arXiv subjects

Daniel Brosch

Publications and source records attributed to Daniel Brosch.

7 recordsLinked to original sources

arXiv2025

Lower and Upper Bounds for Small Canonical and Ordered Ramsey Numbers

In this paper, we investigate three extensions of Ramsey numbers to other combinatorial settings. We first consider ordered Ramsey numbers. Here, we ask for a monochromatic copy of a linearly ordered graph $G$ in every $2$-edge-coloring of a linearly ordered complete graph $K_n$. The smallest such $n$ is denoted by $\vec{R}(G)$. Next, we study canonical Ramsey numbers. A canonical coloring of a linearly ordered graph $G$ is an edge-coloring in which $G$ is monochromatic, rainbow, or min/max-lexicographic. In the latter case, each pair of edges receives the same color if and only if they share the same first (respectively, second) vertex. Erd\H{o}s and Rado showed that for every $p$ there exists $n$ such that every edge-coloring of a linearly ordered $K_n$ contains a canonical copy of $K_p$; the smallest such $n$ is denoted by $ER(G)$. Finally, we examine unordered canonical Ramsey numbers, introduced by Richer. An edge-coloring of $G$ is orderable if there exists a linear ordering of its vertices such that the color of each edge is determined by its first vertex. Unlike lexicographic colorings, this notion also includes monochromatic colorings. Richer proved that for all $s$ and $t$, there exists $n$ such that every edge-coloring of $K_n$ contains an orderable copy of $K_s$ or a rainbow $K_t$. The smallest such $n$ is denoted by $CR(s,t)$. In all three settings, we focus on determining the corresponding Ramsey numbers for small graphs $G$. We use tabu search and integer programming to obtain lower bounds, and flag algebras or integer programming to establish upper bounds. Among other results, we determine $\vec{R}(G)$ for all graphs $G$ on up to four vertices except $K_4^-$, $ER(P_4)$ for all orderings of $P_4$, and the exact values $CR(6,3)=26$ and $CR(3,5)=13$.

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arXiv2025

The Augmented Mixing Method: Computing High-Accuracy Primal-Dual Solutions to Large-Scale SDPs via Column Updates

The Burer-Monteiro factorization has become a powerful tool for solving large-scale semidefinite programs (SDPs), enabling recently developed low-rank solvers to tackle problems previously beyond reach. However, existing methods are typically designed to prioritize scalability over solution accuracy. We introduce the Augmented Mixing Method, a new algorithm that combines the Burer-Monteiro factorization with an inexact augmented Lagrangian framework and a block coordinate descent scheme. Our method emphasizes solving low-dimensional subproblems efficiently and to high precision. Inequality constraints are handled directly, without explicitly maintaining slack variables in the algorithm. A novel dynamic update strategy for the penalty parameter ensures that primal and dual feasibility progress remain balanced. This approach enables our method to compute highly accurate primal-dual solutions, even for large-scale SDPs with over ten million inequality constraints. Despite lacking theoretical convergence guarantees, the Augmented Mixing Method shows strong practical performance with default parameters across a wide range of SDP instances. It often produces more accurate primal-dual solutions than state-of-the-art interior-point methods and scales significantly better. Our open-source Julia implementation is memory-efficient, customizable, and supports arbitrary-precision arithmetic.

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arXiv2024

Getting to the Root of the Problem: Sums of Squares for Limits of Trees

The inducibility of a graph represents its maximum density as an induced subgraph over all possible sequences of graphs of size growing to infinity. This invariant of graphs has been extensively studied since its introduction in $1975$ by Pippenger and Golumbic. In $2017$, Czabarka, Sz\'ekely and Wagner extended this notion to leaf-labeled rooted binary trees, which are objects widely studied in the field of phylogenetics. They obtain the first results and bounds for the densities and inducibilities of such trees. Following up on their work, we apply Razborov's flag algebra theory to this setting, introducing the flag algebra of rooted leaf-labeled binary trees. This framework allows us to use polynomial optimization methods, based on semidefinite programming, to efficiently obtain new upper bounds for the inducibility of trees and to improve existing ones. Additionally, we obtain the first outer approximations of profiles of trees, which represent all possible simultaneous densities of a pair of trees in a sequence of trees of growing sizes. Finally, we are able to prove the non-convexity of some of these profiles.

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arXiv2022

New lower bounds on crossing numbers of $K_{m,n}$ from semidefinite programming

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189--202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624]. We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that $\text{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\text{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\text{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\text{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all $n$. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.

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arXiv2020

Optimizing hypergraph-based polynomials modeling job-occupancy in queueing with redundancy scheduling

We investigate two classes of multivariate polynomials with variables indexed by the edges of a uniform hypergraph and coefficients depending on certain patterns of union of edges. These polynomials arise naturally to model job-occupancy in some queuing problems with redundancy scheduling policy. The question, posed by Cardinaels, Borst and van Leeuwaarden (arXiv:2005.14566, 2020), is to decide whether their global minimum over the standard simplex is attained at the uniform probability distribution. By exploiting symmetry properties of these polynomials we can give a positive answer for the first class and partial results for the second one, where we in fact show a stronger convexity property of these polynomials over the simplex.

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arXiv2020

Jordan symmetry reduction for conic optimization over the doubly nonnegative cone: theory and software

A common computational approach for polynomial optimization problems (POPs) is to use (hierarchies of) semidefinite programming (SDP) relaxations. When the variables in the POP are required to be nonnegative, these SDP problems typically involve nonnegative matrices, i.e. they are conic optimization problems over the doubly nonnegative cone. The Jordan reduction, a symmetry reduction method for conic optimization, was recently introduced for symmetric cones by Parrilo and Permenter [Mathematical Programming 181(1), 2020]. We extend this method to the doubly nonnegative cone, and investigate its application to known relaxations of the quadratic assignment and maximum stable set problems. We also introduce new Julia software where the symmetry reduction is implemented.

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arXiv2019

Minimum energy configurations on a toric lattice as a quadratic assignment problem

We consider three known bounds for the quadratic assignment problem (QAP): an eigenvalue, a convex quadratic programming (CQP), and a semidefinite programming (SDP) bound. Since the last two bounds were not compared directly before, we prove that the SDP bound is stronger than the CQP bound. We then apply these to improve known bounds on a discrete energy minimization problem, reformulated as a QAP, which aims to minimize the potential energy between repulsive particles on a toric grid. Thus we are able to prove optimality for several configurations of particles and grid sizes, complementing earlier results by Bouman, Draisma and Van Leeuwaarden [ SIAM Journal on Discrete Mathematics, 27(3):1295--1312, 2013]. The semidefinite programs in question are too large to solve without pre-processing, and we use a symmetry reduction method by Parrilo and Permenter [Mathematical Programming, 181:51--84, 2020] to make computation of the SDP bounds possible.

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