A technique for obtaining true approximations for k-center with covering constraints

There has been a recent surge of interest in incorporating fairness aspects into classical clustering problems. Two recently introduced variants of the k-Center problem in this spirit are Colorful k-Center, introduced by Bandyapadhyay, Inamdar, Pai, and Varadarajan, and lottery models, such as the Fair Robust k-Center problem introduced by Harris, Pensyl, Srinivasan, and Trinh. To address fairness aspects, these models, compared to traditional k-Center, include additional covering constraints. Prior approximation results for these models require to relax some of the normally hard constraints, like the number of centers to be opened or the involved covering constraints, and therefore, only obtain constant-factor pseudo-approximations. In this paper, we introduce a new approach to deal with such covering constraints that leads to (true) approximations, including a 4-approximation for Colorful k-Center with constantly many colors—settling an open question raised by Bandyapadhyay, Inamdar, Pai, and Varadarajan—and a 4-approximation for Fair Robust k-Center, for which the existence of a (true) constant-factor approximation was also open. We complement our results by showing that if one allows an unbounded number of colors, then Colorful k-Center admits no approximation algorithm with finite approximation guarantee, assuming that $$\mathtt {P}\ne \mathtt {NP}$$ P ≠ NP . Moreover, under the Exponential Time Hypothesis, the problem is inapproximable if the number of colors grows faster than logarithmic in the size of the ground set.

Prophet Inequalities for Bayesian Persuasion

We study an information-structure design problem (i.e., a Bayesian persuasion problem) in an online scenario. Inspired by the classic gambler's problem, consider a set of candidates who arrive sequentially and are evaluated by one agent (the sender). This agent learns the value from hiring the candidate to herself as well as the value to another agent, the receiver. The sender provides a signal to the receiver who, in turn, makes an irrevocable decision on whether or not to hire the candidate. A-priori, for each agent the distribution of valuation is independent across candidates but may not be identical. We design good online signaling schemes for the sender. To assess the performance, we compare the expected utility to that of an optimal offline scheme by a prophet sender who knows all candidate realizations in advance. We show an optimal prophet inequality for online Bayesian persuasion, with a 1/2-approximation when the instance satisfies a "satisfactory-status-quo" assumption. Without this assumption, there are instances without any finite approximation factor. We extend the results to combinatorial domains and obtain prophet inequalities for matching with multiple hires and multiple receivers.

A Linear Spline Markov Approximation Method for Random Maps with Position Dependent Probabilities

We present a rigorous convergence analysis of a linear spline Markov finite approximation method for computing stationary densities of random maps with position dependent probabilities, which consist of several chaotic maps. The whole analysis is based on a new Lasota–Yorke-type inequality for the Markov operator associated with the random map, which is better than the previous one in the literature and much simpler to obtain. We also present numerical results to support our theoretical analysis.

LOCAL KEISLER MEASURES AND NIP FORMULAS

We study generically stable measures in the local, NIP context. We show that in this setting, a measure is generically stable if and only if it admits a natural finite approximation.

Phase transitions in the extended particle systems, Hagedorn temperature and fractal dimension of space, as a confinement phase transition order parameter

We consider a formal definition of New physics, fundamental constants of physics, formulation of the quantum statistical theory including extra integrals, monopole mechanism for hadronization in QCD, finite approximation of the zeta-function and fermion factorization of the bosonic statistical sum.