707 research outputs found
Counting Problems in Parameterized Complexity
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs.
While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way
Bounds on the Complex Zeros of (Di)Chromatic Polynomials and Potts-Model Partition Functions
I show that there exist universal constants such that, for
all loopless graphs of maximum degree , the zeros (real or complex)
of the chromatic polynomial lie in the disc . Furthermore,
. This result is a corollary of a more general result
on the zeros of the Potts-model partition function in the
complex antiferromagnetic regime . The proof is based on a
transformation of the Whitney-Tutte-Fortuin-Kasteleyn representation of to a polymer gas, followed by verification of the
Dobrushin-Koteck\'y-Preiss condition for nonvanishing of a polymer-model
partition function. I also show that, for all loopless graphs of
second-largest degree , the zeros of lie in the disc . Along the way, I give a simple proof of a generalized (multivariate)
Brown-Colbourn conjecture on the zeros of the reliability polynomial for the
special case of series-parallel graphs.Comment: 47 pages (LaTeX). Revised version contains slightly simplified proofs
of Propositions 4.2 and 4.5. Version 3 fixes a silly error in my proof of
Proposition 4.1, and adds related discussion. To appear in Combinatorics,
Probability & Computin
Correlation Decay in Random Decision Networks
We consider a decision network on an undirected graph in which each node
corresponds to a decision variable, and each node and edge of the graph is
associated with a reward function whose value depends only on the variables of
the corresponding nodes. The goal is to construct a decision vector which
maximizes the total reward. This decision problem encompasses a variety of
models, including maximum-likelihood inference in graphical models (Markov
Random Fields), combinatorial optimization on graphs, economic team theory and
statistical physics. The network is endowed with a probabilistic structure in
which costs are sampled from a distribution. Our aim is to identify sufficient
conditions to guarantee average-case polynomiality of the underlying
optimization problem. We construct a new decentralized algorithm called Cavity
Expansion and establish its theoretical performance for a variety of models.
Specifically, for certain classes of models we prove that our algorithm is able
to find near optimal solutions with high probability in a decentralized way.
The success of the algorithm is based on the network exhibiting a correlation
decay (long-range independence) property. Our results have the following
surprising implications in the area of average case complexity of algorithms.
Finding the largest independent (stable) set of a graph is a well known NP-hard
optimization problem for which no polynomial time approximation scheme is
possible even for graphs with largest connectivity equal to three, unless P=NP.
We show that the closely related maximum weighted independent set problem for
the same class of graphs admits a PTAS when the weights are i.i.d. with the
exponential distribution. Namely, randomization of the reward function turns an
NP-hard problem into a tractable one
Structural sparsity of complex networks: bounded expansion in random models and real-world graphs
This research establishes that many real-world networks exhibit bounded expansion, a strong notion of structural sparsity, and demonstrates that it can be leveraged to design efficient algorithms for network analysis. Specifically, we give a new linear-time fpt algorithm for motif counting and linear time algorithms to compute localized variants of several centrality measures. To establish structural sparsity in real-world networks, we analyze several common network models regarding their structural sparsity. We show that, with high probability, (1) graphs sampled with a prescribed sparse degree sequence; (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. In contrast, we show that the Kleinberg and the Barabási–Albert model have unbounded expansion. We support our findings with empirical measurements on a corpus of real-world networks
Hitting forbidden minors: Approximation and Kernelization
We study a general class of problems called F-deletion problems. In an
F-deletion problem, we are asked whether a subset of at most vertices can
be deleted from a graph such that the resulting graph does not contain as a
minor any graph from the family F of forbidden minors.
We obtain a number of algorithmic results on the F-deletion problem when F
contains a planar graph. We give (1) a linear vertex kernel on graphs excluding
-claw , the star with leves, as an induced subgraph, where
is a fixed integer. (2) an approximation algorithm achieving an approximation
ratio of , where is the size of an optimal solution on
general undirected graphs. Finally, we obtain polynomial kernels for the case
when F contains graph as a minor for a fixed integer . The graph
consists of two vertices connected by parallel edges. Even
though this may appear to be a very restricted class of problems it already
encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback
Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is
based on a non-trivial application of protrusion techniques, previously used
only for problems on topological graph classes
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