A multiple knapsack constraint over a set of items is defined by a set of bins of arbitrary capacities, and a weight for each of the items. An assignment for the constraint is an allocation of subsets of items to the bins , such that the total weight of items assigned to each bin is bounded by the bin capacity. We study modular (linear) and submodular maximization problems subject to a constant number of (i.e., d-resource) multiple knapsack constraints, in which a solution is a subset of items, along with an assignment of the selected items for each of the d multiple knapsack constraints. Our results include a polynomial time approximation scheme for modular maximization with a constant number of multiple knapsack constraints and a matroid constraint, thus generalizing the best known results for the classic multiple knapsack problem as well as d-dimensional knapsack, for any d \geq 2. We further obtain a tight (1-e^{-1-\epsilon)-approximation for monotone submodular optimization subject to a constant number of multiple knapsack constraints and a matroid constraint, and a (}  0.385- \epsilon )-approximation for non-monotone submodular optimization subject to a constant number of multiple knapsack constraints. At the heart of our algorithms lies a novel representation of a multiple knapsack constraint as a polytope. We consider this key tool as a main technical contribution of this paper .
A multiple knapsack constraint over a set of items is defined by a set of bins of arbitrary capacities, and a weight for each of the items. An assignment for the constraint is an allocation of subsets of items to the bins which adheres to bin capacities. In this paper we present a unified algorithm that yields efficient approximations for a wide class of submodular and modular optimization problems involving multiple knapsack constraints. One notable example is a polynomial time approximation scheme for Multiple-Choice Multiple Knapsack, improving upon the best known ratio of 2. -\epsilon)-approximation for monotone submodular optimization subject to a constant number of multiple knapsack constraints and a matroid constraint, and a (} Another example is Non-monotone Submodular Multiple Knapsack, for which we obtain a ( 0.385- \varepsilon )-approximation , matching the best known ratio for a single knapsack constraint. The robustness of our algorithm is achieved by applying a novel fractional variant of the classical linear grouping technique, which is of independent interest .
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, such that the total weight of items assigned to each bin is bounded by the bin capacity. We study modular (linear) and submodular maximization problems subject to a constant number of (i.e., d-resource) multiple knapsack constraints, in which a solution is a subset of items, along with an assignment of the selected items for each of the d
which adheres to bin capacities. In this paper we present a unified algorithm that yields efficient approximations for a wide class of submodular and modular optimization problems involving
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Our results include
One notable example is
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modular maximization with a constant number of multiple knapsack constraints and a matroid constraint, thus generalizing
Multiple-Choice Multiple Knapsack, improving upon
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results for the classic multiple knapsack problem as well as d-dimensional knapsack, for any d \geq
ratio of
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We further obtain a tight (1-e^{-1
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Another example is Non-monotone Submodular Multiple Knapsack, for which we obtain a (
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\epsilon
\varepsilon
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for non-monotone submodular optimization subject to a constant number of multiple knapsack constraints. At the heart of our algorithms lies a novel representation of a multiple knapsack constraint as a polytope. We consider this key tool as a main technical contribution of this paper
, matching the best known ratio for a single knapsack constraint. The robustness of our algorithm is achieved by applying a novel fractional variant of the classical linear grouping technique, which is of independent interest