Evdokia Nikolova

Stochastic shortest paths via Quasi-convex maximization

We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact nΘ (log n) algorithm for the case of normally distributed edge lengths, which is based on quasi-convex maximization. We then prove average and smoothed polynomial bounds for this algorithm, which also translate to average and smoothed bounds for the parametric shortest path problem, and extend to a more general non-convex optimization setting. We also consider a number other edge length distributions, giving a range of exact and approximation schemes.

First-price path auctions

We study first-price auction mechanisms for auctioning flow between given nodes in a graph. A first-price auction is any auction in which links on winning paths are paid their bid amount; the designer has flexibility in specifying remaining details. We assume edges are independent agents with fixed capacities and costs, and their objective is to maximize their profit. We characterize all strong ε-Nash equilibria of a first-price auction, and show that the total payment is never significantly more than, and often less than, the well known dominant strategy Vickrey-Clark-Groves mechanism. We then present a randomized version of the first-price auction for which the equilibrium condition can be relaxed to ε-Nash equilibrium. We next consider a model in which the amount of demand is uncertain, but its probability distribution is known. For this model, we show that a simple ex ante first-price auction may not have any ε-Nash equilibria. We then present a modified mechanism with 2-parameter bids which does have an ε-Nash equilibrium. For a randomized version of this 2-parameter mechanism we characterize the set of all eNEs and prove a bound on the total payment in any eNE.

Approximation algorithms for reliable stochastic combinatorial optimization

We consider optimization problems that can be formulated as minimizing the cost of a feasible solution wT x over an arbitrary combinatorial feasible set F ⊂ {0, 1}n. For these problems we describe a broad class of corresponding stochastic problems where the cost vector W has independent random components, unknown at the time of solution. A natural and important objective that incorporates risk in this stochastic setting is to look for a feasible solution whose stochastic cost has a small tail or a small convex combination of mean and standard deviation. Our models can be equivalently reformulated as nonconvex programs for which no efficient algorithms are known. In this paper, we make progress on these hard problems. Our results are several efficient general-purpose approximation schemes. They use as a black-box (exact or approximate) the solution to the underlying deterministic problem and thus immediately apply to arbitrary combinatorial problems. For example, from an available δ-approximation algorithm to the linear problem, we construct a δ(1 + e)-approximation algorithm for the stochastic problem, which invokes the linear algorithm only a logarithmic number of times in the problem input (and polynomial in 1/e), for any desired accuracy level e > 0. The algorithms are based on a geometric analysis of the curvature and approximability of the nonlinear level sets of the objective functions.

A Truthful Mechanism for Offline Ad Slot Scheduling

We consider the Offline Ad Slot Schedulingproblem, where advertisers must be scheduled to sponsored searchslots during a given period of time. Advertisers specify a budget constraint, as well as a maximum cost per click, and may not be assigned to more than one slot for a particular search. We give a truthful mechanism under the utility model where bidders try to maximize their clicks, subject to their personal constraints. In addition, we show that the revenue-maximizing mechanism is not truthful, but has a Nash equilibrium whose outcome is identical to our mechanism. Our mechanism employs a descending-price auction that maintains a solution to a certain machine scheduling problem whose job lengths depend on the price, and hence are variable over the auction.

Practical Route Planning Under Delay Uncertainty: Stochastic Shortest Path Queries.

We describe an algorithm for stochastic path planning and applications to route planning in the presence of traffic delays. We improve on the prior state of the art by designing, analyzing, implementing, and evaluating data structures that answer approximate stochastic shortest-path queries. For example, our data structures can be used to efficiently compute paths that maximize the probability of arriving at a destination before a given time deadline. Our main theoretical result is an algorithm that, given a directed planar network with edge lengths characterized by expected travel time and variance, pre-computes a data structure in quasi-linear time such that approximate stochastic shortestpath queries can be answered in poly-logarithmic time (actual worst-case bounds depend on the probabilistic model). Our main experimental results are two-fold: (i) we provide methods to extract travel-time distributions from a large set of heterogenous GPS traces and we build a stochastic model of an entire city, and (ii) we adapt our algorithms to work for realworld road networks, we provide an efficient implementation, and we evaluate the performance of our method for the model of the aforementioned city.

