Essays on Discrete Optimization

Xingyu Zhang

This thesis studies two discrete optimization problems: ordering problems in optimal stopping theory and popular matchings. The main goal of this thesis is to find the boundary between NP-hardness and tractability for these problems, and whenever possible, designs polynomial-time algorithms for the easy cases and approximation schemes or prophet inequalities for the hard cases. In the first part of the thesis, we study ordering problems in optimal stopping theory. In the optimal stopping problem, a player is presented with 𝓃 random variables 𝑋₁, . . . , 𝑋n, whose distributions are known to the player, but not their realizations. After observing the realization of 𝑋ᵢ, the player can choose to stop and earn reward 𝑋ᵢ, or reject 𝑋ᵢ and probe the next variable 𝑋ᵢ₊₁. If 𝑋ᵢ is rejected, it cannot be accepted in the future. The goal of the player is to maximize the expected reward at...