Chaitanya Swamy

Stephen Arndt, Benjamin Moseley, Kirk Pruhs et al.

We study algorithmic matroid intersection coloring. Given $k$ matroids on a common ground set $U$ of $n$ elements, the goal is to partition $U$ into the fewest number of color classes, where each color class is independent in all matroids. It is known that $2χ_{\max}$ colors suffice to color the intersection of two matroids, $(2k-1)χ_{\max}$ colors suffice for general $k$, where $χ_{\max}$ is the maximum chromatic number of the individual matroids. However, these results are non-constructive, leveraging techniques such as topological Hall's theorem and Sperner's Lemma. We provide the first polynomial-time algorithms to color two or more general matroids where the approximation ratio depends only on $k$ and, in particular, is independent of $n$. For two matroids, we constructively match the $2χ_{\max}$ existential bound, yielding a 2-approximation for the Matroid Intersection Coloring problem. For $k$ matroids we achieve a $(k^2-k)χ_{\max}$ coloring, which is the first $O(1)$-approximation for constant $k$. Our approach introduces a novel matroidal structure we call a \emph{flexible decomposition}. We use this to formally reduce general matroid intersection coloring to graph coloring while avoiding the limitations of partition reduction techniques, and without relying on non-constructive topological machinery. Furthermore, we give a \emph{fully polynomial randomized approximation scheme} (FPRAS) for coloring the intersection of two matroids when $χ_{\max}$ is large. This yields the first polynomial-time constructive algorithm for an asymptotic variant of Rota's Basis Conjecture. This constructivizes Montgomery and Sauermann's recent asymptotic breakthrough and generalizes it to arbitrary matroids.

Jian Li, Yuval Rabani, Leonard J. Schulman et al.

We study the problem of learning from unlabeled samples very general statistical mixture models on large finite sets. Specifically, the model to be learned, $\vartheta$, is a probability distribution over probability distributions $p$, where each such $p$ is a probability distribution over $[n] = \{1,2,\dots,n\}$. When we sample from $\vartheta$, we do not observe $p$ directly, but only indirectly and in very noisy fashion, by sampling from $[n]$ repeatedly, independently $K$ times from the distribution $p$. The problem is to infer $\vartheta$ to high accuracy in transportation (earthmover) distance. We give the first efficient algorithms for learning this mixture model without making any restricting assumptions on the structure of the distribution $\vartheta$. We bound the quality of the solution as a function of the size of the samples $K$ and the number of samples used. Our model and results have applications to a variety of unsupervised learning scenarios, including learning topic models and collaborative filtering.