The Multiple Knapsackproblem (MKP) is a natural and well known generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the Generalized Assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APX-hard and a 2-approximation for it is implicit in the work of Shmoys and Tardos [26], and thus far, this was also the best known approximation for MKP. The main result of this paper is a polynomial time approximation scheme for MKP. Apart from its inherent theoretical interest as a common generalization of the well-studied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APX-hard. We substantiate this by showing that slight generalizations of MKP that are very restricted versions of GAP are APX-hard. Thus our results help demarcate the boundary at which instances of GAP becomeAPX-hard. An interesting and novel aspect of our approach is an approximation preserving reduction from an arbitrary instance of MKP to an instance with O(log n) distinct sizes and profits.

Several combinatorial optimization problems choose elements to minimize the total cost of constructing a feasible solution that satisfies requirements of clients. In the STEINER TREE problem, for example, edges must be chosen to connect terminals (clients); in VERTEX COVER, vertices must be chosen to cover edges (clients); in FACILITY LOCATION, facilities must be chosen and demand vertices (clients) connected to these chosen facilities. We consider a stochastic version of such a problem where the solution is constructed in two stages: Before the actual requirements materialize, we can choose elements in a first stage. The actual requirements are then revealed, drawn from a pre-specified probability distribution π; thereupon, some more elements may be chosen to obtain a feasible solution for the actual requirements. However, in this second (recourse) stage, choosing an element is costlier by a factor of σ> 1. The goal is to minimize the first stage cost plus the expected second stage cost. We give a general yet simple technique to adapt approximation algorithms for several deterministic problems to their stochastic versions via the following method. • First stage: Draw σ independent sets of clients from the distribution π and apply the approximation algorithm to construct a feasible solution for the union of these sets. • Second stage: Since the actual requirements have now been revealed, augment the first-stage solution to be feasible for these requirements.

We consider a stochastic variant of the NP-hard 0/1 knapsack problem in which item values are deterministic and item sizes are independent random variables with known, arbitrary distributions. Items are placed in the knapsack sequentially, and the act of placing an item in the knapsack instantiates its size. Our goal is to compute a solution “policy ” that maximizes the expected value of items placed in the knapsack, and we consider both non-adaptive policies (that designate a priori a fixed sequence of items to insert) and adaptive policies (that can make dynamic choices based on the instantiated sizes of items placed in the knapsack thus far). We show that adaptivity provides only a constant-factor improvement by demonstrating a greedy non-adaptive algorithm that approximates the optimal adaptive policy within a factor of 7. We also design an adaptive polynomial-time algorithm which approximates the optimal adaptive policy within a factor of 5 + ɛ, for any constant ɛ> 0. 1.

Internet search companies sell advertisement slots based on users ’ search queries via an auction. Advertisers have to determine how to place bids on the keywords of their interest in order to maximize their return for a given budget: this is the budget optimization problem. The solution depends on the distribution of future queries. In this paper, we formulate stochastic versions of the budget optimization problem based on natural probabilistic models of distribution over future queries, and address two questions that arise. Evaluation Given a solution, can we evaluate the expected value of the objective function? Optimization Can we find a solution that maximizes the objective function in expectation? Our main results are approximation and complexity results for these two problems in our three stochastic models. In particular, our algorithmic results show that simple prefix strategies that bid on all cheap keywords up to some level are either optimal or good approximations for many cases; we show other cases to be NP-hard. 1

We present the first approximation algorithms for a large class of budgeted learning problems. One classic example of the above is the budgeted multi-armed bandit problem. In this problem each arm of the bandit has an unknown reward distribution on which a prior is specified as input. The knowledge about the underlying distribution can be refined in the exploration phase by playing the arm and observing the rewards. However, there is a budget on the total number of plays allowed during exploration. After this exploration phase, the arm with the highest (posterior) expected reward is chosen for exploitation. The goal is to design the adaptive exploration phase subject to a budget constraint on the number of plays, in order to maximize the expected reward of the arm chosen for exploitation. While this problem is reasonably well understood in the infinite horizon setting or regret bounds, the budgeted version of the problem is NP-Hard. For this problem, and several generalizations, we provide approximate policies that achieve a reward within constant factor of the reward optimal policy. Our algorithms use a novel linear program rounding technique based on stochastic packing.

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We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact n Θ(log n) algorithm for the case of normally distributed edge lengths, which is based on quasi-convex maximization. We then prove average and smoothed polynomial bounds for this algorithm, which also translate to average and smoothed bounds for the parametric shortest path problem, and extend to a more general non-convex optimization setting. We also consider a number other edge length distributions, giving a range of exact and approximation schemes.

Abstract. The multiple knapsack problem (MKP) is a natural and well-known generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the generalized assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APX-hard and a 2-approximation, for it is implicit in the work of Shmoys and Tardos [Math. Program. A, 62 (1993), pp. 461–474], and thus far, this was also the best known approximation for MKP. The main result of this paper is a polynomial time approximation scheme (PTAS) for MKP. Apart from its inherent theoretical interest as a common generalization of the well-studied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APX-hard. We substantiate this by showing that slight generalizations of MKP are APX-hard. Thus our results help demarcate the boundary at which instances of GAP become APX-hard. An interesting aspect of our approach is a PTAS-preserving reduction from an arbitrary instance of MKP to an instance with O(log n) distinct sizes and profits.

In the past decade, the limitations of models considering fixed (worst case) task execution times have been acknowledged for large application classes within soft real-time systems. A more realistic model considers the tasks having varying execution times with given probability distributions. Considering such a model with specified task execution time probability distribution functions, an important performance indicator of the system is the expected deadline miss ratio of the tasks and of the task graphs. This article presents an approach for obtaining this indicator in an analytic way. Our goal is to keep the analysis cost low, in terms of required analysis time and memory, while considering as general classes of target application models as possible. The following main assumptions have been made on the applications which are modelled as sets of task graphs: the tasks are periodic, the task execution times have given generalised probability distribution functions, the task execution deadlines are given and arbitrary, the scheduling policy can belong to practically any class of non-preemptive scheduling policies, and a designer supplied maximum number of concurrent instantiations of the same task graph is tolerated in the system. Experiments show the efficiency of the proposed technique for monoprocessor systems.

In several database applications, parameters like selectivities and load are known only with some associated uncertainty, which is specified, or modeled, as a distribution over values. The performance of query optimizers and monitoring schemes can be improved by spending resources like time or bandwidth in observing or resolving these parameters, so that better query plans can be generated. In a resourceconstrained situation, deciding which parameters to observe in order to best optimize the expected quality of the plan generated (or in general, optimize the expected value of a certain objective function) itself becomes an interesting optimization problem. We present a framework for studying such problems, and present several scenarios arising in anomaly detection in complex systems, monitoring extreme values in sensor networks, load shedding in data stream systems, and estimating rates in wireless channels and minimum latency routes in networks, which can be modeled in this framework with the appropriate objective functions. Even for several simple objective functions, we show the problems are Np-Hard. We present greedy algorithms with good performance bounds. The proof of the performance bounds are via novel sub-modularity arguments.