Robust two-stage combinatorial optimization problems under convex uncertainty

组合纯探索与全臂反馈及其扩展：在有限观测下求解不确定性组合优化

Combinatorial Pure Exploration with Full-bandit Feedback and Beyond: Solving Combinatorial Optimization under Uncertainty with Limited Observation

本文回顾了有限反馈下组合纯探索（CPE）问题的最新方法，该问题旨在解决输入参数不确定时的组合优化，并讨论了半臂反馈受限场景下的技术进展。

Bottleneck combinatorial optimization problems with uncertain costs and the OWA criterion

In this paper a class of bottleneck combinatorial optimization problems with uncertain costs is discussed. The uncertainty is modeled by specifying a discrete scenario set containing a finite number of cost vectors, called scenarios. In order to choose a solution the Ordered Weighted Averaging aggregation operator (shortly OWA) is applied. The OWA operator generalizes traditional criteria in decision making under uncertainty such as the maximum, minimum, average, median, or Hurwicz criterion. New complexity and approximation results in this area are provided. These results are general and rema…

From Convex Optimization to Randomized Mechanisms: Toward Optimal Combinatorial Auctions

We design an expected polynomial-time, truthful-in-expectation, (1-1/e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are m matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions, matroid weighted-rank functions, and convex combinations thereof. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P != NP. Ours is the first truthful-in-expectation …

Adaptive Uncertainty Resolution in Bayesian Combinatorial Optimization Problems

In several applications such as databases, planning, and sensor networks, parameters such as selectivity, load, or sensed values are known only with some associated uncertainty. The performance of such a system (as captured by some objective function over the parameters) is significantly improved if some of these parameters can be probed or observed. In a resource constrained situation, deciding which parameters to observe in order to optimize system performance itself becomes an interesting and important optimization problem. This general problem is the focus of this paper. One of the most im…

On the Reflexivity of Point Sets

We introduce a new measure for planar point sets S that captures a combinatorial distance that S is from being a convex set: The reflexivity rho(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study considers also some closely related quantities, such as the convex cover number kappa_c(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex par…

Rational Convex Programs, Their Feasibility, and the Arrow-Debreu Nash Bargaining Game

Over the last decade, combinatorial algorithms have been obtained for exactly solving several nonlinear convex programs. We first provide a formal context to this activity by introducing the notion of {\em rational convex programs} -- this also enables us to identify a number of questions for further study. So far, such algorithms were obtained for total problems only. Our main contribution is developing the methodology for handling non-total problems, i.e., their associated convex programs may be infeasible for certain settings of the parameters. The specific problem we study pertains to a Na…