2,039 research outputs found
Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth
We give a fixed-parameter tractable algorithm that, given a parameter and
two graphs , either concludes that one of these graphs has treewidth
at least , or determines whether and are isomorphic. The running
time of the algorithm on an -vertex graph is ,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in time that, for a
given graph on vertices, either concludes that the treewidth of is
at least , or: * finds in an isomorphic-invariant way a graph
that is isomorphic to ; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes together with a
tree decomposition of of width .
Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for and are equal.Comment: Full version of a paper presented at FOCS 201
Fully polynomial FPT algorithms for some classes of bounded clique-width graphs
Parameterized complexity theory has enabled a refined classification of the
difficulty of NP-hard optimization problems on graphs with respect to key
structural properties, and so to a better understanding of their true
difficulties. More recently, hardness results for problems in P were achieved
using reasonable complexity theoretic assumptions such as: Strong Exponential
Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to
these assumptions, many graph theoretic problems do not admit truly
subquadratic algorithms, nor even truly subcubic algorithms (Williams and
Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central
technique used to tackle the difficulty of the above mentioned problems is
fixed-parameter algorithms for polynomial-time problems with polynomial
dependency in the fixed parameter (P-FPT). This technique was introduced by
Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016)
and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this
technique to clique-width, another important graph parameter, remained to be
done. In this paper we study several graph theoretic problems for which
hardness results exist such as cycle problems (triangle detection, triangle
counting, girth, diameter), distance problems (diameter, eccentricities, Gromov
hyperbolicity, betweenness centrality) and maximum matching. We provide
hardness results and fully polynomial FPT algorithms, using clique-width and
some of its upper-bounds as parameters (split-width, modular-width and
-sparseness). We believe that our most important result is an -time algorithm for computing a maximum matching where
is either the modular-width or the -sparseness. The latter generalizes
many algorithms that have been introduced so far for specific subclasses such
as cographs, -lite graphs, -extendible graphs and -tidy
graphs. Our algorithms are based on preprocessing methods using modular
decomposition, split decomposition and primeval decomposition. Thus they can
also be generalized to some graph classes with unbounded clique-width
Mixture Selection, Mechanism Design, and Signaling
We pose and study a fundamental algorithmic problem which we term mixture
selection, arising as a building block in a number of game-theoretic
applications: Given a function from the -dimensional hypercube to the
bounded interval , and an matrix with bounded entries,
maximize over in the -dimensional simplex. This problem arises
naturally when one seeks to design a lottery over items for sale in an auction,
or craft the posterior beliefs for agents in a Bayesian game through the
provision of information (a.k.a. signaling).
We present an approximation algorithm for this problem when
simultaneously satisfies two smoothness properties: Lipschitz continuity with
respect to the norm, and noise stability. The latter notion, which
we define and cater to our setting, controls the degree to which
low-probability errors in the inputs of can impact its output. When is
both -Lipschitz continuous and -stable, we obtain an (additive)
PTAS for mixture selection. We also show that neither assumption suffices by
itself for an additive PTAS, and both assumptions together do not suffice for
an additive FPTAS.
We apply our algorithm to different game-theoretic applications from
mechanism design and optimal signaling. We make progress on a number of open
problems suggested in prior work by easily reducing them to mixture selection:
we resolve an important special case of the small-menu lottery design problem
posed by Dughmi, Han, and Nisan; we resolve the problem of revenue-maximizing
signaling in Bayesian second-price auctions posed by Emek et al. and Miltersen
and Sheffet; we design a quasipolynomial-time approximation scheme for the
optimal signaling problem in normal form games suggested by Dughmi; and we
design an approximation algorithm for the optimal signaling problem in the
voting model of Alonso and C\^{a}mara
A Tensor Approach to Learning Mixed Membership Community Models
Community detection is the task of detecting hidden communities from observed
interactions. Guaranteed community detection has so far been mostly limited to
models with non-overlapping communities such as the stochastic block model. In
this paper, we remove this restriction, and provide guaranteed community
detection for a family of probabilistic network models with overlapping
communities, termed as the mixed membership Dirichlet model, first introduced
by Airoldi et al. This model allows for nodes to have fractional memberships in
multiple communities and assumes that the community memberships are drawn from
a Dirichlet distribution. Moreover, it contains the stochastic block model as a
special case. We propose a unified approach to learning these models via a
tensor spectral decomposition method. Our estimator is based on low-order
moment tensor of the observed network, consisting of 3-star counts. Our
learning method is fast and is based on simple linear algebraic operations,
e.g. singular value decomposition and tensor power iterations. We provide
guaranteed recovery of community memberships and model parameters and present a
careful finite sample analysis of our learning method. As an important special
case, our results match the best known scaling requirements for the
(homogeneous) stochastic block model
The world of hereditary graph classes viewed through Truemper configurations
In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms
Canonisation and Definability for Graphs of Bounded Rank Width
We prove that the combinatorial Weisfeiler-Leman algorithm of dimension
is a complete isomorphism test for the class of all graphs of rank
width at most . Rank width is a graph invariant that, similarly to tree
width, measures the width of a certain style of hierarchical decomposition of
graphs; it is equivalent to clique width. It was known that isomorphism of
graphs of rank width is decidable in polynomial time (Grohe and Schweitzer,
FOCS 2015), but the best previously known algorithm has a running time
for a non-elementary function . Our result yields an isomorphism
test for graphs of rank width running in time . Another
consequence of our result is the first polynomial time canonisation algorithm
for graphs of bounded rank width. Our second main result is that fixed-point
logic with counting captures polynomial time on all graph classes of bounded
rank width.Comment: 32 page
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