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Register a new userThe {0,1}-knapsack problem with qualitative levels
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Added by
Luca Elias Sch\\\"afer
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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A variant of the classical knapsack problem is considered in which each item is associated with an integer weight and a qualitative level. We define a dominance relation over the feasible subsets of the given item set and show that this relation defines a preorder. We propose a dynamic programming algorithm to compute the entire set of non-dominated rank cardinality vectors and we state two greedy algorithms, which efficiently compute a single efficient solution.
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We consider the distributed version of the Multiple Knapsack Problem (MKP), where $m$ items are to be distributed amongst $n$ processors, each with a knapsack. We propose different distributed approximation algorithms with a tradeoff between time and message complexities. The algorithms are based on the greedy approach of assigning the best item to the knapsack with the largest capacity. These algorithms obtain a solution with a bound of $frac{1}{n+1}$ times the optimum solution, with either $mathcal{O}left(mlog nright)$ time and $mathcal{O}left(m nright)$ messages, or $mathcal{O}left(mright)$ time and $mathcal{O}left(mn^{2}right)$ messages.
We introduce and study a discrete multi-period extension of the classical knapsack problem, dubbed generalized incremental knapsack. In this setting, we are given a set of $n$ items, each associated with a non-negative weight, and $T$ time periods with non-decreasing capacities $W_1 leq dots leq W_T$. When item $i$ is inserted at time $t$, we gain a profit of $p_{it}$; however, this item remains in the knapsack for all subsequent periods. The goal is to decide if and when to insert each item, subject to the time-dependent capacity constraints, with the objective of maximizing our total profit. Interestingly, this setting subsumes as special cases a number of recently-studied incremental knapsack problems, all known to be strongly NP-hard. Our first contribution comes in the form of a polynomial-time $(frac{1}{2}-epsilon)$-approximation for the generalized incremental knapsack problem. This result is based on a reformulation as a single-machine sequencing problem, which is addressed by blending dynamic programming techniques and the classical Shmoys-Tardos algorithm for the generalized assignment problem. Combined with further enumeration-based self-reinforcing ideas and newly-revealed structural properties of nearly-optimal solutions, we turn our basic algorithm into a quasi-polynomial time approximation scheme (QPTAS). Hence, under widely believed complexity assumptions, this finding rules out the possibility that generalized incremental knapsack is APX-hard.
In this paper, we study the stochastic unbounded min-knapsack problem ($textbf{Min-SUKP}$). The ordinary unbounded min-knapsack problem states that: There are $n$ types of items, and there is an infinite number of items of each type. The items of the same type have the same cost and weight. We want to choose a set of items such that the total weight is at least $W$ and the total cost is minimized. The prob~generalizes the ordinary unbounded min-knapsack problem to the stochastic setting, where the weight of each item is a random variable following a known distribution and the items of the same type follow the same weight distribution. In prob, different types of items may have different cost and weight distributions. In this paper, we provide an FPTAS for $textbf{Min-SUKP}$, i.e., the approximate value our algorithm computes is at most $(1+epsilon)$ times the optimum, and our algorithm runs in $poly(1/epsilon,n,log W)$ time.
We introduce and study a general version of the fractional online knapsack problem with multiple knapsacks, heterogeneous constraints on which items can be assigned to which knapsack, and rate-limiting constraints on the assignment of items to knapsacks. This problem generalizes variations of the knapsack problem and of the one-way trading problem that have previously been treated separately, and additionally finds application to the real-time control of electric vehicle (EV) charging. We introduce a new algorithm that achieves a competitive ratio within an additive factor of one of the best achievable competitive ratios for the general problem and matches or improves upon the best-known competitive ratio for special cases in the knapsack and one-way trading literatures. Moreover, our analysis provides a novel approach to online algorithm design based on an instance-dependent primal-dual analysis that connects the identification of worst-case instances to the design of algorithms. Finally, we illustrate the proposed algorithm via trace-based experiments of EV charging.
The random-order or secretary model is one of the most popular beyond-worst case model for online algorithms. While it avoids the pessimism of the traditional adversarial model, in practice we cannot expect the input to be presented in perfectly random order. This has motivated research on ``best of both worlds (algorithms with good performance on both purely stochastic and purely adversarial inputs), or even better, on inputs that are a mix of both stochastic and adversarial parts. Unfortunately the latter seems much harder to achieve and very few results of this type are known. Towards advancing our understanding of designing such robust algorithms, we propose a random-order model with bursts of adversarial time steps. The assumption of burstiness of unexpected patterns is reasonable in many contexts, since changes (e.g. spike in a demand for a good) are often triggered by a common external event. We then consider the Knapsack Secretary problem in this model: there is a knapsack of size $k$ (e.g., available quantity of a good), and in each of the $n$ time steps an item comes with its value and size in $[0,1]$ and the algorithm needs to make an irrevocable decision whether to accept or reject the item. We design an algorithm that gives an approximation of $1 - tilde{O}(Gamma/k)$ when the adversarial time steps can be covered by $Gamma ge sqrt{k}$ intervals of size $tilde{O}(frac{n}{k})$. In particular, setting $Gamma = sqrt{k}$ gives a $(1 - O(frac{ln^2 k}{sqrt{k}}))$-approximation that is resistant to up to a $frac{ln^2 k}{sqrt{k}}$-fraction of the items being adversarial, which is almost optimal even in the absence of adversarial items. Also, setting $Gamma = tilde{Omega}(k)$ gives a constant approximation that is resistant to up to a constant fraction of items being adversarial.
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