Related papers: Optimal Homologous Cycles, Total Unimodularity, an…

Maximizing submodular functions under cardinality constraints lies at the core of numerous data mining and machine learning applications, including data diversification, data summarization, and coverage problems. In this work, we study this…

We complete the complexity classification by degree of minimizing a polynomial over the integer points in a polyhedron in $\mathbb{R}^2$. Previous work shows that optimizing a quadratic polynomial over the integer points in a polyhedral…

This expository article presents a self-contained introduction to simplicial homology for finite simplicial complexes, emphasizing concrete computation and geometric intuition. Beginning with orientations of simplices and the construction…

We show that any submodular minimization (SM) problem defined on a linear constraint set with constraints having up to two variables per inequality, are 2-approximable in polynomial time. If the constraints are monotone (the two variables…

We study the multiscale simplicial flat norm (MSFN) problem, which computes flat norm at various scales of sets defined as oriented subcomplexes of finite simplicial complexes in arbitrary dimensions. We show that the multiscale simplicial…

In a previous paper it was shown that a machine learning regression problem can be solved within the framework of random function theory, with the optimal kernel analytically derived from symmetry and indifference principles and coinciding…

We conjecture that the balanced complete bipartite graph $K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil}$ contains more cycles than any other $n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent…

Let $G$ be the simple algebraic group $\mathrm{SL}_2$ defined over an algebraically closed field $k$ of characteristic $p > 0$. Using results of A. Parker, we develop a method which gives, for any $q \in \mathbb{N}$, a closed form…

We present a distributed algorithm to compute the first homology of a simplicial complex. Such algorithms are very useful in topological analysis of sensor networks, such as its coverage properties. We employ spanning trees to compute a…

Wall-clock-time is minimized for a solution to a linear-program with block-diagonal-structure, by decomposing the linear-program into as many small-sized subproblems as possible, each block resulting in a separate subproblem, when the…

The multiscale simplicial flat norm (MSFN) of a d-cycle is a family of optimal homology problems indexed by a scale parameter {\lambda} >= 0. Each instance (mSFN) optimizes the total weight of a homologous d-cycle and a bounded (d +…

We describe an essential improvement of our recent algorithm for computing cohomology of Lie (super)algebra based on partition of the whole cochain complex into minimal subcomplexes. We replace the arithmetic of rational numbers or integers…

Let $D$ be a directed graph cellularly embedded in a surface together with non-negative cost on its arcs. Given any integer circulation in $D$, we study the problem of finding a minimum-cost non-negative integer circulation in $D$ that is…

Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an…

It has been shown that a global minimizer of a smooth determinant of a matrix function corresponds to the largest cycle of a graph. When it exists, this is a Hamiltonian cycle. Finding global minimizers even of a smooth function is a…

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a…

Submodular optimization generalizes many classic problems in combinatorial optimization and has recently found a wide range of applications in machine learning (e.g., feature engineering and active learning). For many large-scale…

Topology optimization of frame structures under free-vibration eigenvalue constraints constitutes a challenging nonconvex polynomial optimization problem with disconnected feasible sets. In this article, we first formulate it as a…

We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first…

This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We…