4,529 research outputs found
The Lazy Flipper: MAP Inference in Higher-Order Graphical Models by Depth-limited Exhaustive Search
This article presents a new search algorithm for the NP-hard problem of
optimizing functions of binary variables that decompose according to a
graphical model. It can be applied to models of any order and structure. The
main novelty is a technique to constrain the search space based on the topology
of the model. When pursued to the full search depth, the algorithm is
guaranteed to converge to a global optimum, passing through a series of
monotonously improving local optima that are guaranteed to be optimal within a
given and increasing Hamming distance. For a search depth of 1, it specializes
to Iterated Conditional Modes. Between these extremes, a useful tradeoff
between approximation quality and runtime is established. Experiments on models
derived from both illustrative and real problems show that approximations found
with limited search depth match or improve those obtained by state-of-the-art
methods based on message passing and linear programming.Comment: C++ Source Code available from
http://hci.iwr.uni-heidelberg.de/software.ph
Perron vector optimization applied to search engines
In the last years, Google's PageRank optimization problems have been
extensively studied. In that case, the ranking is given by the invariant
measure of a stochastic matrix. In this paper, we consider the more general
situation in which the ranking is determined by the Perron eigenvector of a
nonnegative, but not necessarily stochastic, matrix, in order to cover
Kleinberg's HITS algorithm. We also give some results for Tomlin's HOTS
algorithm. The problem consists then in finding an optimal outlink strategy
subject to design constraints and for a given search engine.
We study the relaxed versions of these problems, which means that we should
accept weighted hyperlinks. We provide an efficient algorithm for the
computation of the matrix of partial derivatives of the criterion, that uses
the low rank property of this matrix. We give a scalable algorithm that couples
gradient and power iterations and gives a local minimum of the Perron vector
optimization problem. We prove convergence by considering it as an approximate
gradient method.
We then show that optimal linkage stategies of HITS and HOTS optimization
problems verify a threshold property. We report numerical results on fragments
of the real web graph for these search engine optimization problems.Comment: 28 pages, 5 figure
A Variant of the Maximum Weight Independent Set Problem
We study a natural extension of the Maximum Weight Independent Set Problem
(MWIS), one of the most studied optimization problems in Graph algorithms. We
are given a graph , a weight function ,
a budget function , and a positive integer .
The weight (resp. budget) of a subset of vertices is the sum of weights (resp.
budgets) of the vertices in the subset. A -budgeted independent set in
is a subset of vertices, such that no pair of vertices in that subset are
adjacent, and the budget of the subset is at most . The goal is to find a
-budgeted independent set in such that its weight is maximum among all
the -budgeted independent sets in . We refer to this problem as MWBIS.
Being a generalization of MWIS, MWBIS also has several applications in
Scheduling, Wireless networks and so on. Due to the hardness results implied
from MWIS, we study the MWBIS problem in several special classes of graphs. We
design exact algorithms for trees, forests, cycle graphs, and interval graphs.
In unweighted case we design an approximation algorithm for -claw free
graphs whose approximation ratio () is competitive with the approximation
ratio () of MWIS (unweighted). Furthermore, we extend Baker's
technique \cite{Baker83} to get a PTAS for MWBIS in planar graphs.Comment: 18 page
Approximating Spectral Impact of Structural Perturbations in Large Networks
Determining the effect of structural perturbations on the eigenvalue spectra
of networks is an important problem because the spectra characterize not only
their topological structures, but also their dynamical behavior, such as
synchronization and cascading processes on networks. Here we develop a theory
for estimating the change of the largest eigenvalue of the adjacency matrix or
the extreme eigenvalues of the graph Laplacian when small but arbitrary set of
links are added or removed from the network. We demonstrate the effectiveness
of our approximation schemes using both real and artificial networks, showing
in particular that we can accurately obtain the spectral ranking of small
subgraphs. We also propose a local iterative scheme which computes the relative
ranking of a subgraph using only the connectivity information of its neighbors
within a few links. Our results may not only contribute to our theoretical
understanding of dynamical processes on networks, but also lead to practical
applications in ranking subgraphs of real complex networks.Comment: 9 pages, 3 figures, 2 table
Free Energy Approximations for CSMA networks
In this paper we study how to estimate the back-off rates in an idealized
CSMA network consisting of links to achieve a given throughput vector using
free energy approximations. More specifically, we introduce the class of
region-based free energy approximations with clique belief and present a closed
form expression for the back-off rates based on the zero gradient points of the
free energy approximation (in terms of the conflict graph, target throughput
vector and counting numbers). Next we introduce the size clique free
energy approximation as a special case and derive an explicit expression for
the counting numbers, as well as a recursion to compute the back-off rates. We
subsequently show that the size clique approximation coincides with a
Kikuchi free energy approximation and prove that it is exact on chordal
conflict graphs when . As a by-product these results provide us
with an explicit expression of a fixed point of the inverse generalized belief
propagation algorithm for CSMA networks. Using numerical experiments we compare
the accuracy of the novel approximation method with existing methods
From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More
We consider questions that arise from the intersection between the areas of
polynomial-time approximation algorithms, subexponential-time algorithms, and
fixed-parameter tractable algorithms. The questions, which have been asked
several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a
non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and
Minimum Dominating Set (DomSet) problems parameterized by the size of the
optimal solution. In particular, letting be the optimum and be
the size of the input, is there an algorithm that runs in
time and outputs a solution of size
, for any functions and that are independent of (for
Clique, we want )?
In this paper, we show that both Clique and DomSet admit no non-trivial
FPT-approximation algorithm, i.e., there is no
-FPT-approximation algorithm for Clique and no
-FPT-approximation algorithm for DomSet, for any function
(e.g., this holds even if is the Ackermann function). In fact, our results
imply something even stronger: The best way to solve Clique and DomSet, even
approximately, is to essentially enumerate all possibilities. Our results hold
under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which
states that no -time algorithm can distinguish between a satisfiable
3SAT formula and one which is not even -satisfiable for some
constant .
Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for
Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and
Maximum Induced Matching in bipartite graphs. Additionally, we rule out
-FPT-approximation algorithm for Densest -Subgraph although this
ratio does not yet match the trivial -approximation algorithm.Comment: 43 pages. To appear in FOCS'1
Maximum-entropy moment-closure for stochastic systems on networks
Moment-closure methods are popular tools to simplify the mathematical
analysis of stochastic models defined on networks, in which high dimensional
joint distributions are approximated (often by some heuristic argument) as
functions of lower dimensional distributions. Whilst undoubtedly useful,
several such methods suffer from issues of non-uniqueness and inconsistency.
These problems are solved by an approach based on the maximisation of entropy,
which is motivated, derived and implemented in this article. A series of
numerical experiments are also presented, detailing the application of the
method to the Susceptible-Infective-Recovered model of epidemics, as well as
cautionary examples showing the sensitivity of moment-closure techniques in
general.Comment: 20 pages, 7 figure
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