366 research outputs found
The parameterised complexity of counting connected subgraphs and graph motifs
We introduce a family of parameterised counting problems on graphs, p-#Induced Subgraph With Property(Φ), which generalises a number of problems which have previously been studied. This paper focuses on the case in which Φ defines a family of graphs whose edge-minimal elements all have bounded treewidth; this includes the special case in which Φ describes the property of being connected. We show that exactly counting the number of connected induced k-vertex subgraphs in an n-vertex graph is #W[1]-hard, but on the other hand there exists an FPTRAS for the problem; more generally, we show that there exists an FPTRAS for p-#Induced Subgraph With Property(Φ) whenever Φ is monotone and all the minimal graphs satisfying Φ have bounded treewidth. We then apply these results to a counting version of the Graph Motif problem
Exponential Time Complexity of the Permanent and the Tutte Polynomial
We show conditional lower bounds for well-studied #P-hard problems:
(a) The number of satisfying assignments of a 2-CNF formula with n variables
cannot be counted in time exp(o(n)), and the same is true for computing the
number of all independent sets in an n-vertex graph.
(b) The permanent of an n x n matrix with entries 0 and 1 cannot be computed
in time exp(o(n)).
(c) The Tutte polynomial of an n-vertex multigraph cannot be computed in time
exp(o(n)) at most evaluation points (x,y) in the case of multigraphs, and it
cannot be computed in time exp(o(n/polylog n)) in the case of simple graphs.
Our lower bounds are relative to (variants of) the Exponential Time
Hypothesis (ETH), which says that the satisfiability of n-variable 3-CNF
formulas cannot be decided in time exp(o(n)). We relax this hypothesis by
introducing its counting version #ETH, namely that the satisfying assignments
cannot be counted in time exp(o(n)). In order to use #ETH for our lower bounds,
we transfer the sparsification lemma for d-CNF formulas to the counting
setting
Lattices of Graphical Gaussian Models with Symmetries
In order to make graphical Gaussian models a viable modelling tool when the
number of variables outgrows the number of observations, model classes which
place equality restrictions on concentrations or partial correlations have
previously been introduced in the literature. The models can be represented by
vertex and edge coloured graphs. The need for model selection methods makes it
imperative to understand the structure of model classes. We identify four model
classes that form complete lattices of models with respect to model inclusion,
which qualifies them for an Edwards-Havr\'anek model selection procedure. Two
classes turn out most suitable for a corresponding model search. We obtain an
explicit search algorithm for one of them and provide a model search example
for the other.Comment: 29 pages, 18 figures. Restructured Section 5, results unchanged;
added references in Section 6; amended example in Section 6.
Coloring Graphs having Few Colorings over Path Decompositions
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no
time algorithm for
deciding if an -vertex graph with pathwidth
admits a proper vertex coloring with colors unless the Strong Exponential
Time Hypothesis (SETH) is false. We show here that nevertheless, when
, where is the maximum degree in the
graph , there is a better algorithm, at least when there are few colorings.
We present a Monte Carlo algorithm that given a graph along with a path
decomposition of with pathwidth runs in time, that
distinguishes between -colorable graphs having at most proper
-colorings and non--colorable graphs. We also show how to obtain a
-coloring in the same asymptotic running time. Our algorithm avoids
violating SETH for one since high degree vertices still cost too much and the
mentioned hardness construction uses a lot of them.
We exploit a new variation of the famous Alon--Tarsi theorem that has an
algorithmic advantage over the original form. The original theorem shows a
graph has an orientation with outdegree less than at every vertex, with a
different number of odd and even Eulerian subgraphs only if the graph is
-colorable, but there is no known way of efficiently finding such an
orientation. Our new form shows that if we instead count another difference of
even and odd subgraphs meeting modular degree constraints at every vertex
picked uniformly at random, we have a fair chance of getting a non-zero value
if the graph has few -colorings. Yet every non--colorable graph gives a
zero difference, so a random set of constraints stands a good chance of being
useful for separating the two cases.Comment: Strengthened result from uniquely -colorable graphs to graphs with
few -colorings. Also improved running tim
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