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): 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): 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].

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.S.

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Abstract. 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 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 rj,t resources. The scheduler must allocate resources j,t to user j at time t such that each allocation satisfies the requirement (rj,t ≤ j,t) and the combined allocations do not exceed T ( P j j,t ≤ T ). The objective is to minimize the total number of changes to allocated resources (the number of pairs j, t where j,t = j,t+1). We consider online algorithms for RAP whose resource pool is increased to sT and obtain an online algorithm which is O(log s n) 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 [OLW01], 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, inResearch partially supported by NSF Award CCR-0515332, but views expressed are not endorsed by the NSF. Research partially supported by NSF Award CCR-0515332, but views expressed are not endorsed by the NSF. 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 [CYV04,cKA02]. We adapt the online randomized algorithm for the RAP problem to solve TMAV problem with similar competitive ratio: an algori...