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1. Two-Sided Kirszbraun Theorem

   https://doi.org/10.4230/LIPIcs.SoCG.2021.13  

   Backurs, Arturs; Mahabadi, Sepideh; Makarychev, Konstantin; Makarychev, Yury (June 2021, Leibniz international proceedings in informatics)

   Buchin, Kevin; Colin de Verdiere, Eric (Ed.)

   In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y. In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖. 

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2. Approximately Efficient Two-Sided Combinatorial Auctions

   https://doi.org/10.1145/3381523  

   Colini-Baldeschi, Riccardo; Goldberg, Paul W.; Keijzer, Bart de; Leonardi, Stefano; Roughgarden, Tim; Turchetta, Stefano (April 2020, ACM Transactions on Economics and Computation)
3. On the Two-sided Permutation Inversion Problem

   https://doi.org/10.62056/a0qj89n4e  

   Alagic, Gorjan; Bai, Chen; Poremba, Alexander; Shi, Kaiyan (April 2024, IACR Communications in Cryptology)

   In the permutation inversion problem, the task is to find the preimage of some challenge value, given oracle access to the permutation. This fundamental problem in query complexity appears in many contexts, particularly cryptography. In this work, we examine the setting in which the oracle allows for quantum queries to both the forward and the inverse direction of the permutation—except that the challenge value cannot be submitted to the latter. Within that setting, we consider three options for the inversion algorithm: whether it can get quantum advice about the permutation, whether the query algorithm can restrict the distribution with which the challenge input is sampled, and whether it must produce the entire preimage (search) or only the first bit (decision). We prove several theorems connecting the hardness of the resulting variations of the permutation inversion problem and establish lower bounds for them. Our results show that, perhaps surprisingly, the permutation inversion problem does not become significantly easier when the adversary is granted oracle access to the inverse—provided it cannot query the challenge itself. 

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4. A two-sided model of paid peering

   https://doi.org/10.1016/j.telpol.2022.102352  

   Nikkhah, Ali; Jordan, Scott (May 2022, Telecommunications policy)

   Internet users have suffered collateral damage in tussles over paid peering between large ISPs and large content providers. Paid peering is a relationship where two networks exchange traffic with payment, which provides direct access to each other’s customers without having to pay a third party to carry that traffic for them. The issue will arise again when the United States Federal Communications Commission (FCC) considers a new net neutrality order. We first consider the effect of paid peering on broadband prices. We adopt a two-sided market model in which an ISP maximizes profit by setting broadband prices and a paid peering price. We analytically derive the profit-maximizing prices, and show that they satisfy a generalization of the well-known Lerner rule. Our result shows that paid peering fees reduce the premium plan price, increase the video streaming price and the total price for premium tier customers who subscribe to video streaming services; however, the ISP passes on to its customers only a portion of the revenue from paid peering. ISP profit increases but video streaming profit decreases as an ISP moves from settlement-free peering to paid peering price. We next consider the effect of paid peering on consumer surplus. We find that consumer surplus is a uni-modal function of the paid peering fee. The paid peering fee that maximizes consumer surplus depends on elasticities of demand for broadband and for video streaming. However, consumer surplus is maximized when paid peering fees are significantly lower than those that maximize ISP profit. However, it does not follow that settlement-free peering is always the policy that maximizes consumer surplus. The peering price depends critically on the incremental ISP cost per video streaming subscriber; at different costs, it can be negative, zero, or positive. 

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5. Two-sided permutation statistics via symmetric functions

   https://doi.org/10.1017/fms.2024.89  

   Gessel, Ira M; Zhuang, Yan (November 2024, Forum of Mathematics, Sigma)

   Abstract Given a permutation statistic$$\operatorname {\mathrm {st}}$$, define its inverse statistic$$\operatorname {\mathrm {ist}}$$by. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$whenever$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$. Our work leads to a rederivation of Stanley’s generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and$$\gamma $$-positivity. 

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