Related papers: Approximability of the Four-Vertex Model
We study the computational complexity of the map redistricting problem (gerrymandering). Mathematically, the electoral district designer (gerrymanderer) attempts to partition a weighted graph into $k$ connected components (districts) such…
We show the existence of a deep neural network capable of approximating a wide class of high-dimensional approximations. The construction of the proposed neural network is based on a quasi-optimal polynomial approximation. We show that this…
Functions of one or more variables are usually approximated with a basis: a complete, linearly-independent system of functions that spans a suitable function space. The topic of this paper is the numerical approximation of functions using…
We study the problem of computing approximate market equilibria in Fisher markets with separable piecewise-linear concave (SPLC) utility functions. In this setting, the problem was only known to be PPAD-complete for inverse-polynomial…
Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G+F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an…
We address the classical knapsack problem and a variant in which an upper bound is imposed on the number of items that can be selected. We show that appropriate combinations of rounding techniques yield novel and powerful ways of rounding.…
Considering the worst-case scenario, junction tree algorithm remains the most general solution for exact MAP inference with polynomial run-time guarantees. Unfortunately, its main tractability assumption requires the treewidth of a…
Standard sparse pseudo-input approximations to the Gaussian process (GP) cannot handle complex functions well. Sparse spectrum alternatives attempt to answer this but are known to over-fit. We suggest the use of variational inference for…
The Consensus Clustering problem has been introduced as an effective way to analyze the results of different microarray experiments. The problem consists of looking for a partition that best summarizes a set of input partitions (each…
Chance constrained program where one seeks to minimize an objective over decisions which satisfy randomly disturbed constraints with a given probability is computationally intractable. This paper proposes an approximate approach to address…
We propose a novel stochastic algorithm that randomly samples entire rows and columns of the matrix as a way to approximate an arbitrary matrix function using the power series expansion. This contrasts with existing Monte Carlo methods,…
$\newcommand{\eps}{\varepsilon}$ We observe that a $(1-\eps)$-approximation algorithm to Independent Set, that works for any induced subgraph of the input graph, can be used, via a polynomial time reduction, to provide a…
In many safety-critical settings, probabilistic ML systems have to make predictions subject to algebraic constraints, e.g., predicting the most likely trajectory that does not cross obstacles. These real-world constraints are rarely convex,…
We give a polynomial time approximation scheme (PTAS) for computing the supremum of a Gaussian process. That is, given a finite set of vectors $V\subseteq\mathbb{R}^d$, we compute a $(1+\varepsilon)$-factor approximation to $\mathop…
Operator splitting schemes are a class of powerful algorithms that solve complicated monotone inclusion and convex optimization problems that are built from many simpler pieces. They give rise to algorithms in which all simple pieces of the…
We give fully polynomial-time approximation schemes (FPTAS) for the partition function of ferromagnetic 2-spin systems in certain parameter regimes. The threshold we obtain is almost tight up to an integrality gap. Our technique is based on…
Randomized algorithms for low-rank approximation of quaternion matrices have gained increasing attention in recent years. However, existing methods overlook pass efficiency, the ability to limit the number of passes over the input…
Efficiently finding the maximum a posteriori (MAP) configuration of a graphical model is an important problem which is often implemented using message passing algorithms. The optimality of such algorithms is only well established for…
In the F-minor-free deletion problem we want to find a minimum vertex set in a given graph that intersects all minor models of graphs from the family F. The Vertex planarization problem is a special case of F-minor-free deletion for the…
We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem…