How Close Does It Get?
From Near-Optimal Network Algorithms to Suboptimal Equilibrium Outcomes
Ruben Brokkelkamp
Abstract:
The five chapters in this thesis are connected by the following question: How Close Does It Get?
In Chapter 3, we study two pricing problems in networks. We are given a directed graph with edge costs, a set of commodities, and a designated node u. Each commodity has a flow demand that needs to be transported from a source node to a destination node, and it uses shortest paths to do so. If there are multiple shortest paths, the flow splits uniformly. The node u can change the cost of at most a given number of its outgoing edges, which affects the shortest paths in the network and, indirectly, the flows. The node u is interested in either maximizing the flow that goes through it or maximizing the revenue it earns on the flow going through its outgoing edges. The revenue on an outgoing edge of u is equal to the amount of flow going through it times the cost of that edge. The revenue of u is the sum of the revenues of its outgoing edges. We prove that the problem is not only NP-hard but also highly inapproximable in general for both objectives. However, we develop efficient optimal and approximation algorithms for special cases. How close does it get? We show that the guarantees of the approximation algorithms are best possible given our inapproximability results.
In Chapter 4, we look into the notion of the Most Probable Shortest Path (MPSP) in an uncertain network. We model an uncertain network by assigning a probability to each edge with which that edge is available. We show that computing the probability that a path is the shortest path is #P-hard. To compute the MPSP we develop a sampling-based Monte Carlo-type algorithm to quickly find the MPSP . Based on this notion of shortest path, we also define a new betweenness centrality measure and give a sampling-based algorithm for computing it. How close does it get? We show that with high probability, our algorithm returns the MPSP and we conduct extensive experiments to assess its performance in practice.
In Chapter 5, we consider the problem of related machine scheduling in which a set of jobs, each with a processing time, must be scheduled on a set of machines, each with a speed. The time it takes to process a job on a machine is the processing time of the job divided by the speed of the machine. The greedy algorithm that schedules the jobs from shortest to longest on the machine on which it completes earliest has an approximation guarantee equal to the price of anarchy of the game- theoretic version of this problem. In the game, each job is controlled by a player that is interested in minimizing the completion time of their own job. Finding a tight bound on the price of anarchy is a difficult problem and requires new techniques. How close does it get? We outline a technique based on a primal-dual method that is able to recover the best-known bound and has the potential to yield better bounds. Further, we provide better lower-bound instances for a fixed number of jobs and machines.
In Chapter 6, we study a first-price multi-unit auction in which a corrupt auctioneer approaches winning bidders with the offer to lower their bid to the highest losing bid in exchange for a bribe that is equal to a \gamma-fraction of the gains. We show that this auction is equivalent to a \gammy-hybrid auction in which the payments are a convex combination of first-price and second-price payments. We also consider a more general \gamma-approximate first-price auction where the payments recover at least a \gamma-fraction of the first-price payments. We study the social welfare loss that is caused by the corrupt auctioneer as a function of \gamma. How close does it get? We derive tight coarse correlated price of anarchy bounds if players are allowed to overbid. Then, we make a no-overbidding assumption and prove more (almost) tight bounds on the price of anarchy for various equilibrium concepts and specific versions of the auction.
In Chapter 7, we study the problem of designing truthful mechanisms for players that are (partially) altruistic. We model this by extending the standard utility model with other-regarding preferences. Unfortunately, VCG cannot be applied here anymore. We derive characterizations of truthful mechanisms in the new model, exploiting the specific form of the other-regarding preferences. Next, we give a recipe for truthful mechanisms and use it to define two specific models of altruism. Because of the altruistic dispositions, smaller payments are needed to incentivize participants to reveal their true preferences. We show that by using one of our altruistic models, the public project problem can be resolved for moderate altruism levels. Further, we look into redistributing the surplus among the participants in the presence of altruistic players. How close does it get? We show that mechanisms that do not take altruism into account can be transformed into ones that do and, in the process, make sure that less money leaves the group of participants.