Optimal Pricing for MHR and λ-regular Distributions

Let be a class of probability distributions over the discrete domain [n] = {1, …, n}. We show that if satisfies a rather general condition – essentially, that each distribution in can be well-approximated by a variable-width histogram with few bins – then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of k unknown distributions from . We analyze several natural types of distributions over [n], including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated by a histogram with few bins. Applying our general algorithm, we obtain near-optimally efficient algorithms for all these mixture learning problems as described below. More precisely, Log-concave distributions: We learn any mixture of k log-concave distributions over [n] using k · Õ(1/ε4) samples (independent of n) and running in time Õ(k log(n)/ε4) bit-operations (note that reading a single sample from [n] takes Θ(log n) bit operations). For the special case k = 1 we give an efficient algorithm using Õ(1/ε3) samples; this generalizes the main result of [DDS12b] from the class of Poisson Binomial distributions to the much broader class of all log-concave distributions. Our upper bounds are not far from optimal since any algorithm for this learning problem requires Ω(k/ε5/2) samples. Monotone hazard rate (MHR) distributions: We learn any mixture of k MHR distributions over [n] using O(k log(n/ε)/ε4) samples and running in time Õ(k log (n)/ε4) bit-operations. Any algorithm for this learning problem must use Ω(k log(n)/ε3) samples. Unimodal distributions: We give an algorithm that learns any mixture of k unimodal distributions over [n] using O(k log(n)/ε4) samples and running in time Õ(k log2(n)/ε4) bit-operations. Any algorithm for this problem must use Ω(k log(n)/ε3) samples.

Consider a seller who seeks to provide service to a collection of interested parties, subject to feasibility constraints on which parties may be simultaneously served. Assuming that a distribution is known on the value of each party for service---arguably a strong assumption---Myerson's seminal work provides revenue optimizing auctions [12]. We show instead that, for very general feasibility constraints, only knowledge of the median of each party's value distribution, or any other quantile of these distributions, or approximations thereof, suffice for designing simple auctions that simultaneously approximate both the optimal revenue and the optimal welfare. Our results apply to all downward-closed feasibility constraints under the assumption that the underlying, unknown value distributions are monotone hazard rate, and to all matroid feasibility constraints under the weaker assumption of regularity of the underlying distributions. Our results jointly generalize the single-item results obtained by Azar and Micali [2] on parametric auctions, and Daskalakis and Pierrakos [6] for simultaneously approximately optimal and efficient auctions.

The tight asymptotic approximation ratio of First Fit for bin packing with cardinality constraints

This article presents applicable bounds for distributions with monotone log-odds rate. The bounds are based on results obtained by Wang et al. (2003) and Zimmer et al. (1998). Due to the complexity of some of the equations, numerical analysis and computer algorithms were necessary to calculate some of the results of these bounds. Five different types of bounds are tabled in this research. Numerical examples of their application and comparisons to the results obtained by Barlow and Marshall (1965) of bounds on distributions with monotone hazard rate are also provided. The first type of tables is for lower bounds of increasing log-odds rate distributions with knowledge of two values of the reliability of the distribution. The second and third types of tables are for upper bounds under the same conditions. The fourth type of tables is for lower bounds of increasing log-odds rate distributions with knowledge of the mean and the median of the distribution. Finally, the fifth type of tables is for lower bounds of increasing log-odds rate distributions with knowledge of the mean and the variance of the distribution.

SUMMARY The generalized Pareto distribution as a model for survival data is considered. The three parameter model is general enough so as to include increasing, decreasing and constant failure rate distributions. Maximum likelihood estimation for the progressively censored case is discussed at length and conditions for existence and uniqueness of estimates are given. Examples are presented to illustrate its appropriateness for a variety of situations. We consider a population of items whose times to failure from a well-defined origin are being observed. Time to failure is assumed to be a random variable T having survivor function S(t) = pr (T > t), density f(t), and hazard rate h(t) = -d log S(t)/dt. Although the most widely used model for h(t) is constant hazard, giving rise to the exponential distribution, this may not be a realistic assumption when modelling chronic disease populations. Also, when the origin corresponds to an event which may dramatically influence survival, such as surgery, the objection to constant hazard may be especially strong. In situations where constant hazard is not a reasonable assumption, monotone hazard usually is. The Weibull and most other parametric families used to model survival data have monotone hazard rates. Another family of monotone hazard rates which forms the basis of this paper is given by

