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.

Rian Neogi, Kanstantsin Pashkovich, Chaitanya Swamy

In budget-feasible mechanism design, a buyer wishes to procure a set of items of maximum value from self-interested players. We have a valuation function $v:2^U \to \mathbb{R}_+$, where $U$ is the set of all items, where $v(S)$ specifies the value obtained from set $S$ of items. The entirety of current work on budget-feasible mechanisms has focused on the single-dimensional setting, wherein each player holds a single item $e$ and incurs a private cost $c_e$ for supplying item $e$. We introduce multidimensional budget feasible mechanism design: the universe $U$ is now partitioned into item-sets $\{G_i\}$ held by the different players, and each player $i$ incurs a private cost $c_i(S_i)$ for supplying the set $S_i\subseteq G_i$ of items. A budget-feasible mechanism is a mechanism that is truthful, and where the total payment made to the players is at most some given budget $B$. The goal is to devise a budget-feasible mechanism that procures a set of items of large value. We obtain the first approximation guarantees for multidimensional budget feasible mechanism design. Our contributions are threefold. First, we prove an impossibility result showing that the standard benchmark used in single-dimensional budget-feasible mechanism design, namely the algorithmic optimum is inadequate in that no budget-feasible mechanism can achieve good approximation relative to this. We identify that the chief underlying issue here is that there could be a monopolist which prevents a budget-feasible mechanism from obtaining good guarantees. Second, we devise an alternate benchmark, $OPT_{Bench}$, that allows for meaningful approximation guarantees, thereby yielding a metric for comparing mechanisms. Third, we devise budget-feasible mechanisms that achieve constant-factor approximation guarantees with respect to this benchmark for XOS valuations.

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.