Algebraic combinatorial optimization on the degree of determinants of noncommutative symbolic matrices

We address the computation of the degrees of minors of a noncommutative symbolic matrix of form $$A[c] := \sum_{k=1}^m A_k t^{c_k} x_k,$$ where $A_k$ are matrices over a field $\mathbb{K}$, $x_i$ are noncommutative variables, $c_k$ are integer weights, and $t$ is a commuting variable specifying the degree. This problem extends noncommutative Edmonds' problem (Ivanyos et al. 2017), and can formulate various combinatorial optimization problems. Extending the study by Hirai 2018, and Hirai, Ikeda 2022, we provide novel duality theorems and polyhedral characterization for the maximum degrees of minors of $A[c]$ of all sizes, and develop a strongly polynomial-time algorithm for computing them. This algorithm is viewed as a unified algebraization of the classical Hungarian method for bipartite matching and the weight-splitting algorithm for linear matroid intersection. As applications, we provide polynomial-time algorithms for weighted fractional linear matroid matching and linear optimization over rank-2 Brascamp-Lieb polytopes

Weighted Linear Matroid Parity

The matroid parity (or matroid matching) problem, introduced as a common generalization of matching and matroid intersection problems, is so general that it requires an exponential number of oracle calls. Nevertheless, Lovasz (1978) showed that this problem admits a min-max formula and a polynomial algorithm for linearly represented matroids. Since then efficient algorithms have been developed for the linear matroid parity problem. This talk presents a recently developed polynomial-time algorithm for the weighted linear matroid parity problem. The algorithm builds on a polynomial matrix formulation using Pfaffian and adopts a primal-dual approach based on the augmenting path algorithm of Gabow and Stallmann (1986) for the unweighted problem

Some Applications of the Weighted Combinatorial Laplacian

The weighted combinatorial Laplacian of a graph is a symmetric matrix which is the discrete analogue of the Laplacian operator. In this thesis, we will study a new application of this matrix to matching theory yielding a new characterization of factor-criticality in graphs and matroids. Other applications are from the area of the physical design of very large scale integrated circuits. The placement of the gates includes the minimization of a quadratic form given by a weighted Laplacian. A method based on the dual constrained subgradient method is proposed to solve the simultaneous placement and gate-sizing problem. A crucial step of this method is the projection to the flow space of an associated graph, which can be performed by minimizing a quadratic form given by the unweighted combinatorial Laplacian.Andwendungen der gewichteten kombinatorischen Laplace-Matrix Die gewichtete kombinatorische Laplace-Matrix ist das diskrete Analogon des Laplace-Operators. In dieser Arbeit stellen wir eine neuartige Charakterisierung von Faktor-Kritikalität von Graphen und Matroiden mit Hilfe dieser Matrix vor. Wir untersuchen andere Anwendungen im Bereich des Entwurfs von höchstintegrierten Schaltkreisen. Die Platzierung basiert auf der Minimierung einer quadratischen Form, die durch eine gewichtete kombinatorische Laplace-Matrix gegeben ist. Wir präsentieren einen Algorithmus für das allgemeine simultane Platzierungs- und Gattergrößen-Optimierungsproblem, der auf der dualen Subgradientenmethode basiert. Ein wichtiger Bestandteil dieses Verfahrens ist eine Projektion auf den Flussraum eines assoziierten Graphen, die als die Minimierung einer durch die Laplace-Matrix gegebenen quadratischen Form aufgefasst werden kann