Graph parameters from symplectic group invariants

In this paper we introduce, and characterize, a class of graph parameters obtained from tensor invariants of the symplectic group. These parameters are similar to partition functions of vertex models, as introduced by de la Harpe and Jones, [P. de la Harpe, V.F.R. Jones, Graph invariants related to statistical mechanical models: examples and problems, Journal of Combinatorial Theory, Series B 57 (1993) 207-227]. Yet they give a completely different class of graph invariants. We moreover show that certain evaluations of the cycle partition polynomial, as defined by Martin [P. Martin, Enum\'erations eul\'eriennes dans les multigraphes et invariants de Tutte-Grothendieck, Diss. Institut National Polytechnique de Grenoble-INPG; Universit\'e Joseph-Fourier-Grenoble I, 1977], give examples of graph parameters that can be obtained this way.Comment: Some corrections have been made on the basis of referee comments. 21 pages, 1 figure. Accepted in JCT

New characterizations of partition functions using connection matrices

In this thesis we expand upon a line of research pioneered by Freedman, Lovász and Schrijver, and Szegedy, that uses algebraic methods to characterize families of partition functions. We introduce two new types of partition functions: skew partition functions and mixed partition functions. We give two algebraic characterizations of skew partition functions and we show that a mixed partition functions satisfy certain algebraic relationships that are related to the invariant of the symmetric group and to the invariant theory of the Orthosymplectic Supergroup. We furthermore give a characterization of real-valued partition functions for 3-graphs and for virtual link diagrams in terms of positive semidefiniteness of the associated connection matrices

Counting Problems on Quantum Graphs: Parameterized and Exact Complexity Classifications

Quantum graphs, as defined by Lovász in the late 60s, are formal linear combinations of simple graphs with finite support. They allow for the complexity analysis of the problem of computing finite linear combinations of homomorphism counts, the latter of which constitute the foundation of the structural hardness theory for parameterized counting problems: The framework of parameterized counting complexity was introduced by Flum and Grohe, and McCartin in 2002 and forms a hybrid between the classical field of computational counting as founded by Valiant in the late 70s and the paradigm of parameterized complexity theory due to Downey and Fellows which originated in the early 90s. The problem of computing homomorphism numbers of quantum graphs subsumes general motif counting problems and the complexity theoretic implications have only turned out recently in a breakthrough regarding the parameterized subgraph counting problem by Curticapean, Dell and Marx in 2017. We study the problems of counting partially injective and edge-injective homomorphisms, counting induced subgraphs, as well as counting answers to existential first-order queries. We establish novel combinatorial, algebraic and even topological properties of quantum graphs that allow us to provide exhaustive parameterized and exact complexity classifications, including necessary, sufficient and mostly explicit tractability criteria, for all of the previous problems.Diese Arbeit befasst sich mit der Komplexit atsanalyse von mathematischen Problemen die als Linearkombinationen von Graphhomomorphismenzahlen darstellbar sind. Dazu wird sich sogenannter Quantengraphen bedient, bei denen es sich um formale Linearkombinationen von Graphen handelt und welche von Lov asz Ende der 60er eingef uhrt wurden. Die Bestimmung der Komplexit at solcher Probleme erfolgt unter dem von Flum, Grohe und McCartin im Jahre 2002 vorgestellten Paradigma der parametrisierten Z ahlkomplexit atstheorie, die als Hybrid der von Valiant Ende der 70er begr undeten klassischen Z ahlkomplexit atstheorie und der von Downey und Fellows Anfang der 90er eingef uhrten parametrisierten Analyse zu verstehen ist. Die Berechnung von Homomorphismenzahlen zwischen Quantengraphen und Graphen subsumiert im weitesten Sinne all jene Probleme, die das Z ahlen von kleinen Mustern in gro en Strukturen erfordern. Aufbauend auf dem daraus resultierenden Durchbruch von Curticapean, Dell und Marx, das Subgraphz ahlproblem betre end, behandelt diese Arbeit die Analyse der Probleme des Z ahlens von partiell- und kanteninjektiven Homomorphismen, induzierten Subgraphen, und Tre ern von relationalen Datenbankabfragen die sich als existentielle Formeln ausdr ucken lassen. Insbesondere werden dabei neue kombinatorische, algebraische und topologische Eigenschaften von Quantengraphen etabliert, die hinreichende, notwendige und meist explizite Kriterien f ur die Existenz e zienter Algorithmen liefern

Equality on all #CSP Instances Yields Constraint Function Isomorphism via Interpolation and Intertwiners

A fundamental result in the study of graph homomorphisms is Lov\'asz's theorem that two graphs are isomorphic if and only if they admit the same number of homomorphisms from every graph. A line of work extending Lov\'asz's result to more general types of graphs was recently capped by Cai and Govorov, who showed that it holds for graphs with vertex and edge weights from an arbitrary field of characteristic 0. In this work, we generalize from graph homomorphism -- a special case of #CSP with a single binary function -- to general #CSP by showing that two sets $\mathcal{F}$ and $\mathcal{G}$ of arbitrary constraint functions are isomorphic if and only if the partition function of any #CSP instance is unchanged when we replace the functions in $\mathcal{F}$ with those in $\mathcal{G}$. We give two very different proofs of this result. First, we demonstrate the power of the simple Vandermonde interpolation technique of Cai and Govorov by extending it to general #CSP. Second, we give a proof using the intertwiners of the automorphism group of a constraint function set, a concept from the representation theory of compact groups. This proof is a generalization of a classical version of the recent proof of the Lov\'asz-type result by Man\v{c}inska and Roberson relating quantum isomorphism and homomorphisms from planar graphs.Comment: 21 pages, 2 figure