A seminal result of Bulow and Klemperer [1989] demonstrates the power of competition for extracting revenue: when selling a single item to n bidders whose values are drawn i.i.d. from a regular distribution, the simple welfare-maximizing VCG mechanism (in this case, a second price-auction) with one additional bidder extracts at least as much revenue in expectation as the optimal mechanism. The beauty of this theorem stems from the fact that VCG is a prior-independent mechanism, where the seller possesses no information about the distribution, and yet, by recruiting one additional bidder it performs better than any prior-dependent mechanism tailored exactly to the distribution at hand (without the additional bidder). In this work, we establish the first full Bulow-Klemperer results in multi-dimensional environments, proving that by recruiting additional bidders, the revenue of the VCG mechanism surpasses that of the optimal (possibly randomized, Bayesian incentive compatible) mechanism. For a given environment with i.i.d. bidders, we term the number of additional bidders needed to achieve this guarantee the environment's competition complexity. Using the recent duality-based framework of Cai et al. [2016] for reasoning about optimal revenue, we show that the competition complexity of n bidders with additive valuations over m independent, regular items is at most n+2m-2 and at least log(m). We extend our results to bidders with additive valuations subject to downward-closed constraints, showing that these significantly more general valuations increase the competition complexity by at most an additive m-1 factor. We further improve this bound for the special case of matroid constraints, and provide additional extensions as well.

We consider the problem of maintaining an approximately maximum (fractional) matching and an approximately minimum vertex cover in a dynamic graph. Starting with the seminal paper by Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. There remains, however, a polynomial gap between the best known worst case update time and the best known amortised update time for this problem, even after allowing for randomisation. Specifically, Bernstein and Stein [ICALP 2015, SODA 2016] have the best known worst case update time. They present a deterministic data structure with approximation ratio $(3/2+\epsilon)$ and worst case update time $O(m^{1/4}/\epsilon^2)$, where $m$ is the number of edges in the graph. In recent past, Gupta and Peng [FOCS 2013] gave a deterministic data structure with approximation ratio $(1+\epsilon)$ and worst case update time $O(\sqrt{m}/\epsilon^2)$. No known randomised data structure beats the worst case update times of these two results. In contrast, the paper by Onak and Rubinfeld [STOC 2010] gave a randomised data structure with approximation ratio $O(1)$ and amortised update time $O(\log^2 n)$, where $n$ is the number of nodes in the graph. This was later improved by Baswana, Gupta and Sen [FOCS 2011] and Solomon [FOCS 2016], leading to a randomised date structure with approximation ratio $2$ and amortised update time $O(1)$. We bridge the polynomial gap between the worst case and amortised update times for this problem, without using any randomisation. We present a deterministic data structure with approximation ratio $(2+\epsilon)$ and worst case update time $O(\log^3 n)$, for all sufficiently small constants $\epsilon$.

Designing revenue optimal auctions for selling an item to $n$ symmetric bidders is a fundamental problem in mechanism design. Myerson (1981) shows that the second price auction with an appropriate reserve price is optimal when bidders' values are drawn i.i.d. from a known regular distribution. A cornerstone in the prior-independent revenue maximization literature is a result by Bulow and Klemperer (1996) showing that the second price auction without a reserve achieves (n-1)/n of the optimal revenue in the worst case. We construct a randomized mechanism that strictly outperforms the second price auction in this setting. Our mechanism inflates the second highest bid with a probability that varies with $n$. For two bidders we improve the performance guarantee from 0.5 to 0.512 of the optimal revenue. We also resolve a question in the design of revenue optimal mechanisms that have access to a single sample from an unknown distribution. We show that a randomized mechanism strictly outperforms all deterministic mechanisms in terms of worst case guarantee.

Two classes of distributions that are widely used in the analysis of Bayesian auctions are the Monotone Hazard Rate MHR and Regular distributions. They can both be characterized in terms of the rate of change of the associated virtual value functions: for MHR distributions the condition is that for values $$v < v'$$, $$\phi v' - \phi v \ge v' - v$$, and for regular distributions, $$\phi v' - \phi v \ge 0$$. Cole and Roughgarden introduced the interpolating class of $$\alpha $$-Strongly Regular distributions $$\alpha $$-SR distributions for short, for which $$\phi v' - \phi v \ge \alpha v' - v$$, for $$0 \le \alpha \le 1$$. In this paper, we investigate five distinct auction settings for which good expected revenue bounds are known when the bidders' valuations are given by MHR distributions. In every case, we show that these bounds degrade gracefully when extended to $$\alpha $$-SR distributions. For four of these settings, the auction mechanism requires knowledge of these distributions in the other setting, the distributions are needed only to ensure good bounds on the expected revenue. In these cases we also investigate what happens when the distributions are known only approximately via samples, specifically how to modify the mechanisms so that they remain effective and how the expected revenue depends on the number of samples.