Risk sensitivity of price of anarchy under uncertainty

In algorithmic game theory, the price of anarchy framework studies efficiency loss in decentralized environments. In optimization and decision theory, the price of robustness framework explores the tradeoffs between optimality and robustness in the case of single agent decision making under uncertainty. We establish a connection between the two that provides a novel analytic framework for proving tight performance guarantees for distributed systems in uncertain environments. We present applications of this framework to novel variants of atomic congestion games with uncertain costs, for which we provide tight performance bounds under a wide range of risk attitudes. Our results establish that the individuals attitude towards uncertainty has a critical effect on system performance and should therefore be a subject of close and systematic investigation.

High-Performance heuristics for optimization in stochastic traffic engineering problems

We consider a stochastic routing model in which the goal is to find the optimal route that incorporates a measure of risk The problem arises in traffic engineering, transportation and even more abstract settings such as task planning (where the time to execute tasks is uncertain), etc The stochasticity is specified in terms of arbitrary edge length distributions with given mean and variance values in a graph The objective function is a positive linear combination of the mean and standard deviation of the route Both the nonconvex objective and exponentially sized feasible set of available routes present a challenging optimization problem for which no efficient algorithms are known In this paper we evaluate the practical performance of algorithms and heuristic approaches which show very promising results in terms of both running time and solution accuracy.

Incentive-Compatible Interdomain Routing with Linear Utilities

We revisit the problem of incentive-compatible interdomain routing, examining the quite realistic special case in which the utilities of autonomous systems (ASes) are linear functions of the traffic in the incident links and the traffic leaving each AS. We show that incentive-compatibility toward maximizing total welfare is achievable efficiently, and in the uncapacitated case, by an algorithm that can be easily implemented by the border gateway protocol (BGP), the standard protocol for interdomain routing.

On the Expected VCG Overpayment in Large Networks

Motivated by the increasing need to price networks, we study the prices resulting from of a variant of the VCG mechanism, specifically defined for networks by J. Feigenbaum, et al (2002). This VCG mechanism is the unique efficient and strategyproof mechanism, however it is not budget-balanced and in fact it is known to result in arbitrarily bad overpayments for some graphs by A. Archer and E. Tardos (2002), In contrast, we study more common types of graphs and show that the VCG overpayment is not too high, so it is still an attractive pricing candidate. We prove that the average overpayment in Erdos-Renyi random graphs with unit costs is p/(2 - p) for any n, when the average degree is higher than a given threshold. Our simulations show that the overpayment is greater than p/(2 - p) below this threshold, hence, together with the constant upper bound from Mihail et al. (2003), the overpayment is constant regardless of graph size. We then present simulation results which show that power-law graphs with unit costs has overpayments that decrease with graph size and finally, power-law graphs with uniformly random costs has a small constant overpayment

Brief announcement: on the expected overpayment of VCG mechanisms in large networks

The VCG mechanism for buying a path from s to t in a network with selfish edges, selects the lowest-cost path (LCP) and pays each edge e on the path the edge’s cost plus a bonus equal to the increase in cost of the lowest-cost path from s to t if edge e were omitted from the graph G, that is payment(e, s, t) = cost(e)+LCP (G− e, s, t)− LCP (G, s, t). These payments are unacceptably high in some graphs containing two node-disjoint paths, and there are few positive results which prove small upper bounds on the expected payments in large and more realistic networks [4, 1]. In this paper, we give an improvement on the VCG overpayment bound of Mihail, Papadimitriou and Saberi [4] for random graphs. As a corollary, we show that the overpayment is not necessarily correlated with the competitiveness of a graph (measured, for example, by the number of node-disjoint paths between a source-destination pair [1]), and it is not necessarily correlated with the size of the graph, contrary to the conjecture in Mihail et al. [4] that the overpayment in random graphs Gn,p is Θ(1/np). Our results show that the overpayment is low when there are two almost equally short paths between the source and destination; the presence of more edges or more competition does not necessarily reduce it and may even hurt it.