Primal-Dual Algorithms for Deterministic Inventory Problems

Primal-Dual Algorithms for Deterministic Inventory Problem

Designing cost-sharing methods for Bayesian games

We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria, have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE).We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players

A simple randomised algorithm for convex optimisation: Application to two-stage stochastic programming

We consider maximising a concave function over a convex set by a simple randomised algorithm. The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set. Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability. As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems. These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models. Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models. Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron

Some Recent Developments in the Design and Analysis of Approximation Algorithms

Some Recent Developments in the Design and Analysis of Approximation Algorithm

Computing Near-Optimal Solutions to Combinatorial Optimization Problems

Computing Near-Optimal Solutions to Combinatorial Optimization Problem

Scheduling unrelated parallel machines with costs

Scheduling unrelated parallel machines with cost

Computational complexity

Computational complexity . . . insightintothequestionwhycertaincomputationalproblems appeartobemoredicultthan others.Computationhasaddedadimensiontothestudyof combinatorics.Thetheoremthat, 4.InsideP exists?Althoughsuchanalgorithmisknownforthematchingproblem, thisisnotthecasefor manycombinatorialproblems.Indeed,thegreatestchallenge confrontingcomplexitytheoryisto providetechniquestoprovethatnoecientalgorithmexistsforagivenproblem. path,isnotthecompleteanswer;isitpossibletoecientlyconstructalargermatchingifone referredtothetextbooksbyGarey&Johnson(1979)andHopcroft&Ullman(1979)aswellasto thesequestions,andwhilesubstantialprogresshasbeenmadetowardsdistinguishingthediculty ofcomputationalproblems,mostofthebasicissuesremainunresolved.Inthischapter,wewill describethefundamentalsofthistheoryandgiveabriefsurveyoftheresultsthathavebeen obtainedinitsrstquartercentury.Foramoredetailedandcompleteexposition,thereaderis Computationalcomplexitytheoryprovidesthemathematicalframeworkinwhichtodiscuss themorerecentHandbookofTheoreticalComputerScienceeditedbyvanLeeuwen(1990). 1ComplexityofComputationalProblems computationalproblem,settingupamathematicalmodelofcomputation,andthenformalizing complexityofcomputationalproblems.Thisinvolvesdescribingwhatispreciselymeantbya thenotionofthecomputationalresourcesrequiredforaproblemwithrespecttothatmodel. Unfortunately,thereisnoonestandardizedspecicationinwhichtodiscussthesequestions.For thistheorytoproducemeaningfulresults,itisessentialthatthedenitionsberobustenoughsotha

Fast approximation algorithms for fractional packing and covering problems

This paper published in "Computational Optimization and Applications" 2 (1993) 301-31