14,219 research outputs found
Accelerating Parametric Probabilistic Verification
We present a novel method for computing reachability probabilities of
parametric discrete-time Markov chains whose transition probabilities are
fractions of polynomials over a set of parameters. Our algorithm is based on
two key ingredients: a graph decomposition into strongly connected subgraphs
combined with a novel factorization strategy for polynomials. Experimental
evaluations show that these approaches can lead to a speed-up of up to several
orders of magnitude in comparison to existing approache
A bivariate chromatic polynomial for signed graphs
We study Dohmen--P\"onitz--Tittmann's bivariate chromatic polynomial
which counts all -colorings of a graph such
that adjacent vertices get different colors if they are . Our first
contribution is an extension of to signed graphs, for which we
obtain an inclusion--exclusion formula and several special evaluations giving
rise, e.g., to polynomials that encode balanced subgraphs. Our second goal is
to derive combinatorial reciprocity theorems for and its
signed-graph analogues, reminiscent of Stanley's reciprocity theorem linking
chromatic polynomials to acyclic orientations.Comment: 8 pages, 4 figure
Evaluations of topological Tutte polynomials
We find new properties of the topological transition polynomial of embedded
graphs, . We use these properties to explain the striking similarities
between certain evaluations of Bollob\'as and Riordan's ribbon graph
polynomial, , and the topological Penrose polynomial, . The general
framework provided by also leads to several other combinatorial
interpretations these polynomials. In particular, we express , ,
and the Tutte polynomial, , as sums of chromatic polynomials of graphs
derived from ; show that these polynomials count -valuations of medial
graphs; show that counts edge 3-colourings; and reformulate the Four
Colour Theorem in terms of . We conclude with a reduction formula for the
transition polynomial of the tensor product of two embedded graphs, showing
that it leads to additional relations among these polynomials and to further
combinatorial interpretations of and .Comment: V2: major revision, several new results, and improved expositio
Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials
In this paper we show a new way of constructing deterministic polynomial-time
approximation algorithms for computing complex-valued evaluations of a large
class of graph polynomials on bounded degree graphs. In particular, our
approach works for the Tutte polynomial and independence polynomial, as well as
partition functions of complex-valued spin and edge-coloring models.
More specifically, we define a large class of graph polynomials
and show that if and there is a disk centered at zero in the
complex plane such that does not vanish on for all bounded degree
graphs , then for each in the interior of there exists a
deterministic polynomial-time approximation algorithm for evaluating at
. This gives an explicit connection between absence of zeros of graph
polynomials and the existence of efficient approximation algorithms, allowing
us to show new relationships between well-known conjectures.
Our work builds on a recent line of work initiated by. Barvinok, which
provides a new algorithmic approach besides the existing Markov chain Monte
Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In
particular a tiny error in Proposition 4.4 has been fixed. The introduction
and concluding remarks have also been rewritten to incorporate the most
recent developments. Accepted for publication in SIAM Journal on Computatio
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