Essays on The Online Multiple Knapsack Problem & The Online Reservation Problem

Abstract

We study two problems in this thesis, viz. the online multiple knapsack problem (OMKP) and the online reservation problem (ORP). The OMKP has applications in revenue management and scheduling. There are multiple identical knapsacks. Items arrive one at a time, each having a value-density and a size. Upon arrival, an item must either be placed into a knapsack or turned away irrevocably, without any information about future items, except the largest possible values of item size and value-density. An accepted item yields a reward equal to its value. The decision maker seeks decision rules that have the best competitive-ratio (CR), which is the worst-case ratio of the optimal reward in the full-information case to the reward earned by the algorithm. We derive lower bounds on CR of any algorithm, analyze CRs of some known algorithms, and propose three new algorithms that have CRs significantly better than the known algorithms for specific ranges of parameter values. We also study the online revenue management problem where all items have sizes equal to the knapsack capacities. In the ORP, we model many problem features faced by resource owner in sharing-economy platforms, many of which operate as follows. Owners list availability of resources (such as apartments, cars or tutoring services), prices, and contract-length limits. Customers propose contract start times and lengths. Owners decide immediately (or within a short time window) whether to accept or decline each proposal, even if the contract is for a future date. Accepted proposals generate a revenue for the owner. Declined proposals are lost. At any decision epoch, the owner has no information about future requests. The owner seeks easy-to-implement algorithms that have the best CR. We derive lower bounds on CR of any algorithm, analyze CRs of intuitive algorithms that are fair in the sense that they always accept a proposal whenever a resource is available, and propose two new algorithms that have CRs significantly better than any fair algorithm for specific ranges of parameter values. We also study a special case of the ORP where all proposals arrive exactly at their start times.