The Equivalence of Sampling and Searching

In a sampling problem, we are given an input x, and asked to sample approximately from a probability distribution D_x. In a search problem, we are given an input x, and asked to find a member of a nonempty set A_x with high probability. (An example is finding a Nash equilibrium.) In this paper, we use tools from Kolmogorov complexity and algorithmic information theory to show that sampling and search problems are essentially equivalent. More precisely, for any sampling problem S, there exists a search problem R_S such that, if C is any "reasonable" complexity class, then R_S is in the search version of C if and only if S is in the sampling version. As one application, we show that SampP=SampBQP if and only if FBPP=FBQP: in other words, classical computers can efficiently sample the output distribution of every quantum circuit, if and only if they can efficiently solve every search problem that quantum computers can solve. A second application is that, assuming a plausible conjecture, there exists a search problem R that can be solved using a simple linear-optics experiment, but that cannot be solved efficiently by a classical computer unless the polynomial hierarchy collapses. That application will be described in a forthcoming paper with Alex Arkhipov on the computational complexity of linear optics.Comment: 16 page

Computer-aided verification in mechanism design

In mechanism design, the gold standard solution concepts are dominant strategy incentive compatibility and Bayesian incentive compatibility. These solution concepts relieve the (possibly unsophisticated) bidders from the need to engage in complicated strategizing. While incentive properties are simple to state, their proofs are specific to the mechanism and can be quite complex. This raises two concerns. From a practical perspective, checking a complex proof can be a tedious process, often requiring experts knowledgeable in mechanism design. Furthermore, from a modeling perspective, if unsophisticated agents are unconvinced of incentive properties, they may strategize in unpredictable ways. To address both concerns, we explore techniques from computer-aided verification to construct formal proofs of incentive properties. Because formal proofs can be automatically checked, agents do not need to manually check the properties, or even understand the proof. To demonstrate, we present the verification of a sophisticated mechanism: the generic reduction from Bayesian incentive compatible mechanism design to algorithm design given by Hartline, Kleinberg, and Malekian. This mechanism presents new challenges for formal verification, including essential use of randomness from both the execution of the mechanism and from the prior type distributions. As an immediate consequence, our work also formalizes Bayesian incentive compatibility for the entire family of mechanisms derived via this reduction. Finally, as an intermediate step in our formalization, we provide the first formal verification of incentive compatibility for the celebrated Vickrey-Clarke-Groves mechanism

Electrochemical Investigation of Phenethylammonium Bismuth Iodide as Anode in Aqueous Zn2+ Electrolytes

Despite the high potential impact of aqueous battery systems, fundamental characteristics such as cost, safety, and stability make them less feasible for large-scale energy storage systems. One of the main barriers encountered in the commercialization of aqueous batteries is the development of large-scale electrodes with high reversibility, high rate capability, and extended cycle stability at low operational and maintenance costs. To overcome some of these issues, the current research work is focused on a new class of material based on phenethylammonium bismuth iodide on fluorine doped SnO2-precoated glass substrate via aerosol-assisted chemical vapor deposition, a technology that is industrially competitive. The anode materials were electrochemically investigated in Zn2+ aqueous electrolytes as a proof of concept, which presented a specific capacity of 220 mAh g−1 at 0.4 A g−1 with excellent stability after 50 scans and capacity retention of almost 100%

Spatial and Temporal Coherence in Strongly Coupled Plasmonic Bose-Einstein Condensates

We report first-order spatial and temporal correlations in strongly coupled plasmonic Bose-Einstein condensates. The condensate is large, more than twenty times the intrinsic spatial coherence length of the polaritons and hundred times the healing length, making plasmonic lattices an attractive platform for studying long-range spatial correlations in two dimensions. We find that both spatial and temporal coherence display non-exponential decay; the results suggest power-law or stretched exponential behaviour with different exponents for spatial and temporal correlation decays.Comment: 11 pages, 11 figure

