We study a constrained version of the knapsack problem in which dependencies between items are given by the adjacencies of a graph. In the 1-neighbour knapsack problem, an item can be selected only if at least one of its neighbours is also selected. In the all-neighbours knapsack problem, an item can be selected only if all its neighbours are also selected. We give approximation algorithms and hardness results when the nodes have both uniform and arbitrary weight and profit functions, and when the dependency graph is directed and undirected.
As we show in , the general, undirected 1-neighbour knapsack problem generalizes several maximum coverage problems including the budgeted variant considered by Khuller, Moss, and Naor @cite_3 which has a tight @math -approximation unless P=NP. Our algorithm for the general 1-neighbour problem follows the approach taken by Khuller, Moss, and Naor but, because of the dependency graph, requires several new technical ideas. In particular, our analysis of the greedy step represents a non-trivial generalization of the standard greedy algorithm for submodular maximization.
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The multi-armed bandit (MAB) problem features the classical tradeoff between exploration and exploitation. The input specifies several stochastic arms which evolve with each pull, and the goal is to maximize the expected reward after a fixed budget of pulls. The celebrated work of , surveyed in [8], presumes a condition on the arms called the martingale assumption. In [9], A. obtained an LP-based 1 48-approximation for the problem with the martingale assumption removed. We improve the algorithm to a 4 27-approximation, with simpler analysis. Our algorithm also generalizes to the case of MAB superprocesses with (stochastic) multi-period actions. This generalization captures the framework introduced by Guha and Munagala in [11], and yields new results for their budgeted learning problems. Also, we obtain a (1 2 -- e)-approximation for the variant of MAB where preemption (playing an arm, switching to another arm, then coming back to the first arm) is not allowed. This contains the stochastic knapsack problem of Dean, Goemans, and Vondrak in [6] with correlated rewards, where we are given a knapsack of fixed size, and a set of jobs each with a joint distribution for its size and reward. The actual size and reward of a job can only be discovered in real-time as it is being scheduled, and the objective is to maximize expected reward before the knapsack size is exhausted. Our (1 2 -- e)-approximation improves the 1 16 and 1 8 approximations in [9] for correlated stochastic knapsack with cancellation and no cancellation, respectively, providing to our knowledge the first tight algorithm for these problems that matches the integrality gap of 2. We sample probabilities from an exponential-sized dynamic programming solution, whose existence is guaranteed by an LP projection argument. We hope this technique can also be applied to other dynamic programming problems which can be projected down onto a small LP. Given @math elements with non-negative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most @math . We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error @math ). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows. In the classic k-center problem, we are given a metric graph, and the objective is to open k nodes as centers such that the maximum distance from any vertex to its closest center is minimized. In this paper, we consider two important generalizations of k-center, the matroid center problem and the knapsack center problem. Both problems are motivated by recent content distribution network applications. Our contributions can be summarized as follows: 1 We consider the matroid center problem in which the centers are required to form an independent set of a given matroid. We show this problem is NP-hard even on a line. We present a 3-approximation algorithm for the problem on general metrics. We also consider the outlier version of the problem where a given number of vertices can be excluded as the outliers from the solution. We present a 7-approximation for the outlier version. 2 We consider the (multi-)knapsack center problem in which the centers are required to satisfy one (or more) knapsack constraint(s). It is known that the knapsack center problem with a single knapsack constraint admits a 3-approximation. However, when there are at least two knapsack constraints, we show this problem is not approximable at all. To complement the hardness result, we present a polynomial time algorithm that gives a 3-approximate solution such that one knapsack constraint is satisfied and the others may be violated by at most a factor of 1+e. We also obtain a 3-approximation for the outlier version that may violate the knapsack constraint by 1+e. We consider the two-dimensional geometric knapsack problem defined as follows. Given a collection of rectangular axis-parallel items with weights, we want to find a maximum weight subset of the items that can be packed into a rectangular knapsack, i.e., which can be assigned positions within the knapsack such that the items are pairwise non-overlapping. The goal is to compute the optimal collection of items together with a feasible packing. The problem arises naturally in several applications, and various special cases of the problem have been studied. For the general case the best known result is a (2 + e)-approximation algorithm, while the only hardness result is NP-hardness. Our main result is a (1 + e)-approximation algorithm that runs in quasi-polynomial time, provided that the input data consists of (quasi-)polynomially bounded integers. We achieve this result in the setting with and without allowing rotation of the items. Our key technical contribution is to show the existence of a partition for the knapsack into a small number of rectangular boxes. Intuitively, this partition describes the segmentation of the knapsack in some near-optimal solution into large items, areas containing items that are high and thin, and areas containing items that are small and wide. Handling the interaction between these three types of items is a core bottleneck in the design of approximation algorithms for the problem and our partition allows to control this at only marginal cost. In particular, it is so powerful that we do not even need to round the sizes of the items, which is a canonical step in algorithms for geometric knapsack, geometric bin-packing, etc. Finally, we present new algorithms for two-dimensional knapsack and two-dimensional bin-packing under (1 + e)-resource augmentation with a substantially better running time dependence on e (although at the expense of requiring quasi-polynomial time). The objective value of our computed solutions is as good as the optimum without additional resources. Here we exploit a recent result showing the existence of certain low-complexity low weight separators for sets of non-overlapping rectangles.