Related papers: Automated Lower Bounds for Small Matrix Multiplica…
Submodular function minimization (SFM) and matroid intersection are fundamental discrete optimization problems with applications in many fields. It is well known that both of these can be solved making $\mathrm{poly}(N)$ queries to a…
In safety-critical applications that rely on the solution of an optimization problem, the certification of the optimization algorithm is of vital importance. Certification and suboptimality results are available for a wide range of…
The {\it matrix-chain multiplication} problem is a classic problem that is widely taught to illustrate dynamic programming. The textbook solution runs in $\theta(n^3)$ time. However, there is a complex $O(n \log n)$-time method \cite{HU82},…
Computational problem certificates are additional data structures for each output, which can be used by a-possibly randomized-verification algorithm that proves the correctness of each output. In this paper, we give an algorithm that…
Border complexity captures functions that can be approximated by low-complexity ones. Debordering is the task of proving an upper bound on some non-border complexity measure in terms of a border complexity measure, thus getting rid of…
Assembling stiffness matrices represents a significant cost in many finite element computations. We address the question of optimizing the evaluation of these matrices. By finding redundant computations, we are able to significantly reduce…
Random constraint satisfaction problems (CSPs) such as random $3$-SAT are conjectured to be computationally intractable. The average case hardness of random $3$-SAT and other CSPs has broad and far-reaching implications on problems in…
We analyze rank decompositions of the $3\times 3$ matrix multiplication tensor over $\mathbb{Z}/2\mathbb{Z}$. We restrict our attention to decompositions of rank $\le 21$, as only those decompositions will yield an asymptotically faster…
We develop primal-dual coordinate methods for solving bilinear saddle-point problems of the form $\min_{x \in \mathcal{X}} \max_{y\in\mathcal{Y}} y^\top A x$ which contain linear programming, classification, and regression as special cases.…
The problem of planar registration consists in finding the transformation that better aligns two point sets. In our setting, the search domain is the set of planar rigid transformations and the objective function is the sum of the distances…
In this paper we present complexity certification results for a distributed Augmented Lagrangian (AL) algorithm used to solve convex optimization problems involving globally coupled linear constraints. Our method relies on the Accelerated…
We present a formally verified global optimization framework. Given a semialgebraic or transcendental function $f$ and a compact semialgebraic domain $K$, we use the nonlinear maxplus template approximation algorithm to provide a certified…
We consider the problem of multiplying sparse matrices (over a semiring) where the number of non-zero entries is larger than main memory. In the classical paper of Hong and Kung (STOC '81) it was shown that to compute a product of dense $U…
The goal of this work is to prove an analogue of a recent result of Harper on almost sure lower bounds of random multiplicative functions, in a setting that can be thought of as a simplified function field analogue. It answers a question…
Arithmetic complexity is considered simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic…
This paper studies lower bounds for fundamental optimization problems in the CONGEST model. We show that solving problems exactly in this model can be a hard task, by providing $\tilde{\Omega}(n^2)$ lower bounds for cornerstone problems,…
Additive Fourier Transform is sdudied. A fast multiplication algorithm for polynomials over the binary field is given. The bit complexity of the algorithm is $O(n(log n)(\log\log n)^2)$.
We implement an Augmented Lagrangian method to minimize a constrained least-squares cost function designed to find polyadic decompositions of the matrix multiplication tensor. We use this method to obtain new discrete decompositions and…
A common approach in the design of MapReduce algorithms is to minimize the number of rounds. Indeed, there are many examples in the literature of monolithic MapReduce algorithms, which are algorithms requiring just one or two rounds.…
We study the robustness verification problem for tree-based models, including decision trees, random forests (RFs) and gradient boosted decision trees (GBDTs). Formal robustness verification of decision tree ensembles involves finding the…