem can or cannot be solved by dynamic programming. After defining the model, we prove three main results: (i) that certain types of natural restrictions of our seemingly more powerful model can be simulated by the BT model; (ii) that in general our model is stronger than the BT model—a fact which is witnessed by the classical shortest paths problem; (iii) that our model has very real limitations, namely that bipartite matching cannot be efficiently computed in it, hence suggesting that there are problems that can be solved efficiently by network flow algorithms and by simple linear programming that cannot be solved by natural dynamic programming approaches.

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A Stronger Model of Dynamic Programming Algorithms

In this paper, we consider two new online optimization problems (each with several variants), present similar online algorithms for both, and show that one reduces to the other. Both problems involve a control trying to minimize the number of changes that need to be made in maintaining a state that satisfies each of many users' requirements. Our algorithms have the property that the control only needs to be informed of a change in a users needs when the current state no longer satisfies the user. This is particularly important when the application is one of trying to minimize communication between the users and the control.

The Resource Allocation Problem (RAP) is an abstraction of scheduling malleable and evolving jobs on multiprocessor machines. A scheduler has a fixed pool of resources of total size T. There are n users, and each user j has a resource requirement for r$_{j,{\it t}}$ resources. The scheduler must allocate resources l$_{j,{\it t}}$ to user j at time t such that each allocation satisfies the requirement (r$_{j,{\it t}}$ ≤l$_{j,{\it t}}$) and the combined allocations do not exceed T (∑j lj,t ≤T). The objective is to minimize the total number of changes to allocated resources (the number of pairs j,t where l$_{j,{\it t}}$ ≠l$_{j, {\it t}+1}$).

We consider online algorithms for RAP whose resource pool is increased to sT and obtain an online algorithm which is O(logsn)- competitive. Further we show that the increased resource pool is crucial to the performance of the algorithm by proving that there is no online algorithm using T resources which is f(n)-competitive for any f(n). Note that our upper bounds all have the property that the algorithms only know the list of users whose requirements are currently unsatisfied and never learn the precise requirements of users. We feel this is important for many applications, since users rarely report underutilized resources as readily as they do unmet requirements. On the other hand, our lower bounds apply to online algorithms that have complete knowledge about past requirements.

The Transmission-Minimizing Approximate Value problem is a generalization of one defined in [1], in which low-power sensors monitor real-time events in a distributed wireless network and report their results to a centralized cache. In order to minimize network traffic, the cache is allowed to maintain approximations to the values at the sensors, in the form of intervals containing the values, and to vary the lengths of intervals for the different sensors so that sensors with fluctuating values are measured less precisely than more stable ones. A constraint for the cache is that the sum of the lengths of the intervals must be within some precision parameter T. Similar models are described in [2,3]. We adapt the online randomized algorithm for the RAP problem to solve TMAV problem with similar competitive ratio: an algorithm can maintain sT precision and be O(logsn)-competitive in transmissions against an adversary maintaining precision T.

Further we show that solving TMAV is as hard as solving RAP, by reducing RAP to TMAV. This proves similar lower bounds for TMAV as we had for RAP, when s is near 1.

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Sashka Davis

Papers

Models of Greedy Algorithms for Graph Problems

Models of greedy algorithms for graph problems

Models of Greedy Algorithms for Graph Problems

A Stronger Model of Dynamic Programming Algorithms

Online algorithms to minimize resource reallocations and network communication