There has been much recent work on the revenue-raising properties of truthful mechanisms for selling goods to selfish bidders. Typically the revenue of a mechanism is compared against a benchmark (such as, the maximum revenue obtainable by an omniscient seller selling at a fixed price to at least two customers), with a view to understanding how much lower the mechanism's revenue is than the benchmark, in the worst case. We study this issue in the context of lotteries, where the seller may sell a probability of winning an item. We are interested in two general issues. Firstly, we aim at using the true optimum revenue as benchmark for our auctions. Secondly, we study the extent to which the expressive power resulting from lotteries, helps to improve the worst-case ratio. We study this in the well-known context of digital goods. We show that in this scenario, collusion-resistant lotteries (these are lotteries for which no coalition of bidders exchanging side payments has an advantage in lying) are as powerful as truthful ones.

#SAT problem is NP-complete, so the small improvement of #SAT problems at the worst case (such as from O(ck) to O((c-e)k)) will make the efficiency of the algorithm improved in the level of exponent. In this paper, we present a new #2-SAT algorithm based on DPLL regarding the number of clauses as parameter. In order to improve the upper bound, we propose two new transformation rules and make a more elaborate analysis of the constraint graph for choosing better variables to branch. By analyzing, we obtain the new worst case upper bound O(1.1740m), which is the best result up to now within our knowledge.

Dynamic Pricing in an Unknown and Sales-dependent Evolving Marketplace

The problem of merging sorted lists in the least number of pairwise comparisons has been solved completely only for a few special cases. Graham and Karp (Sorting Search 3:197–207, 1999) independently discovered that the tape merge algorithm is optimal in the worst case when the two lists have the same size. In their seminal papers, Stockmeyer and Yao (SIAM J Comput 9(1):85–90, 1980), Murphy and Paull (Inf Control 42(1):87–96, 1979), and Christen (On the optimality of the straight merging algorithm, 1978) independently showed when the lists to be merged are of size m and n satisfying $$m\le n\le \lfloor \frac{3}{2}m\rfloor +1$$, the tape merge algorithm is optimal in the worst case. This paper extends this result by showing that the tape merge algorithm is optimal in the worst case whenever the size of one list is no larger than 1.52 times the size of the other. The main tool we use to prove the lower bound is Knuth’s (1999) adversary methods. In addition, we show that the lower bound cannot be improved to 1.8 via Knuth’s adversary methods. Moreover, we design a simple procedure, and by invoking this procedure recursively until the remaining subproblem can be solved efficiently by another known algorithm, we achieve constant improvement of the upper bound for $$2m-2\le n\le 3m $$.

The problem of merging sorted lists in the least number of pairwise comparisons has been solved completely only for a few special cases. Graham and Karp [TAOCP, 1999] independently discovered that the tape merge algorithm is optimal in the worst case when the two lists have the same size. Stockmeyer and Yao [SICOMP, 1980], Murphy and Paull [Inform. Control, 1979], and Christen [1978] independently showed when the lists to be merged are of size m and n satisfying m leq n leq floor(3/2 m) + 1, the tape merge algorithm is optimal in the worst case. This paper extends this result by showing that the tape merge algorithm is optimal in the worst case whenever the size of one list is no larger than 1.52 times the size of the other. The main tool we used to prove lower bounds is Knuth’s adversary methods [TAOCP, 1999]. In addition, we show that the lower bound cannot be improved to 1.8 via Knuth's adversary methods. We also develop a new inequality about Knuth's adversary methods, which might be interesting in its own right. Moreover, we design a simple procedure to achieve constant improvement of the upper bounds for 2m - 2 leq n leq 3m.

We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in mathsf {Quasi}text {-}mathsf {NP} = mathsf {NTIME}[n^{(log n)^{O(1)}}] and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes mathcal { C}, by showing that mathcal { C} admits non-trivial satisfiability and/or # SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial # SAT algorithm for a circuit class {mathcal C}. Say that a symmetric Boolean function f(x1,…,xn) is sparse if it outputs 1 on O(1) values of {sum }_{i} x_{i}. We show that for every sparse f, and for all “typical” mathcal { C}, faster # SAT algorithms for mathcal { C} circuits imply lower bounds against the circuit class f circ mathcal { C}, which may be stronger than mathcal { C} itself. In particular: # SAT algorithms for nk-size mathcal { C}-circuits running in 2n/nk time (for all k) imply NEXP does not have (f circ mathcal { C})-circuits of polynomial size.# SAT algorithms for 2^{n^{{varepsilon }}}-size mathcal { C}-circuits running in 2^{n-n^{{varepsilon }}} time (for some ε > 0) imply Quasi-NP does not have (f circ mathcal { C})-circuits of polynomial size.Applying # SAT algorithms from the literature, one immediate corollary of our results is that Quasi-NP does not have EMAJ ∘ ACC0 ∘ THR circuits of polynomial size, where EMAJ is the “exact majority” function, improving previous lower bounds against ACC0 [Williams JACM’14] and ACC0 ∘THR [Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.