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The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we…
We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to…
Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term polynomial method to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of…
There is a growing interest in harnessing the potential of the Rydberg-atom system to address complex combinatorial optimization challenges. Here we present an experimental demonstration of how the quadratic unconstrained binary…
$(1^a, 2^b)$-coloring is the problem of partitioning the vertex set of a graph into $a$ independent sets and $b$ 2-independent sets. This problem was recently introduced by Choi and Liu. We study the computational complexity and extremal…
We consider the following problem: for a given graph $G$ and two integers $k$ and $d$, can we apply a fixed graph operation at most $k$ times in order to reduce a given graph parameter $\pi$ by at least $d$? We show that this problem is…
We give a quasipolynomial time algorithm for the graph matching problem (also known as noisy or robust graph isomorphism) on correlated random graphs. Specifically, for every $\gamma>0$, we give a $n^{O(\log n)}$ time algorithm that given a…
Identifying the sets of operations that can be executed simultaneously is an important problem appearing in many parallel applications. By modeling the operations and their interactions as a graph, one can identify the independent…
We present a new algorithm for solving the reduction problem in the context of holonomic integrals, which in turn provides an approach to integration with parameters. Our method extends the Griffiths--Dwork reduction technique to holonomic…
Combinatorial optimization problems are ubiquitous in various disciplines and applications. Many heuristic algorithms have been devoted to solve these types of problems. In order to increase the efficiency for finding the optimal solutions,…
We consider sequential and parallel decomposition methods for a dual problem of a general total variation minimization problem with applications in several image processing tasks, like image inpainting, estimation of optical flow and…
We propose a novel variational method for solving the sub-graph isomorphism problem on a gate-based quantum computer. The method relies (1) on a new representation of the adjacency matrices of the underlying graphs, which requires a number…
Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization…
In this paper, we consider a bilevel polynomial optimization problem where the objective and the constraint functions of both the upper and the lower level problems are polynomials. We present methods for finding its global minimizers and…
Probabilistic graphical models compactly represent joint distributions by decomposing them into factors over subsets of random variables. In Bayesian networks, the factors are conditional probability distributions. For many problems, common…
In this paper, we develop a new deflation technique for refining or verifying the isolated singular zeros of polynomial systems. Starting from a polynomial system with an isolated singular zero, by computing the derivatives of the input…
Sample complexity bounds are a common performance metric in the Reinforcement Learning literature. In the discounted cost, infinite horizon setting, all of the known bounds have a factor that is a polynomial in $1/(1-\gamma)$, where $\gamma…
The graph partitioning problem is a well-known NP-hard problem. In this paper, we formulate a 0-1 quadratic integer programming model for the graph partitioning problem with vertex weight constraints and fixed vertex constraints, and…
We give an especially simple proof of a theorem in graph theory that forms the key part of the solution to a problem in commutative algebra, on how to characterize the integral closure of a polynomial ring generated by quadratic monomials.
The 3-domatic number problem asks whether a given graph can be partitioned intothree dominating sets. We prove that this problem can be solved by a deterministic algorithm in time 2.695^n (up to polynomial factors). This result improves the…