Contracts for Density and Packing Functions

Abstract

We study contracts for combinatorial problems in multi-agent settings. In this problem, a principal designs a contract with several agents, whose actions the principal is unable to observe. The principal is able to see only the outcome of the agents' collective actions. All agents that decided to exert effort incur costs, and so naturally all agents expect a fraction of the principal's reward as a compensation. The principal needs to decide what fraction of their reward to give to each agent so that the principal's expected utility is maximized. One of our focuses is on the case when the principal's reward function is supermodular and is based on some graph. Recently, Deo-Campo Vuong et al. showed that for this problem it is impossible to provide any finite multiplicative approximation or additive FPTAS unless P=NP. On a positive note, Deo-Campo Vuong et al. provided an additive PTAS for the case when all agents have the same cost. Deo-Campo Vuong et al. asked whether an additive PTAS can be obtained for the general case, i.e for the case when agents potentially have different costs. In this thesis, we answer this open question in positive. Additionally, we provide multiplicative approximation algorithms for functions that are based on hypergraphs and encode packing constraints. This family of functions provides a generalization for XOS functions.