45 research outputs found
Index Reduction for Differential-Algebraic Equations with Mixed Matrices
Differential-algebraic equations (DAEs) are widely used for modeling of
dynamical systems. The difficulty in solving numerically a DAE is measured by
its differentiation index. For highly accurate simulation of dynamical systems,
it is important to convert high-index DAEs into low-index DAEs. Most of
existing simulation software packages for dynamical systems are equipped with
an index-reduction algorithm given by Mattsson and S\"{o}derlind.
Unfortunately, this algorithm fails if there are numerical cancellations.
These numerical cancellations are often caused by accurate constants in
structural equations. Distinguishing those accurate constants from generic
parameters that represent physical quantities, Murota and Iri introduced the
notion of a mixed matrix as a mathematical tool for faithful model description
in structural approach to systems analysis. For DAEs described with the use of
mixed matrices, efficient algorithms to compute the index have been developed
by exploiting matroid theory.
This paper presents an index-reduction algorithm for linear DAEs whose
coefficient matrices are mixed matrices, i.e., linear DAEs containing physical
quantities as parameters. Our algorithm detects numerical cancellations between
accurate constants, and transforms a DAE into an equivalent DAE to which
Mattsson--S\"{o}derlind's index-reduction algorithm is applicable. Our
algorithm is based on the combinatorial relaxation approach, which is a
framework to solve a linear algebraic problem by iteratively relaxing it into
an efficiently solvable combinatorial optimization problem. The algorithm does
not rely on symbolic manipulations but on fast combinatorial algorithms on
graphs and matroids. Furthermore, we provide an improved algorithm under an
assumption based on dimensional analysis of dynamical systems.Comment: A preliminary version of this paper is to appear in Proceedings of
the Eighth SIAM Workshop on Combinatorial Scientific Computing, Bergen,
Norway, June 201
Market Pricing for Matroid Rank Valuations
In this paper, we study the problem of maximizing social welfare in
combinatorial markets through pricing schemes. We consider the existence of
prices that are capable to achieve optimal social welfare without a central
tie-breaking coordinator. In the case of two buyers with rank valuations, we
give polynomial-time algorithms that always find such prices when one of the
matroids is a simple partition matroid or both matroids are strongly base
orderable. This result partially answers a question raised by D\"uetting and
V\'egh in 2017. We further formalize a weighted variant of the conjecture of
D\"uetting and V\'egh, and show that the weighted variant can be reduced to the
unweighted one based on the weight-splitting theorem for weighted matroid
intersection by Frank. We also show that a similar reduction technique works
for M-concave functions, or equivalently, gross substitutes
functions
Matchings and Flows in Hypergraphs
In this dissertation, we study matchings and flows in hypergraphs using combinatorial methods. These two problems are among the best studied in the field of combinatorial optimization. As hypergraphs are a very general concept, not many results on graphs can be generalized to arbitrary hypergraphs. Therefore, we consider special classes of hypergraphs, which admit more structure, to transfer results from graph theory to hypergraph theory.
In Chapter 2, we investigate the perfect matching problem on different classes of hypergraphs generalizing bipartite graphs. First, we give a polynomial time approximation algorithm for the maximum weight matching problem on so-called partitioned hypergraphs, whose approximation factor is best possible up to a constant. Afterwards, we look at the theorems of König and Hall and their relation. Our main result is a condition for the existence of perfect matchings in normal hypergraphs that generalizes Hall’s condition for bipartite graphs.
In Chapter 3, we consider perfect f-matchings, f-factors, and (g,f)-matchings. We prove conditions for the existence of (g,f)-matchings in unimodular hypergraphs, perfect f-matchings in uniform Mengerian hypergraphs, and f-factors in uniform balanced hypergraphs. In addition, we give an overview about the complexity of the (g,f)-matching problem on different classes of hypergraphs generalizing bipartite graphs.
In Chapter 4, we study the structure of hypergraphs that admit a perfect matching. We show that these hypergraphs can be decomposed along special cuts. For graphs it is known that the resulting decomposition is unique, which does not hold for hypergraphs in general. However, we prove the uniqueness of this decomposition (up to parallel hyperedges) for uniform hypergraphs.
In Chapter 5, we investigate flows on directed hypergraphs, where we focus on graph-based directed hypergraphs, which means that every hyperarc is the union of a set of pairwise disjoint ordinary arcs. We define a residual network, which can be used to decide whether a given flow is optimal or not. Our main result in this chapter is an algorithm that computes a minimum cost flow on a graph-based directed hypergraph. This algorithm is a generalization of the network simplex algorithm.Diese Arbeit untersucht Matchings und Flüsse in Hypergraphen mit Hilfe kombinatorischer Methoden. In Graphen gehören diese Probleme zu den grundlegendsten der kombinatorischen Optimierung. Viele Resultate lassen sich nicht von Graphen auf Hypergraphen verallgemeinern, da Hypergraphen ein sehr abstraktes Konzept bilden. Daher schauen wir uns bestimmte Klassen von Hypergraphen an, die mehr Struktur besitzen, und nutzen diese aus um Resultate aus der Graphentheorie zu übertragen. In Kapitel 2 betrachten wir das perfekte Matchingproblem auf Klassen von „bipartiten“ Hypergraphen, wobei es verschiedene Möglichkeiten gibt den Begriff „bipartit“ auf Hypergraphen zu definieren. Für sogenannte partitionierte Hypergraphen geben wir einen polynomiellen Approximationsalgorithmus an, dessen Gütegarantie bis auf eine Konstante bestmöglich ist. Danach betrachten wir die Sätze von Konig und Hall und untersuchen deren Zusammenhang. Unser Hauptresultat ist eine Bedingung für die Existenz von perfekten Matchings auf normalen Hypergraphen, die Halls Bedingung für bipartite Graphen verallgemeinert. Als Verallgemeinerung von perfekten Matchings betrachten wir in Kapitel 3 perfekte f-Matchings, f-Faktoren und (g, f)-Matchings. Wir beweisen Bedingungen für die Existenz von (g, f)-Matchings auf unimodularen Hypergraphen, perfekten f-Matchings auf uniformen Mengerschen Hypergraphen und f-Faktoren auf uniformen balancierten Hypergraphen. Außerdem geben wir eine Übersicht über die Komplexität des (g, f)-Matchingproblems auf verschiedenen Klassen von Hypergraphen an, die bipartite Graphen verallgemeinern. In Kapitel 4 untersuchen wir die Struktur von Hypergraphen, die ein perfektes Matching besitzen. Wir zeigen, dass diese Hypergraphen entlang spezieller Schnitte zerlegt werden können. Für Graphen weiß man, dass die so erhaltene Zerlegung eindeutig ist, was im Allgemeinen für Hypergraphen nicht zutrifft. Wenn man jedoch uniforme Hypergraphen betrachtet, dann liefert jede Zerlegung die gleichen unzerlegbaren Hypergraphen bis auf parallele Hyperkanten. Kapitel 5 beschäftigt sich mit Flüssen in gerichteten Hypergraphen, wobei wir Hypergraphen betrachten, die auf gerichteten Graphen basieren. Das bedeutet, dass eine Hyperkante die Vereinigung einer Menge von disjunkten Kanten ist. Wir definieren ein Residualnetzwerk, mit dessen Hilfe man entscheiden kann, ob ein gegebener Fluss optimal ist. Unser Hauptresultat in diesem Kapitel ist ein Algorithmus, um einen Fluss minimaler Kosten zu finden, der den Netzwerksimplex verallgemeinert