NP-hardness and approximation algorithms for solving Euclidean problem of finding a maximum total weight subset of vectors

We present a formal study of first-look and preferred deals that are a recently introduced generation of contracts for selling online advertisements, which generalize traditional reservation contracts and are suitable for advertisers with advanced targeting capabilities. Under these deals, one or more advertisers gain priority access to an inventory of impressions before others and can choose to purchase in real time. In particular, we propose constant-factor approximation algorithms for maximizing the revenue that can be obtained from these deals when offered to all or a subset of the advertisers, whose valuation distributions can be independent or correlated through a common value component. We evaluate our algorithms using data from Google’s ad exchange platform and show they perform better than the approximation guarantees and obtain significantly higher revenue than auctions; in certain cases, the observed revenue is 85%–96% of the optimal revenue achievable. We also prove the NP-hardness of designing deals when advertisers’ valuations are arbitrarily correlated and the optimality of menus of deals among a certain class of selling mechanisms in an incomplete distributional information setting.

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Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

Journal of Artificial Intelligence Research ◽

10.1613/jair.1.12222 ◽

2020 ◽

Vol 69 ◽

pp. 109-141

Author(s):

Paul Goldberg ◽

Alexandros Hollender ◽

Warut Suksompong

Keyword(s):

Approximation Algorithms ◽

Efficient Algorithms ◽

Decision Problems ◽

Fair Allocation ◽

Prior Work ◽

Cake Cutting ◽

Np Hardness

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results extending and strengthening those from prior work. Finally, we investigate connections between approximate and exact envy-freeness, as well as between continuous and discrete cake cutting.

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Contiguous Cake Cutting: Hardness Results and Approximation Algorithms

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v34i02.5570 ◽

2020 ◽

Vol 34 (02) ◽

pp. 1990-1997

Author(s):

Paul W. Goldberg ◽

Alexandros Hollender ◽

Warut Suksompong

Keyword(s):

Approximation Algorithms ◽

Efficient Algorithms ◽

Decision Problems ◽

Fair Allocation ◽

Prior Work ◽

Cake Cutting ◽

Np Hardness

We study the fair allocation of a cake, which serves as a metaphor for a divisible resource, under the requirement that each agent should receive a contiguous piece of the cake. While it is known that no finite envy-free algorithm exists in this setting, we exhibit efficient algorithms that produce allocations with low envy among the agents. We then establish NP-hardness results for various decision problems on the existence of envy-free allocations, such as when we fix the ordering of the agents or constrain the positions of certain cuts. In addition, we consider a discretized setting where indivisible items lie on a line and show a number of hardness results strengthening those from prior work.

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NP-Hardness and Approximation Algorithms for Iterative Pricing on Social Networks with Externalities

International Journal of Foundations of Computer Science ◽

10.1142/s0129054121500313 ◽

2021 ◽

pp. 1-23

Author(s):

Chenli Shen ◽

Wensong Lin

Keyword(s):

Intrinsic Value ◽

Complete Information ◽

Total Weight ◽

Negative Externality ◽

Total Degree ◽

Negative Externalities ◽

Intrinsic Values ◽

Positive Externality ◽

Np Hardness ◽

Over Time

We study how a monopolist seller should price an indivisible product iteratively to the consumers who are connected by a known link-weighted directed social network. For two consumers [Formula: see text] and [Formula: see text], there is an arc directed from [Formula: see text] to [Formula: see text] if and only if [Formula: see text] is a fashion leader of [Formula: see text]. Assuming complete information about the network, the seller offers consumers a sequence of prices over time and the goal is to obtain the maximum revenue. We assume that the consumers buy the product as soon as the seller posts a price not greater than their valuations of the product. The product’s value for a consumer is determined by three factors: a fixed consumer specified intrinsic value and a variable positive (resp. negative) externality that is exerted from the consumer’s out(resp. in)-neighbours. The setting of positive externality is that the influence of fashion leaders on a consumer is the total weight of links from herself to her fashion leaders who have owned the product, and more fashion leaders of a consumer owning the product will increase the influence (external value) on the consumer. And the setting of negative externalities is that the product’s value of showing off for a consumer is the total weight of links from her followers who do not own the product to herself, and more followers of a consumer owning the product will decrease this external value for the consumer. We confirm that finding an optimal iterative pricing is NP-hard even for acyclic networks with maximum total degree [Formula: see text] and with all intrinsic values zero. We design a greedy algorithm which achieves [Formula: see text]-approximation for networks with all intrinsic values zero and show that the approximation ratio [Formula: see text] is tight. Complementary to the hardness result, we design a [Formula: see text]-approximation algorithm for Barabási–Albert networks.

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Composite Solutions for Deck Catamarans

Materiale Plastice ◽

10.37358/mp.18.1.4951 ◽

2018 ◽

Vol 55 (1) ◽

pp. 1-4

Author(s):

Elena Felicia Beznea ◽

Ionel Chirica ◽

Adrian Presura ◽

Ionel Iacob

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Sandwich Structure ◽

High Strength ◽

Total Weight ◽

External Loading ◽

Strength Analysis ◽

Allowable Stress ◽

The Finite Element Method ◽

Inland Navigation

The paper is treating the strength analysis of the main deck structure of an inland navigation catamaran for 30 passengers. The main deck should have high stiffness and high strength to resist to external loading and endure high stresses from combined bending and torsion loads. Different materials for sandwich structure of the deck have been analysed by using the Finite Element Method in order to determine the solution which accomplish better designing criteria regarding allowable stress and deformations and total weight.

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