Ordered k-Median with Outliers

We study a natural generalization of the celebrated ordered k-median problem, named robust ordered k-median, also known as ordered k-median with outliers. We are given facilities ? and clients ? in a metric space (???,d), parameters k,m ? ?_+ and a non-increasing non-negative vector w ? ?_+^m. We seek to open k facilities F ? ? and serve m clients C ? ?, inducing a service cost vector c = {d(j,F):j ? C}; the goal is to minimize the ordered objective w^?c^?, where d(j,F) = min_{i ? F}d(j,i) is the minimum distance between client j and facilities in F, and c^? ? ?_+^m is the non-increasingly sorted version of c. Robust ordered k-median captures many interesting clustering problems recently studied in the literature, e.g., robust k-median, ordered k-median, etc. We obtain the first polynomial-time constant-factor approximation algorithm for robust ordered k-median, achieving an approximation guarantee of 127. The main difficulty comes from the presence of outliers, which already causes an unbounded integrality gap in the natural LP relaxation for robust k-median. This appears to invalidate previous methods in approximating the highly non-linear ordered objective. To overcome this issue, we introduce a novel yet very simple reduction framework that enables linear analysis of the non-linear objective. We also devise the first constant-factor approximations for ordered matroid median and ordered knapsack median using the same framework, and the approximation factors are 19.8 and 41.6, respectively

Nature of proton transport in a water-filled carbon nanotube and in liquid water

Proton transport (PT) in bulk liquid water and within a thin water-filled carbon nanotube has been examined with ab initio pathintegral molecular dynamics (PIMD). Barrierless proton transfer is observed in each case when quantum nuclear effects (QNEs) are accounted for. The key difference between the two systems is that in the nanotube facile PT is facilitated by a favorable prealignment of water molecules, whereas in bulk liquid water solvent reorganization is required prior to PT. Configurations where the quantum excess proton is delocalized over several adjacent water molecules along with continuous interconversion between different hydration states reveals that, as in liquid water, the hydrated proton under confinement is best described as a fluxional defect, rather than any individual idealized hydration state such as Zundel, Eigen, or the so-called linear H7O3+ complex along the water chain. These findings highlight the importance of QNEs in intermediate strength hydrogen bonds (HBs) and explain why H+ diffusion through nanochannels is impeded much less than other cations.Comment: 6 pages, 4 figure

Practical algorithms and experimentally validated incentives for equilibrium-based fair division (A-CEEI)

Approximate Competitive Equilibrium from Equal Incomes (A-CEEI) is an equilibrium-based solution concept for fair division of discrete items to agents with combinatorial demands. In theory, it is known that in asymptotically large markets: 1. For incentives, the A-CEEI mechanism is Envy-Free-but-for-Tie-Breaking (EF-TB), which implies that it is Strategyproof-in-the-Large (SP-L). 2. From a computational perspective, computing the equilibrium solution is unfortunately a computationally intractable problem (in the worst-case, assuming $\textsf{PPAD}\ne \textsf{FP}$). We develop a new heuristic algorithm that outperforms the previous state-of-the-art by multiple orders of magnitude. This new, faster algorithm lets us perform experiments on real-world inputs for the first time. We discover that with real-world preferences, even in a realistic implementation that satisfies the EF-TB and SP-L properties, agents may have surprisingly simple and plausible deviations from truthful reporting of preferences. To this end, we propose a novel strengthening of EF-TB, which dramatically reduces the potential for strategic deviations from truthful reporting in our experiments. A (variant of) our algorithm is now in production: on real course allocation problems it is much faster, has zero clearing error, and has stronger incentive properties than the prior state-of-the-art implementation.Comment: To appear in EC 202