20 research outputs found
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
The multivariate Tutte polynomial (known to physicists as the Potts-model partition function) can be defined on an arbitrary finite graph G, or more generally on an arbitrary matroid M, and encodes much important combinatorial information about the graph (indeed, in the matroid case it encodes the full structure of the matroid). It contains as a special case the familiar two-variable Tutte polynomial -- and therefore also its one-variable specializations such as the chromatic polynomial, the flow polynomial and the reliability polynomial -- but is considerably more flexible. I begin by giving an introduction to all these problems, stressing the advantages of working with the multivariate version. I then discuss some questions concerning the complex zeros of the multivariate Tutte polynomial, along with their physical interpretations in statistical mechanics (in connection with the Yang--Lee approach to phase transitions) and electrical circuit theory. Along the way I mention numerous open problems. This survey is intended to be understandable to mathematicians with no prior knowledge of physics
Delta-matroids as subsystems of sequences of Higgs Lifts
In [30], Tardos studied special delta-matroids obtained from sequences of
Higgs lifts; these are the full Higgs lift delta-matroids that we treat and around which
all of our results revolve. We give an excluded-minor characterization of the class of
full Higgs lift delta-matroids within the class of all delta-matroids, and we give similar
characterizations of two other minor-closed classes of delta-matroids that we define using
Higgs lifts. We introduce a minor-closed, dual-closed class of Higgs lift delta-matroids
that arise from lattice paths. It follows from results of Bouchet that all delta-matroids can
be obtained from full Higgs lift delta-matroids by removing certain feasible sets; to address
which feasible sets can be removed, we give an excluded-minor characterization of deltamatroids
within the more general structure of set systems. Many of these excluded minors
occur again when we characterize the delta-matroids in which the collection of feasible
sets is the union of the collections of bases of matroids of different ranks, and yet again
when we require those matroids to have special properties, such as being paving
Sampling Arborescences in Parallel
We study the problem of sampling a uniformly random directed rooted spanning tree, also known as an arborescence, from a possibly weighted directed graph. Classically, this problem has long been known to be polynomial-time solvable; the exact number of arborescences can be computed by a determinant [Tutte, 1948], and sampling can be reduced to counting [Jerrum et al., 1986; Jerrum and Sinclair, 1996]. However, the classic reduction from sampling to counting seems to be inherently sequential. This raises the question of designing efficient parallel algorithms for sampling. We show that sampling arborescences can be done in RNC.
For several well-studied combinatorial structures, counting can be reduced to the computation of a determinant, which is known to be in NC [Csanky, 1975]. These include arborescences, planar graph perfect matchings, Eulerian tours in digraphs, and determinantal point processes. However, not much is known about efficient parallel sampling of these structures. Our work is a step towards resolving this mystery
Paths and walks, forests and planes : arcadian algorithms and complexity
This dissertation is concerned with new results in the area of parameterized algorithms and complexity. We develop a new technique for hard graph problems that generalizes and unifies established methods such as Color-Coding, representative families, labelled walks and algebraic fingerprinting. At the heart of the approach lies an algebraic formulation of the problems, which is effected by means of a suitable exterior algebra. This allows us to estimate the number of simple paths of given length in directed graphs faster than before. Additionally, we give fast deterministic algorithms for finding paths of given length if the input graph contains only few of such paths. Moreover, we develop faster deterministic algorithms to find spanning trees with few leaves. We also consider the algebraic foundations of our new method. Additionally, we investigate the fine-grained complexity of determining the precise number of forests with a given number of edges in a given undirected graph. To wit, this happens in two ways. Firstly, we complete the complexity classification of the Tutte plane, assuming the exponential time hypothesis. Secondly, we prove that counting