Risk-Averse Matchings over Uncertain Graph Databases

A large number of applications such as querying sensor networks, and analyzing protein-protein interaction (PPI) networks, rely on mining uncertain graph and hypergraph databases. In this work we study the following problem: given an uncertain, weighted (hyper)graph, how can we efficiently find a (hyper)matching with high expected reward, and low risk? This problem naturally arises in the context of several important applications, such as online dating, kidney exchanges, and team formation. We introduce a novel formulation for finding matchings with maximum expected reward and bounded risk under a general model of uncertain weighted (hyper)graphs that we introduce in this work. Our model generalizes probabilistic models used in prior work, and captures both continuous and discrete probability distributions, thus allowing to handle privacy related applications that inject appropriately distributed noise to (hyper)edge weights. Given that our optimization problem is NP-hard, we turn our attention to designing efficient approximation algorithms. For the case of uncertain weighted graphs, we provide a $\frac{1}{3}$-approximation algorithm, and a $\frac{1}{5}$-approximation algorithm with near optimal run time. For the case of uncertain weighted hypergraphs, we provide a $\Omega(\frac{1}{k})$-approximation algorithm, where $k$ is the rank of the hypergraph (i.e., any hyperedge includes at most $k$ nodes), that runs in almost (modulo log factors) linear time. We complement our theoretical results by testing our approximation algorithms on a wide variety of synthetic experiments, where we observe in a controlled setting interesting findings on the trade-off between reward, and risk. We also provide an application of our formulation for providing recommendations of teams that are likely to collaborate, and have high impact.Comment: 25 page

On-line Non-stationary Inventory Control using Champion Competition

The commonly adopted assumption of stationary demands cannot actually reflect fluctuating demands and will weaken solution effectiveness in real practice. We consider an On-line Non-stationary Inventory Control Problem (ONICP), in which no specific assumption is imposed on demands and their probability distributions are allowed to vary over periods and correlate with each other. The nature of non-stationary demands disables the optimality of static (s,S) policies and the applicability of its corresponding algorithms. The ONICP becomes computationally intractable by using general Simulation-based Optimization (SO) methods, especially under an on-line decision-making environment with no luxury of time and computing resources to afford the huge computational burden. We develop a new SO method, termed "Champion Competition" (CC), which provides a different framework and bypasses the time-consuming sample average routine adopted in general SO methods. An alternate type of optimal solution, termed "Champion Solution", is pursued in the CC framework, which coincides the traditional optimality sense under certain conditions and serves as a near-optimal solution for general cases. The CC can reduce the complexity of general SO methods by orders of magnitude in solving a class of SO problems, including the ONICP. A polynomial algorithm, termed "Renewal Cycle Algorithm" (RCA), is further developed to fulfill an important procedure of the CC framework in solving this ONICP. Numerical examples are included to demonstrate the performance of the CC framework with the RCA embedded.Comment: I just identified a flaw in the paper. It may take me some time to fix it. I would like to withdraw the article and update it once I finished. Thank you for your kind suppor

Correlation Decay in Random Decision Networks

We consider a decision network on an undirected graph in which each node corresponds to a decision variable, and each node and edge of the graph is associated with a reward function whose value depends only on the variables of the corresponding nodes. The goal is to construct a decision vector which maximizes the total reward. This decision problem encompasses a variety of models, including maximum-likelihood inference in graphical models (Markov Random Fields), combinatorial optimization on graphs, economic team theory and statistical physics. The network is endowed with a probabilistic structure in which costs are sampled from a distribution. Our aim is to identify sufficient conditions to guarantee average-case polynomiality of the underlying optimization problem. We construct a new decentralized algorithm called Cavity Expansion and establish its theoretical performance for a variety of models. Specifically, for certain classes of models we prove that our algorithm is able to find near optimal solutions with high probability in a decentralized way. The success of the algorithm is based on the network exhibiting a correlation decay (long-range independence) property. Our results have the following surprising implications in the area of average case complexity of algorithms. Finding the largest independent (stable) set of a graph is a well known NP-hard optimization problem for which no polynomial time approximation scheme is possible even for graphs with largest connectivity equal to three, unless P=NP. We show that the closely related maximum weighted independent set problem for the same class of graphs admits a PTAS when the weights are i.i.d. with the exponential distribution. Namely, randomization of the reward function turns an NP-hard problem into a tractable one