Circular Networks from Distorted Metrics

Trees have long been used as a graphical representation of species relationships. However complex evolutionary events, such as genetic reassortments or hybrid speciations which occur commonly in viruses, bacteria and plants, do not fit into this elementary framework. Alternatively, various network representations have been developed. Circular networks are a natural generalization of leaf-labeled trees interpreted as split systems, that is, collections of bipartitions over leaf labels corresponding to current species. Although such networks do not explicitly model specific evolutionary events of interest, their straightforward visualization and fast reconstruction have made them a popular exploratory tool to detect network-like evolution in genetic datasets. Standard reconstruction methods for circular networks, such as Neighbor-Net, rely on an associated metric on the species set. Such a metric is first estimated from DNA sequences, which leads to a key difficulty: distantly related sequences produce statistically unreliable estimates. This is problematic for Neighbor-Net as it is based on the popular tree reconstruction method Neighbor-Joining, whose sensitivity to distance estimation errors is well established theoretically. In the tree case, more robust reconstruction methods have been developed using the notion of a distorted metric, which captures the dependence of the error in the distance through a radius of accuracy. Here we design the first circular network reconstruction method based on distorted metrics. Our method is computationally efficient. Moreover, the analysis of its radius of accuracy highlights the important role played by the maximum incompatibility, a measure of the extent to which the network differs from a tree.Comment: Submitte

The significance of GATA3 expression in breast cancer: a 10-year follow-up study.

GATA3 is a transcription factor closely associated with estrogen receptor alpha in breast carcinoma, with a potential prognostic utility. This study investigated the immunohistochemical expression of GATA3 in estrogen receptor alpha-positive and estrogen receptor alpha-negative breast carcinomas. One hundred sixty-six cases of invasive breast carcinomas with 10-year follow-up information were analyzed. Positive GATA3 and estrogen receptor alpha cases were defined as greater than 20% of cells staining. Time to cancer recurrence and time to death were analyzed with survival methods. Of 166 patients, 40 were estrogen receptor alpha negative and 121 estrogen receptor alpha positive. Thirty-eight (23%) recurrences and 51 (31%) deaths were observed. In final multivariable analyses, GATA3-positive tumors had about two thirds the recurrence risk of GATA3-negative tumors (hazard ratio = 0.65, P = .395) and comparable mortality risk (hazard ratio = 0.86, P = .730). In prespecified subgroup analyses, the protective effect of GATA3 expression was most pronounced among estrogen receptor alpha-positive patients who received tamoxifen (hazard ratio = 0.57 for recurrence and 0.68 for death). We found no statistically significant differences in recurrence or survival rates between GATA3-positive and GATA3-negative tumors. However, there was a suggestion of a modest-to-strong protective effect of GATA3 expression among estrogen receptor alpha-positive patients receiving hormone therapy

Sorting and Selection in Posets

Classical problems of sorting and searching assume an underlying linear ordering of the objects being compared. In this paper, we study these problems in the context of partially ordered sets, in which some pairs of objects are incomparable. This generalization is interesting from a combinatorial perspective, and it has immediate applications in ranking scenarios where there is no underlying linear ordering, e.g., conference submissions. It also has applications in reconstructing certain types of networks, including biological networks. Our results represent significant progress over previous results from two decades ago by Faigle and Turán. In particular, we present the first algorithm that sorts a width-w poset of size n with query complexity O(n(w+\log n)) and prove that this query complexity is asymptotically optimal. We also describe a variant of Mergesort with query complexity O(wn log n/w) and total complexity O(w2n log n/w); an algorithm with the same query complexity was given by Faigle and Turán, but no efficient implementation of that algorithm is known. Both our sorting algorithms can be applied with negligible overhead to the more general problem of reconstructing transitive relations. We also consider two related problems: finding the minimal elements, and its generalization to finding the bottom k “levels,” called the k-selection problem. We give efficient deterministic and randomized algorithms for finding the minimal elements with query complexity and total complexity O(wn). We provide matching lower bounds for the query complexity up to a factor of 2 and generalize the results to the k-selection problem. Finally, we present efficient algorithms for computing a linear extension of a poset and computing the heights of all elements

The Complexity of Nash Equilibria in Simple Stochastic Multiplayer Games

We analyse the computational complexity of finding Nash equilibria in simple stochastic multiplayer games. We show that restricting the search space to equilibria whose payoffs fall into a certain interval may lead to undecidability. In particular, we prove that the following problem is undecidable: Given a game G, does there exist a pure-strategy Nash equilibrium of G where player 0 wins with probability 1. Moreover, this problem remains undecidable if it is restricted to strategies with (unbounded) finite memory. However, if mixed strategies are allowed, decidability remains an open problem. One way to obtain a provably decidable variant of the problem is restricting the strategies to be positional or stationary. For the complexity of these two problems, we obtain a common lower bound of NP and upper bounds of NP and PSPACE respectively.Comment: 23 pages; revised versio