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Set Cover is a fundamental problem in combinatorial optimization which has been studied for many decades due to its various applications across multiple domains. In many of these domains, the input data consists of locations, relationships, and other sensitive information of individuals which may leaked due to the set cover output. Attempts have been made to design privacy-preserving algorithms to solve the Set Cover problem under privacy constraints. Under differential privacy, it has been proved that the Set Cover problem has strong impossibility results and no explicit forms of the output can be released to the public. In this work, we observe that these hardness results dissolve when we turn to the Partial Set Cover problem, where we only need to cover a constant fraction of the elements. We show that this relaxation enables us to avoid the impossibility results, and give the first algorithm which outputs an explicit form of set cover with non-trivial utility guarantees under differential privacy. Using our algorithm as a subroutine, we design a differentially-private bicriteria algorithm to solve a recently-proposed facility-location problem for vaccine distribution which generalizes k-supplier with outliers. Our analysis shows that relaxing the covering requirement also allows us to circumvent the inherent hardness of k-supplier and give the first nontrivial guarantees. 

This work probed the thermal “switchability” from ethylene coordination/insertion to controlled radical polymerization of methyl acrylate (MA) for Brookhart-type α-diimine PdII catalysts. The investigation focused on the extremely bulky 2,6-bis(3,5-dimethylphenyl)-4-methylphenyl (Xyl4Ph) α-diimine N-substituents to probe reversible PdII–C bond activation in the MA-quenched Pd-capped PE intermediate and reversible trapping during radical MA polymerization. The substituent steric effect on the relative stability of various [PE–MA–PdII(ArN═CMeCMe═NAr)]+ chain-end structures and on the bond dissociation-free energy (BDFE) for the homolytic PdII–C bond cleavage has been assessed by DFT calculations at the full quantum mechanics (QM) and QM/molecular mechanics (QM/MM) methods. The structures comprise ester-chelated forms with the Pd atom bonded to the α, β, and γ C atoms as a result of 2,1 MA insertion into the PE–Pd bond and of subsequent chain walking, as well as related monodentate (ring-opened) forms resulting from the addition of MA or acetonitrile. The opened Cα-bonded form is electronically favored for smaller N-substituents, including 2,6-diisopropylphenyl (Dipp), particularly when MeCN is added, but the open Cγ-bonded form is preferred for the extremely bulky system with Ar = Xyl4Ph. The Pdα–C bond is the weakest one to cleave, with the BDFE decreasing as the Ar steric bulk is increased (31.8, 25.8, and 12.6 kcal mol–1 for Ph, Dipp, and Xyl4Ph, respectively). However, experimental investigations on the [PE–MA–PdII(ArN═CMeCMe═NAr)]+ (Ar = Xyl4Ph) macroinitiator do not show any evidence of radical formation under thermal activation conditions, while photolytic activation produces both TEMPO-trapped (TEMPO = 2,2,6,6-tetramethylpiperidinyloxy) and unsaturated MA-containing PE chains. The DFT investigation has highlighted a low-energy pathway for termination of the PE–MA• radicals by disproportionation, promoted by β-H elimination/dissociation and H-atom abstraction from the PdII–H intermediate by a second radical. This phenomenon appears to be the main reason for the failure of this PdII system to control the radical polymerization of MA by the OMRP (OMRP = organometallic-mediated radical polymerization) mechanism. 

In this paper, we study the first eigenvalue of the Jacobi operator on an integral n-varifold with constant mean curvature in the unit sphere Sn+1. We found the optimal upper bound and prove a rigidity result characterizing the case when it is attained. This gives a new characterization for certain singular Clifford tori.