Computing graph gonality is hard

There are several notions of gonality for graphs. The divisorial gonality dgon(G) of a graph G is the smallest degree of a divisor of positive rank in the sense of Baker-Norine. The stable gonality sgon(G) of a graph G is the minimum degree of a finite harmonic morphism from a refinement of G to a tree, as defined by Cornelissen, Kato and Kool. We show that computing dgon(G) and sgon(G) are NP-hard by a reduction from the maximum independent set problem and the vertex cover problem, respectively. Both constructions show that computing gonality is moreover APX-hard.Comment: The previous version only dealt with hardness of the divisorial gonality. The current version also shows hardness of stable gonality and discusses the relation between the two graph parameter

Boundary Value Problems on Planar Graphs and Flat Surfaces with integer cone singularities, I: The Dirichlet Problem

Consider a planar, bounded, $m$-connected region $\Omega$, and let \bord\Omega be its boundary. Let $\mathcal{T}$ be a cellular decomposition of \Omega\cup\bord\Omega, where each 2-cell is either a triangle or a quadrilateral. From these data and a conductance function we construct a canonical pair $(S,f)$ where $S$ is a genus $(m-1)$ singular flat surface tiled by rectangles and $f$ is an energy preserving mapping from ${\mathcal T}^{(1)}$ onto $S$.Comment: 27 pages, 11 figures; v2 - revised definition (now denoted by the flux-gradient metric (1.9)) in section 1 and minor modifications of proofs; corrected typo

Recognizing hyperelliptic graphs in polynomial time

Recently, a new set of multigraph parameters was defined, called "gonalities". Gonality bears some similarity to treewidth, and is a relevant graph parameter for problems in number theory and multigraph algorithms. Multigraphs of gonality 1 are trees. We consider so-called "hyperelliptic graphs" (multigraphs of gonality 2) and provide a safe and complete sets of reduction rules for such multigraphs, showing that for three of the flavors of gonality, we can recognize hyperelliptic graphs in O(n log n+m) time, where n is the number of vertices and m the number of edges of the multigraph.Comment: 33 pages, 8 figure

High-Quality Shared-Memory Graph Partitioning

Partitioning graphs into blocks of roughly equal size such that few edges run between blocks is a frequently needed operation in processing graphs. Recently, size, variety, and structural complexity of these networks has grown dramatically. Unfortunately, previous approaches to parallel graph partitioning have problems in this context since they often show a negative trade-off between speed and quality. We present an approach to multi-level shared-memory parallel graph partitioning that guarantees balanced solutions, shows high speed-ups for a variety of large graphs and yields very good quality independently of the number of cores used. For example, on 31 cores, our algorithm partitions our largest test instance into 16 blocks cutting less than half the number of edges than our main competitor when both algorithms are given the same amount of time. Important ingredients include parallel label propagation for both coarsening and improvement, parallel initial partitioning, a simple yet effective approach to parallel localized local search, and fast locality preserving hash tables

Power-counting theorem for non-local matrix models and renormalisation

Solving the exact renormalisation group equation a la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties--typically arising from orthogonal polynomials--which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative R^D in matrix formulation.Comment: 35 pages, 70 figures, LaTeX with svjour macros. v2: proof simplified because a discussion originally designed for \phi^4 on noncommutative R^2 was actually not necessary, see hep-th/0307017. v3: consistency conditions removed because models of interest relate automatically the relevant/marginal interactions to a finite number of base couplings, see hep-th/0401128. v4: integration procedure improved so that the initial cut-off can be directly removed; to appear in Commun. Math. Phy

Lifting harmonic morphisms II: tropical curves and metrized complexes

In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve. This article is the second in a series of two.Comment: 35 pages, 18 figures. This article used to be the second half of arXiv:1303.4812, and is now its seque

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure

Morse theory of harmonic forms

We consider the problem of whether it is possible to improve the Novikov inequalities for closed 1-forms, or any other inequalities of a similar nature, if we assume, additionally, that the given 1-form is harmonic with respect to some Riemannian metric. We show that, under suitable assumptions, it is impossible. We use, in an essential way, a theorem of E.Calabi characterizing 1-forms which are harmonic with respect to some metric. We also study some interesting examples illustrating our results.Comment: 16 pages, AMSTex, 12 figure