Settling the Sample Complexity of Single-parameter Revenue Maximization

This paper settles the sample complexity of single-parameter revenue maximization by showing matching upper and lower bounds, up to a poly-logarithmic factor, for all families of value distributions that have been considered in the literature. The upper bounds are unified under a novel framework, which builds on the strong revenue monotonicity by Devanur, Huang, and Psomas (STOC 2016), and an information theoretic argument. This is fundamentally different from the previous approaches that rely on either constructing an $\epsilon$-net of the mechanism space, explicitly or implicitly via statistical learning theory, or learning an approximately accurate version of the virtual values. To our knowledge, it is the first time information theoretical arguments are used to show sample complexity upper bounds, instead of lower bounds. Our lower bounds are also unified under a meta construction of hard instances.Comment: 49 pages, Accepted by STOC1

On the complexity of optimal homotopies

In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves $\gamma_1$ and $\gamma_2$ on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between $\gamma_1$ and $\gamma_2$ where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems. We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fr\'echet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting

Paradigm and paradox in topology control of power grids

Corrective Transmission Switching can be used by the grid operator to relieve line overloading and voltage violations, improve system reliability, and reduce system losses. Power grid optimization by means of line switching is typically formulated as a mixed integer programming problem (MIP). Such problems are known to be computationally intractable, and accordingly, a number of heuristic approaches to grid topology reconfiguration have been proposed in the power systems literature. By means of some low order examples (3-bus systems), it is shown that within a reasonably large class of “greedy” heuristics, none can be found that perform better than the others across all grid topologies. Despite this cautionary tale, statistical evidence based on a large number of simulations using IEEE 118-bus systems indicates that among three heuristics, a globally greedy heuristic is the most computationally intensive, but has the best chance of reducing generation costs while enforcing N-1 connectivity. It is argued that, among all iterative methods, the locally optimal switches at each stage have a better chance in not only approximating a global optimal solution but also greatly limiting the number of lines that are switched.First author draf

Paradigm and Paradox in Topology Control of Power Grids

Corrective Transmission Switching can be used by the grid operator to relieve line overloading and voltage violations, improve system reliability, and reduce system losses. Power grid optimization by means of line switching is typically formulated as a mixed integer programming problem (MIP). Such problems are known to be computationally intractable, and accordingly, a number of heuristic approaches to grid topology reconfiguration have been proposed in the power systems literature. By means of some low order examples (3-bus systems), it is shown that within a reasonably large class of greedy heuristics, none can be found that perform better than the others across all grid topologies. Despite this cautionary tale, statistical evidence based on a large number of simulations using using IEEE 118- bus systems indicates that among three heuristics, a globally greedy heuristic is the most computationally intensive, but has the best chance of reducing generation costs while enforcing N-1 connectivity. It is argued that, among all iterative methods, the locally optimal switches at each stage have a better chance in not only approximating a global optimal solution but also greatly limiting the number of lines that are switched