LP Relaxation and Tree Packing for Minimum k-cuts

Karger used spanning tree packings [Karger, 2000] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [Karger, 1995; Karger and Stein, 1996]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [Thorup, 2008]. In this paper we revisit properties of an LP relaxation for k-cut proposed by Naor and Rabani [Naor and Rabani, 2001], and analyzed in [Chekuri et al., 2006]. We show that the dual of the LP yields a tree packing, that when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger\u27s analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup\u27s algorithm by a factor of n. We also improve the bound on the number of alpha-approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1-1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Barahona, 2000] and Ravi and Sinha [Ravi and Sinha, 2008]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP. This work arose from an effort to understand and simplify the results of Thorup [Thorup, 2008]

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