The Maximum Likelihood Threshold of a Graph

The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph parameter is connected to the theory of combinatorial rigidity. In particular, if the edge set of a graph $G$ is an independent set in the $n-1$-dimensional generic rigidity matroid, then the maximum likelihood threshold of $G$ is less than or equal to $n$. This connection allows us to prove many results about the maximum likelihood threshold.Comment: Added Section 6 and Section

Weighted Linear Matroid Parity

The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle calls. Nevertheless, Lovasz (1978) showed that this problem admits a min-max formula and a polynomial algorithm for linearly represented matroids. Since then efficient algorithms have been developed for the linear matroid parity problem. This talk presents a recently developed polynomial-time algorithm for the weighted linear matroid parity problem. The algorithm builds on a polynomial matrix formulation using Pfaffian and adopts a primal-dual approach based on the augmenting path algorithm of Gabow and Stallmann (1986) for the unweighted problem

Bi-Criteria and Approximation Algorithms for Restricted Matchings

In this work we study approximation algorithms for the \textit{Bounded Color Matching} problem (a.k.a. Restricted Matching problem) which is defined as follows: given a graph in which each edge $e$ has a color $c_e$ and a profit $p_e \in \mathbb{Q}^+$, we want to compute a maximum (cardinality or profit) matching in which no more than $w_j \in \mathbb{Z}^+$ edges of color $c_j$ are present. This kind of problems, beside the theoretical interest on its own right, emerges in multi-fiber optical networking systems, where we interpret each unique wavelength that can travel through the fiber as a color class and we would like to establish communication between pairs of systems. We study approximation and bi-criteria algorithms for this problem which are based on linear programming techniques and, in particular, on polyhedral characterizations of the natural linear formulation of the problem. In our setting, we allow violations of the bounds $w_j$ and we model our problem as a bi-criteria problem: we have two objectives to optimize namely (a) to maximize the profit (maximum matching) while (b) minimizing the violation of the color bounds. We prove how we can "beat" the integrality gap of the natural linear programming formulation of the problem by allowing only a slight violation of the color bounds. In particular, our main result is \textit{constant} approximation bounds for both criteria of the corresponding bi-criteria optimization problem

Submodular Maximization Meets Streaming: Matchings, Matroids, and More

We study the problem of finding a maximum matching in a graph given by an input stream listing its edges in some arbitrary order, where the quantity to be maximized is given by a monotone submodular function on subsets of edges. This problem, which we call maximum submodular-function matching (MSM), is a natural generalization of maximum weight matching (MWM), which is in turn a generalization of maximum cardinality matching (MCM). We give two incomparable algorithms for this problem with space usage falling in the semi-streaming range---they store only $O(n)$ edges, using $O(n\log n)$ working memory---that achieve approximation ratios of $7.75$ in a single pass and $(3+\epsilon)$ in $O(\epsilon^{-3})$ passes respectively. The operations of these algorithms mimic those of Zelke's and McGregor's respective algorithms for MWM; the novelty lies in the analysis for the MSM setting. In fact we identify a general framework for MWM algorithms that allows this kind of adaptation to the broader setting of MSM. In the sequel, we give generalizations of these results where the maximization is over "independent sets" in a very general sense. This generalization captures hypermatchings in hypergraphs as well as independence in the intersection of multiple matroids.Comment: 18 page