forests with a given number of edges is at least as hard as counting cliques of a given size.Diese Dissertation befasst sich mit neuen Ergebnissen auf dem Gebiet parametrisierter Algorithmen und KomplexitĂ€tstheorie. Wir entwickeln eine neue Technik fĂŒr schwere Graphprobleme, die etablierte Methoden wie Color-Coding, representative families, labelled walks oder algebraic fingerprinting verallgemeinert und vereinheitlicht. Kern der Herangehensweise ist eine algebraische Formulierung der Probleme, die vermittels passender GraĂmannalgebren geschieht. Das erlaubt uns, die Anzahl einfacher Pfade gegebener LĂ€nge in gerichteten Graphen schneller als bisher zu schĂ€tzen. AuĂerdem geben wir schnelle deterministische Verfahren an, Pfade gegebener LĂ€nge zu finden, falls der Eingabegraph nur wenige solche Pfade enthĂ€lt. Ăbrigens entwickeln wir schnellere deterministische Algorithmen, um SpannbĂ€ume mit wenigen BlĂ€ttern zu finden. Wir studieren auĂerdem die algebraischen Grundlagen unserer neuen Methode. Weiters untersuchen wir die fine-grained-KomplexitĂ€t davon, die genaue Anzahl von WĂ€ldern einer gegebenen Kantenzahl in einem gegebenen ungerichteten Graphen zu bestimmen. Und zwar erfolgt das auf zwei verschiedene Arten. Erstens vervollstĂ€ndigen wir die KomplexitĂ€tsklassifizierung der Tutte-Ebene unter Annahme der Expo- nentialzeithypothese. Zweitens beweisen wir, dass WĂ€lder mit gegebener Kantenzahl zu zĂ€hlen, wenigstens so schwer ist, wie Cliquen gegebener GröĂe zu zĂ€hlen.Cluster of Excellence (Multimodal Computing and Interaction
Combinatorial and Computational Methods for the Properties of Homogeneous Polynomials
In this manuscript, we provide foundations of properties of homogeneous polynomials such as the half-plane property, determinantal representability, being weakly determinantal, and having a spectrahedral hyperbolicity cone. One of the motivations for studying those properties comes from the ``generalized Lax conjecture'' stating that every hyperbolicity cone is spectrahedral. The conjecture has particular importance in convex optimization and has curious connections to other areas.
We take a combinatorial approach, contemplating the properties on matroids with a particular focus on operations that preserve these properties. We show that the spectrahedral representability of hyperbolicity cones and being weakly determinantal are minor-closed properties. In addition, they are preserved under passing to the faces of the Newton polytopes of homogeneous polynomials. We present a proved-to-be computationally feasible algorithm to test the half-plane property of matroids and another one for testing being weakly determinantal. Using the computer algebra system Macaulay2 and Julia, we implement these algorithms and conduct tests. We classify matroids on at most 8 elements with respect to the half-plane property and provide our test results on matroids with 9 elements. We provide 14 matroids on 8 elements of rank 4, including the VĂĄmos matroid, that are potential candidates for the search of a counterexample for the conjecture.:1 Background 1
1.1 Some Properties of Homogeneous Polynomials . . . . . . . . . . 1
Hyperbolic Polynomials . . . . . . . . . . . . . . . . . . . . . . 1
The Half-Plane Property and Stability . . . . . . . . . . . . . . 8
Determinantal Representability . . . . . . . . . . . . . . . . . . 15
Spectrahedral Representability . . . . . . . . . . . . . . . . . . 19
1.2 Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Some Operations on Matroids . . . . . . . . . . . . . . . . . . . 29
The Half-Plane Property of Matroids . . . . . . . . . . . . . . . 36
2 Some Operations 43
2.1 Determinantal Representability of Matroids . . . . . . . . . . . 43
A Criterion for Determinantal Representability . . . . . . . . . 46
2.2 Spectrahedral Representability of Matroids . . . . . . . . . . . 50
2.3 Matroid Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . 54
Newton Polytopes of Stable Polynomials . . . . . . . . . . . . . 59
3 Testing the Properties: an Algorithm 61
The Half-Plane Property . . . . . . . . . . . . . . . . . . . . . . 61
Being SOS-Rayleigh and Weak Determinantal Representability 65
4 Test Results on Matroids on 8 and 9 Elements 71
4.1 Matroids on 8 Elements . . . . . . . . . . . . . . . . . . . . . . 71
SOS-Rayleigh and Weakly Determinantal Matroids . . . . . . . 76
4.2 Matroids on 9 Elements . . . . . . . . . . . . . . . . . . . . . . 80
5 Conclusion and Future Perspectives 85
5.1 Spectrahedral Matroids . . . . . . . . . . . . . . . . . . . . . . 86
5.2 Non-negative Non-SOS Polynomials . . . . . . . . . . . . . . . 88
5.3 Completing the Classification of Matroids on 9 Elements and More 89
Bibliography 9