Some new directions in infinite-combinatorial topology

We give a light introduction to selection principles in topology, a young subfield of infinite-combinatorial topology. Emphasis is put on the modern approach to the problems it deals with. Recent results are described, and open problems are stated. Some results which do not appear elsewhere are also included, with proofs.Comment: Small update

Selection principles in mathematics: A milestone of open problems

We survey some of the major open problems involving selection principles, diagonalizations, and covering properties in topology and infinite combinatorics. Background details, definitions and motivations are also provided.Comment: Small update

The Classical Complexity of Boson Sampling

We study the classical complexity of the exact Boson Sampling problem where the objective is to produce provably correct random samples from a particular quantum mechanical distribution. The computational framework was proposed by Aaronson and Arkhipov in 2011 as an attainable demonstration of `quantum supremacy', that is a practical quantum computing experiment able to produce output at a speed beyond the reach of classical (that is non-quantum) computer hardware. Since its introduction Boson Sampling has been the subject of intense international research in the world of quantum computing. On the face of it, the problem is challenging for classical computation. Aaronson and Arkhipov show that exact Boson Sampling is not efficiently solvable by a classical computer unless $P^{\#P} = BPP^{NP}$ and the polynomial hierarchy collapses to the third level. The fastest known exact classical algorithm for the standard Boson Sampling problem takes $O({m + n -1 \choose n} n 2^n )$ time to produce samples for a system with input size $n$ and $m$ output modes, making it infeasible for anything but the smallest values of $n$ and $m$. We give an algorithm that is much faster, running in $O(n 2^n + \operatorname{poly}(m,n))$ time and $O(m)$ additional space. The algorithm is simple to implement and has low constant factor overheads. As a consequence our classical algorithm is able to solve the exact Boson Sampling problem for system sizes far beyond current photonic quantum computing experimentation, thereby significantly reducing the likelihood of achieving near-term quantum supremacy in the context of Boson Sampling.Comment: 15 pages. To appear in SODA '1

Mechanism Design for Team Formation

Team formation is a core problem in AI. Remarkably, little prior work has addressed the problem of mechanism design for team formation, accounting for the need to elicit agents' preferences over potential teammates. Coalition formation in the related hedonic games has received much attention, but only from the perspective of coalition stability, with little emphasis on the mechanism design objectives of true preference elicitation, social welfare, and equity. We present the first formal mechanism design framework for team formation, building on recent combinatorial matching market design literature. We exhibit four mechanisms for this problem, two novel, two simple extensions of known mechanisms from other domains. Two of these (one new, one known) have desirable theoretical properties. However, we use extensive experiments to show our second novel mechanism, despite having no theoretical guarantees, empirically achieves good incentive compatibility, welfare, and fairness.Comment: 12 page